Schrödinger Equation – Potential Applications

#### **Chapter 5**

## Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class of Multiparameter Exponential-Type Potentials

*José Juan Peña, Jesús Morales and Jesús García-Ravelo*

#### **Abstract**

By using the exactly-solvable Schrödinger equation for a class of multi-parameter exponential-type potential, the analytical bound state solutions of the Klein-Gordon equation are presented. The proposal is based on the fact that the Klein-Gordon equation can be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal. The proposal has the advantage of avoiding the use of a specialized method to solve the Klein-Gordon equation for a specific exponential potential due that it can be derived by means of an appropriate choice of the involved parameters. For this, to show the usefulness of the method, the relativistic treatment of spinless particles subject to some already published exponential potentials are directly deduced and given as examples. So, beyond the particular cases considered in this work, this approach can be used to solve the Klein-Gordon equation for new exponential-type potentials having hypergeometric eigenfunctions. Also, it can be easily adapted to other approximations of the centrifugal term different to the Green-Aldrich used in this work.

**Keywords:** Schrödinger-type equation, Klein-Gordon equation, exponential-type potentials, Greene-Aldrich approximation, hypergeometric equation

#### **1. Introduction**

At high energy levels, the study of physical phenomena is carried out by means of equations invariant under Lorentz transformations. That is, it requires relativistic wave equations that may be used as a starting point to evaluate the spin-orbit interactions and relativistic effective core potential in the Schrödinger Hamiltonian. The energy levels from these calculations are aimed to find the positions of experimental spectral lines and to predict lines not heretofore observed in the systems under consideration. To that purpose, the Dirac and Klein-Gordon equations are used for the dynamic description of particles with and without spin, respectively. For that, the

solutions of these equations have been an important field of research by employing several methods as well as different physical potential models; usually a solution method for a specific potential. In this regard, exponential-type potentials are significant in the study of various physical systems, particularly for modeling diatomic molecules. Within the different exponential potential models, stand out the proposals of Hulthén, Eckart, Manning-Rosen, Rosen-Morse, Deng-Fan, Hyllerass, etc., as well as mixed models with two or more of the above potential models. These latter, have been developed with the aim to solve, as particular cases, the specific potentials that are involved. In any case, the common feature of any exponential-type potential is that wave functions are of hypergeometric-type. For this reason, in the quantum mechanics treatment of this kind of potentials, the method that is most often used to find the bound states solutions is the Nikiforov-Uvarov method [1], which is based on solving a hypergeometric-type differential equation (DE) by means of special orthogonal functions. Albeit, other procedures such as Asymptotic Iteration [2], Supersymmetric Quantum Mechanics [3], He's Variational iteration [4], large-N solutions [5] or Quantization-rule [6], among many other methods, have been also employed in both non-relativistic and relativistic studies; obviously, including numerical solutions [7]. In the relativistic studies of spinless particles, it is well known that the Klein-Gordon equation [8, 9] can always be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal [10]. This fact is used in the present research, devoted to obtaining approximate bound state energy eigenvalues and the corresponding eigenfunctions of the Klein-Gordon equation for exponential-type potentials. The method is based on a direct approach applied to the exactly solvable Schrödinger equation with hypergeometric solutions for exponential-type potential [11], which is given in Section 2. With this result, the corresponding analytical bound state solutions of the Klein-Gordon equation are found in the frame of the Green and Aldrich approximation to the centrifugal term [12] as shown in Section 4. Advantageously, according to the method, several specific potentials are derived as particular cases from the proposal such as those given in Section 5.

#### **2. Direct approach to the exactly solvable Schrödinger equation with hypergeometric solutions**

For finding exactly-solvable quantum exponential-type potentials, the Schrödinger equation must be transformed into a hypergeometric differential equation. To do so, let us consider the Schrödinger equation (ℏ**<sup>2</sup>** <sup>¼</sup> **<sup>2</sup>***<sup>m</sup>* <sup>¼</sup> **<sup>1</sup>**)

$$\frac{-d^2\psi(r)}{dr^2} + V(r)\psi(r) = E\psi(r) \tag{1}$$

such that, after using the transformation

$$
\mu(r) = e^{-ar}u(r) \tag{2}
$$

it is written as

$$\frac{-d^2u(r)}{dr^2} + 2a\frac{du(r)}{dr} + \left[V(r) - \left(E + a^2\right)\right]u(r) = 0\tag{3}$$

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

With the aim of relating the above equation with a hypergeometric DE, we use the coordinate transformation

$$\mathbf{x} = q e^{-r/k}, k > \mathbf{0}, q \neq \mathbf{0} \tag{4}$$

such that Eq. (3) is written as

$$x(1-x)\frac{d^2u(\mathbf{x})}{d\mathbf{x}^2} - (1+2ab)(1-\mathbf{x})\frac{du(\mathbf{x})}{d\mathbf{x}} - k^2\left(\frac{1-\mathbf{x}}{\mathbf{x}}\right)[V(\mathbf{x}) - (E+a^2)]u(\mathbf{x}) = \mathbf{0}.\tag{5}$$

Then, the similarity transformation

$$u(\varkappa)(\mathbf{1}-\varkappa)^{d}F(\varkappa) \tag{6}$$

with *d* being a real parameter, gives rise to

$$[\mathbf{x}(\mathbf{1}-\mathbf{x})\mathbf{F}''(\mathbf{x}) + [(\mathbf{1}+2ak) - (\mathbf{1}+2ak+2d)\mathbf{x}]\mathbf{F}'(\mathbf{x}) + R(\mathbf{x})\mathbf{F}(\mathbf{x}) = \mathbf{0} \tag{7}$$

where

$$R(\mathbf{x}) = \frac{\mathbf{x}d(d-1)}{1-\mathbf{x}} - d(\mathbf{1} + 2ab) - k^2 \left(\frac{\mathbf{1}-\mathbf{x}}{\mathbf{x}}\right) \left[V(\mathbf{x}) - \left(E + a^2\right)\right];\tag{8}$$

Hence, Eq. (7) can be compared with a hypergeometric DE

$$[\mathfrak{x}(\mathbf{1}-\mathbf{x})y''(\mathbf{x}) + [c - (\mathbf{1}+a+b)\mathfrak{x}]y'(\mathbf{x}) - aby(\mathbf{x}) = \mathbf{0} \tag{9}$$

provided that

$$11 + 2ak = c,\\
2(ka + d) = a + b,\\
E = -a^2 \tag{10}$$

and

$$R(\mathfrak{x}) = -ab \wedge F(\mathfrak{x}) = \mathfrak{y}(\mathfrak{x}) = {}\_2F\_1(a, b; c: \mathfrak{x}),\tag{11}$$

where <sup>2</sup>*F*<sup>1</sup> is the hypergeometric function. So, from *<sup>R</sup>* <sup>¼</sup> *ab* and *<sup>E</sup>* ¼ �*α*<sup>2</sup> in Eq. (8), it is possible to identify the potential

$$V(\mathbf{x}) = \frac{\mathbf{1}}{k^2} \left[ \frac{(ab - (\mathbf{1} + 2ka)d)\mathbf{x}}{\mathbf{1} - \mathbf{x}} + \frac{d(d-\mathbf{1})\mathbf{x}^2}{\left(\mathbf{1} - \mathbf{x}\right)^2} \right] \tag{12}$$

which, by using Eqs. (4) and (10), can be written as

$$V(r) = \frac{[4ab - 2c(a+b+1-c)]qe^{-r/k} + \left[\left(a-b\right)^2 - \left(c-1\right)^2\right]q^2e^{-2r/k}}{4k^2\left(1 - qe^{-r/k}\right)^2} \tag{13}$$

with eigenfunction given from Eqs. (2) and (6) by

$$\psi(r) = e^{-ar} \left( \mathbf{1} - q e^{-r/k} \right)^d \,\_2F\_1 \left( a, b; c: q e^{-r/k} \right) \tag{14}$$

and eigenvalue

$$E = -a^2 = -\left(\frac{c-1}{2k}\right)^2. \tag{15}$$

At this point, it is convenient to introduce the new parameters *A*, *B*, and *C* such that 4*k*<sup>2</sup> ð Þ¼ *<sup>A</sup>* <sup>þ</sup> *<sup>B</sup>* <sup>4</sup>*ab* � <sup>2</sup>*c a*ð Þ <sup>þ</sup> *<sup>b</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>c</sup>* and 4*k*<sup>2</sup> ð Þ¼ *<sup>C</sup>* � *<sup>A</sup>* ð Þ *<sup>a</sup>* � *<sup>b</sup>* <sup>2</sup> � ð Þ *<sup>c</sup>* � <sup>1</sup> <sup>2</sup> in order to rewrite the potential as a multi-parameter exponential-type potential

$$V(r) = \frac{Aqe^{-r/k}}{1 - qe^{-r/k}} + \frac{Bqe^{-r/k}}{\left(1 - qe^{-r/k}\right)^2} + \frac{Cq^2e^{-2r/k}}{\left(1 - qe^{-r/k}\right)^2} \tag{16}$$

By solving the condition *<sup>d</sup> drV r*ð Þ¼ 0, there will be a minimum value for the potential with sufficient depth for the existence of bound states, namely

$$V(r\_{min}) = \frac{-(A+B)^2}{4(B+C)}$$

$$withr\_{min} = k \ln\left(\frac{q(A-B-2C)}{A+B}\right),\tag{17}$$

provided that

$$A + B < 0 \le B + C \tag{18}$$

which ensures that *V r*ð Þ is an attractive potential with an infinite wall at its singular point *rs* ¼ *kln q*ð Þ.

Regarding the wave function, in order to have a node at *rs* it is necessary to apply the condition *ψ*ð Þ¼ *rs* 0, which is achieved if *d*> 0. Furthermore, by combining the identities 4*k*<sup>2</sup> ð Þ *<sup>A</sup>* <sup>þ</sup> *<sup>B</sup>* and 4*k*<sup>2</sup> ð Þ *C* � *A* given above, we have

$$b = \frac{(h+1)(h-a) - 2k^2(A+B)}{h+1-2a}, c = \frac{2a(h-a) - 2k^2(A+B)}{h+1-2a} \tag{19}$$

such that

$$c = a + b - h \, with \, h = \sqrt{1 + 4k^2(B + C)}\tag{20}$$

besides, if the parameter *a* ¼ �*n*, *n* ¼ 0, 1, 2, 3 … *:*, the hypergeometric function appearing in the eigenfunction given by Eq. (14) becomes a polynomial of *n* � *th* degree in the variable *qe*�*r=<sup>k</sup>*. Additionally, the condition *<sup>ψ</sup>*ð Þ!*<sup>r</sup>* 0 when *<sup>r</sup>* ! <sup>∞</sup> implies that *<sup>α</sup>* <sup>¼</sup> *<sup>c</sup>*�<sup>1</sup> *<sup>k</sup>* > 0, from which, the number of states is

$$0 \le n < k\sqrt{C - A} - \frac{h + 1}{2} \tag{21}$$

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

In short, the above equations lead to a well define wave function for a legitimate Schrödinger equation with potential *V r*ð Þ. Likewise, by using Eqs. (19) and (20) in Eq. (15) the energy spectrum will be

$$E\_n = \frac{-1}{4k^2} \left[ \frac{\left(n + \frac{(h+1)}{2}\right)^2 + k^2(A-C)}{n + \frac{(h+1)}{2}} \right]^2 \tag{22}$$

with wave-functions

$$\Psi\_n = \left( e^{-r/k} \right)^{\frac{c-1}{2}} \left( 1 - q e^{-r/k} \right)^{\frac{k+1}{2}} {}\_2F\_1 \left( -n, b; c : q e^{-r/k} \right) \tag{23}$$

#### **3. Klein-Gordon equation in arbitrary dimensions**

For a spinless particle with energy *Enl* and mass *M*, the D-dimensional Klein-Gordon (KG) equation is given (ℏ ¼ *c* ¼ 1) by [13].

$$-\nabla\_D^2 \mathcal{Y}\_{nlm}(r, \mathfrak{U}) + \left\{ \left[ \mathcal{M} + \mathcal{S}(r) \right]^2 - \left[ E\_{nl} + V(r) \right]^2 \right\} \mathcal{Y}\_{nlm}(r, \mathfrak{U}) = 0 \tag{24}$$

where *V r*ð Þ and *S r*ð Þ are respectively the Lorentz vector and the scalar interaction potentials. The D-dimensional Laplacian operator in the space ð Þ¼ *r*, *Ω* ð Þ *r*, *θ*1, *θ*2, *θ*3, … *θ<sup>D</sup>*�2, *ϕ* is defined ase

$$\nabla\_D^2 = r^{1-D} \frac{\partial}{\partial r} \left[ r^{D-1} \frac{\partial}{\partial r} \right] - \frac{\Lambda^2(\mathfrak{Q})}{r^2} \tag{25}$$

with *Λ Ω*ð Þ the angular momentum operator. Hence, the functione

$$
\Psi\nu\_{nlm}(r,\mathfrak{Q}) = R\_{nl}(r)Y\_{l}^{m}(\mathfrak{Q})\tag{26}
$$

leads to the radial part of Eq. (24)

$$\frac{-d^2R\_{nl}(r)}{dr^2} - \frac{D-1}{r}\frac{dR\_{nl}(r)}{dr} + \frac{l\_D(l\_D+1)}{r^2}R\_{nl}(r)$$

$$+ \left\{ \left[ M + \mathcal{S}(r) \right]^2 - \left[ E\_{nl} + V(r) \right]^2 \right\} R\_{nl}(r) = E R\_{nl}(r) \tag{27}$$

where we have used *lD* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 <sup>4</sup> þ *l l*ð Þ þ *D* � 2 q � 1 <sup>2</sup> such that

$$
\Lambda^2(\mathfrak{Q}) Y\_l^m(\mathfrak{Q}) = l\_D(l\_D + \mathfrak{I}) Y\_l^m(\mathfrak{Q}) \tag{28}
$$

Likewise, with *Rnl*ð Þ¼ *r r* �*D*�<sup>1</sup> <sup>2</sup> *<sup>ψ</sup>nL*ð Þ*<sup>r</sup>* Eq. (27) becomes

$$\frac{1 - d^2 \varphi\_{nL}(r)}{dr^2} + \left\{ \left[ M + \mathcal{S}(r) \right]^2 - \left[ E\_{nL} + V(r) \right]^2 + \frac{L\_t}{r^2} \right\} \boldsymbol{\nu}\_{nL}(r) = E\_{nL} \boldsymbol{\nu}\_{nL}(r) \tag{29}$$

where *Ls* <sup>¼</sup> *L L*ð Þ <sup>þ</sup> <sup>1</sup> , *<sup>L</sup>* <sup>¼</sup> *lD* <sup>þ</sup> *<sup>D</sup>*�<sup>3</sup> <sup>2</sup> such that the case *D* ¼ 3 implies *L* ¼ *lD* ¼ *l:*

At this point, as already mentioned, the D-dimensional KG equation given in Eq. (29) can be reduced to a Schrödinger-like equation, provided that the Lorentz vector and scalar potential are equal [10]. In fact, if *V r*ð Þ¼ *S r*ð Þ the corresponding KG equation is given by

$$\frac{-d^2\psi\_{nL}(r)}{dr^2} + \left\{ \left(E\_{nL}^2 - M^2\right) - 2(E\_{nL} + M)V(r) + \frac{L\_t}{r^2} \right\} \psi\_{nL}(r) = 0\tag{30}$$

Different methods of the solution have been applied for solving the above equation with many models of interaction potentials; see for example Ikhdair [14] and references therein. Hence, to provide a unified treatment to the bound states solution of the KG equation for equal vector and scalar exponential-type potentials, in the next paragraph the results of Section 2 are extended to consider the D-dimensional case. As we will see, this is done by a simple redefinition of the parameters that appear in *V r*ð Þ as defined in Eq. (16).

#### **4. Klein-Gordon equation for exponential-type potentials in arbitrary dimensions**

To deal with the *'*-state approximate solutions for the D-dimensional KG equation with the multi-parameter exponential-type potential given in Eq. (16) we define *q* ¼ 1 and

$$A = A + \frac{aL\_t}{k^2}, B = B + \frac{\beta L\_t}{k^2}, \ C = C + \frac{\gamma L\_t}{k^2} \tag{31}$$

such that

$$V(r) = \frac{Ae^{-r/k}}{1 - e^{-r/k}} + \frac{Be^{-r/k}}{\left(1 - e^{-r/k}\right)^2} + \frac{Ce^{-2r/k}}{\left(1 - e^{-r/k}\right)^2} + L\_r T\_c \tag{32}$$

with

$$T\_c = \frac{(a+\beta)e^{-r/k} + (\chi - a)e^{-2r/k}}{k^2(1 - e^{-r/k})^2} \tag{33}$$

where, according to the values of the parameters *α*, *β*, and *γ*, the function *Tc* would be approximate to the centrifugal term. Consequently, this method accepts different approximation schemes, such as recently shown in [15]. For example, if we consider the case *α* ¼ *γ* ¼ 0, and *β* ¼ 1, it leads to the standard Green-Aldrich approximation [12]

$$T\_c = \frac{e^{-r/k}}{k^2(1 - e^{-r/k})^2} \approx \frac{1}{r^2} \tag{34}$$

however, if we add the constant *c*0*L=k*<sup>2</sup> in both sides of the Eq. (32) one has

$$V(r) + \frac{c\_0 L}{k^2} = \frac{A e^{-r/k}}{1 - e^{-r/k}} + \frac{B e^{-r/k}}{\left(1 - e^{-r/k}\right)^2} + \frac{C q e^{-2r/k}}{\left(1 - e^{-r/k}\right)^2} + L\_s \left[T\_c + \frac{c\_0}{k^2}\right] \tag{35}$$

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

which means that any improvement to the centrifugal term through an additive constant will be reflected as an additional term in the energy spectrum. In fact, the improved Green-Aldrich approximation to the centrifugal term is achieved when *α* ¼ *γ* ¼ 0, *β* ¼ 1 and *c*<sup>0</sup> ¼ 1*=*12, that is [16]

$$T\_c + \frac{c\_0}{k^2} = \frac{e^{-r/k}}{k^2(1 - e^{-r/k})} + \frac{1}{12k^2} \approx \frac{1}{r^2} \tag{36}$$

Another typical improved approximation used to <sup>1</sup> *<sup>r</sup>*<sup>2</sup> is when *α* ¼ *C*1; *β* ¼ 0, and *γ* ¼ *C*<sup>2</sup> and *c*<sup>0</sup> ¼ *C*<sup>0</sup> where the parameters *C*1, *C*2, and *C*<sup>0</sup> are adjustable parameters [17], leading to

$$T\_c + \frac{c\_0}{k^2} = \frac{1}{k^2} \left[ C\_0 + \frac{C\_1 e^{-r/k}}{1 - e^{-r/k}} + \frac{C\_2 e^{-2r/k}}{\left(1 - e^{-r/k}\right)^2} \right] \approx \frac{1}{r^2} \tag{37}$$

However, for the sake of simplicity, we will use the standard Green-Aldrich approximation, *C*<sup>0</sup> ¼ 0, such that

$$V(r) = V(r) + \frac{L\_s}{r^2} \tag{38}$$

with

$$V(r) = \frac{Ae^{-r/k}}{\mathbf{1} - e^{-r/k}} + \frac{Be^{-r/k}}{\left(\mathbf{1} - e^{-r/k}\right)^2} + \frac{Ce^{-2r/k}}{\left(\mathbf{1} - e^{-r/k}\right)^2} \tag{39}$$

Since the solutions of Eq. (1) with potential *V r*ð Þ are given by Eqs. (22) and (23), the energy spectrum and the eigenfunctions will be

$$E\_n = \frac{-1}{4k^2} \left[ \frac{\left(n + \frac{(h\_L + 1)}{2}\right)^2 + k^2(A - C)}{n + \frac{(h\_L + 1)}{2}} \right]^2 \tag{40}$$

and

$$\mathcal{Y}\_{nL}(r) = \left(e^{-r/k}\right)^{\frac{C\_L-1}{2}} \left(1 - e^{-r/k}\right)^{\frac{h\_L+1}{2}} {}\_2F\_1\left(-n, b\_L; c\_L: e^{-r/k}\right) \tag{41}$$

where the new parameters defined in Eq. (31) have been used. Besides, according with Eqs. (19) and (20)

$$h\_L = \sqrt{\left(2L+1\right)^2 + 4k^2(B+C)},\tag{42}$$

$$b\_L = \frac{(h\_L + \mathbf{1})(h\_L + n) - 2k^2(A + B) - 2L\_t}{2n + h\_L + \mathbf{1}}\tag{43}$$

and

$$\text{cc}\_{L} = \frac{-2n(h\_{L} + n) - 2k^{2}(A + B) - 2L\_{s}}{2n + h\_{L} + 1} \tag{44}$$

Furthermore, the number of states will be determined by Eq. (21) as

$$0 \le n < k\sqrt{C - A} - \frac{h\_L + 1}{2} \tag{45}$$

Likewise, in accordance with Eq. (17)

$$V(r\_{min}) = \frac{-\left(k^2(A+B) + L\_s\right)^2}{4k^2\left(k^2(B+C) + L\_s\right)}\tag{46}$$

with

$$r\_{\min} = k \ln \left( \frac{k^2 (A - B - 2C) - L\_s}{k^2 (A + B) + L\_s} \right), \tag{47}$$

such that

$$k^2(A+B) < L\_s \le k^2(B+C) \tag{48}$$

or more explicitly

$$k^2(A+B) + l(l+1) < \frac{(3-D)(4l+D-1)}{4} \le k^2(B+C) + l(l+1) \tag{49}$$

where all possible values of *l* ¼ 0, 1, 2, 3, … *lmax* fulfill the above inequality.

With these elements, the KG equation in arbitrary dimensions given in Eq. (30), for *S r*ð Þ¼ *V r*ð Þ¼ *V r*ð Þ; becomes a Schrödinger-type equation

$$\frac{-d^2\psi\_{nL}(r)}{dr^2} + \left\{\Delta EV(r) + \frac{L\_s}{r^2}\right\}\mu\_{nL}(r) = \tilde{E}\_{nL}\mu\_{nL}(r) \tag{50}$$

with <sup>Δ</sup>*<sup>E</sup>* <sup>¼</sup> <sup>2</sup>ð Þ *EnL* <sup>þ</sup> *<sup>M</sup>* and *<sup>E</sup>*~*nL* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> *nL* � *<sup>M</sup>*<sup>2</sup> Hence, within the frame of the standard Green-Aldrich approximation, directly from Eqs. (40) and (41), the energy spectrum and wave function are respectively

$$\tilde{E}\_{nL} = \frac{-1}{16k^2} \left[ \frac{(2n + h\_L + 1)^2 + 4k^2 \Delta E (A - C)}{2n + h\_L + 1} \right]^2 \tag{51}$$

$$\Psi\_{nL}(r) = \left(e^{-r/k}\right)^{\frac{C\_L-1}{2}} \left(1 - e^{-r/k}\right)^{\frac{h\_L+1}{2}} {}\_2F\_1\left(-n, b\_L; c\_L: e^{-r/k}\right) \tag{52}$$

where

$$h\_L = \sqrt{\left(2L+1\right)^2 + 4k^2\Delta E(B+C)},\tag{53}$$

$$b\_L = \frac{(h\_L + 1)(h\_L + n) - 2k^2 \Delta E(A+B) - 2L\_s}{2n + h\_L + 1} \tag{54}$$

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

and

$$c\_L = \frac{-2n(h\_L + n) - 2k^2 \Delta E(A+B) - 2L\_t}{2n + h\_L + \mathbf{1}} \tag{55}$$

The usefulness of our alternative approach for the calculation of bound state solutions of the D-dimensional KG equation with exponential-type potentials is exemplified in the next section.

#### **5. Applications**

The choice of particular values for the parameters *A*, *B*, and *C* appearing in the multiparameter exponential-type potential of Eq. (39), leads to the solutions of the KG equation in arbitrary dimensions for specific potentials. So, without being exhaustive, at the following we are going to consider only some well-known special cases, it being understood the existence of many others that can be treated in a similar way.

#### **5.1 The Eckart+Hultén potential**

If we assume the parameters *<sup>A</sup>* ¼ �ð Þ *<sup>V</sup>*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*<sup>3</sup> ; *<sup>B</sup>* <sup>¼</sup> <sup>4</sup>*V*1;*<sup>C</sup>* <sup>¼</sup> 0 and *<sup>k</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*<sup>a</sup>* �<sup>1</sup> the potential *V r*ð Þ in Eq. (39) will be

$$V(r) = \frac{-(V\_2 + V\_3)e^{-2ar}}{1 - e^{-2ar}} + \frac{4V\_1 e^{-2ar}}{\left(1 - e^{-2ar}\right)^2} \tag{56}$$

such that the Eckart-type potential which also includes the Hulthén potential is written as

$$V\_{EH} = V(r) - V\_2 = \frac{-V\_2}{1 - e^{-2ar}} - \frac{V\_3 e^{-2ar}}{1 - e^{-2ar}} + \frac{4V\_1 e^{-2ar}}{\left(1 - e^{-2ar}\right)^2} + \frac{L\_t}{r^2} \tag{57}$$

Hence, from Eq. (51) this potential has an energy spectrum given by

$$\tilde{E}\_{nL}^{EH} = \frac{-\alpha^2}{4} \left[ \frac{(2n + h\_L + 1)^2 + \alpha^{-2} \Delta E (V\_2 + V\_3)}{2n + h\_L + 1} \right]^2 - \Delta EV\_2 \tag{58}$$

which agrees with the transcendental Eq. (38) of the reference [14], after considering some algebraic steps on it and the displacement �*V*<sup>2</sup> in the potential *V r*ð Þ. At this point, we want to notice that the term Δ*EV*<sup>2</sup> in above equation appears from *V r*ð Þ¼ *VEH* þ *V*<sup>2</sup> given in Eq. (57). That is,

$$\frac{-d^2\varphi\_{nL}(r)}{dr^2} + \left[\Delta E(V\_{EH} + V\_2) + \frac{L\_s}{r^2}\right]\varphi\_{nL}(r) = \tilde{E}\_{nL}\varphi\_{nL}(r) \tag{59}$$

Implies

$$\frac{-d^2\varphi\_{nL}(r)}{dr^2} + \left[\Delta EV\_{EH} + \frac{L\_l}{r^2}\right]\varphi\_{nL}(r) = \left(\tilde{E}\_{nL} - \Delta EV\_2\right)\varphi\_{nL}(r) \tag{60}$$

Likewise from Eq. (52), the corresponding wave functions are

$$\Psi\_{nL}(r) = \left(e^{-2ar}\right)^{\frac{c\_{EH(L)}-1}{2}} \left(1 - e^{-2ar}\right)^{\frac{h\_{EH(L)}+1}{2}} {}\_2F\_1\left(-n, b\_{EH(L)}; c\_{EH(L)}; e^{-2ar}\right) \tag{61}$$

where

$$h\_L = \sqrt{\left(2L+1\right)^2 + a^{-2}\Delta EV\_1},\tag{62}$$

$$h\_{EH(L)} = \left(h\_{EH(L)} + \mathbf{1}\right)\left(h\_{EH(L)} + n\right) - \tag{63}$$

and

$$\mathcal{L}\_{\rm EH(L)} = -2\mathfrak{n}\left(h\_{\rm EH(L)} + \mathfrak{n}\right) - 2 \tag{64}$$

#### **5.2 The standard Hultén potential**

In this case, for the potential *V r*ð Þ given in Eq. (39), one can apply the selection

$$A = -Za,\\ B = \mathbb{C} = \mathbf{0},\\ k = a^{-1} \tag{65}$$

such that the Hultén potential is

$$V\_H = \frac{-Z\alpha e^{-ar}}{1 - e^{-ar}} + \frac{L\_s}{r^2} \tag{66}$$

Similarly to the above case, from Eq. (51), this potential has an energy spectrum

$$\bar{E}\_{nL}^{H} = \frac{-\alpha^2}{4} \left[ n + L + \mathbf{1} - \frac{Za\Delta E}{n + L + \mathbf{1}} \right]^2 \tag{67}$$

that agrees with Eq. (47) of the reference [14] under the identification of their *ν* ¼ ð Þ *<sup>D</sup>* <sup>þ</sup> <sup>2</sup>*<sup>l</sup>* � <sup>1</sup> *<sup>=</sup>*2 with our *<sup>L</sup>* <sup>¼</sup> *<sup>ν</sup>* � 1 i.e. *<sup>ν</sup>* <sup>¼</sup> *<sup>L</sup>* <sup>þ</sup> 1. Additionally, our *<sup>E</sup>*~*<sup>H</sup> nL* result coincides with that of Saad [18] when the parameters *V*<sup>0</sup> ¼ *S*<sup>0</sup> and *q* ¼ 1 are used. Similarly, from Eq. (52), the corresponding wavefunctions will be

$$\boldsymbol{\Psi}\_{nL}^{H}(\boldsymbol{r}) = (\boldsymbol{e}^{-ar})^{\frac{\boldsymbol{c}\_{H(L)}^{\boldsymbol{c}}-1}{2}} (\mathbf{1} - \boldsymbol{e}^{-ar})^{L+1} \,\_2F\_1(-n, b\_{H(L)}; \boldsymbol{c}\_{H(L)}; \boldsymbol{e}^{-ar}) \tag{68}$$

where, according to Eqs. (54) and (55), the *bL* and *bL* parameters are now

$$b\_{H(L)} = \frac{(L+1)(2L+n+1) + Za^{-1} - L\_t}{n+L+1} \tag{69}$$

and

$$\mathcal{L}\_{H(L)} = \frac{-n(2L + n + 1) + Za^{-1} - L\_t}{n + L + 1} \tag{70}$$

with *hH L*ð Þ ¼ 2*L* þ 1.

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

#### **5.3 The standard Eckart potential**

To obtain this potential model, one selects

$$A = -V\_1, B = V\_2, C = 0; k = b \tag{71}$$

such that

$$V\_E = \frac{-V\_1 e^{-r/b}}{1 - e^{-r/b}} + \frac{V\_2 e^{-r/b}}{\left(1 - e^{-r/b}\right)^2} + \frac{L\_s}{r^2} \tag{72}$$

corresponds to the Eckart potential. So, from Eq. (51), its corresponding energy spectrum results in

$$\tilde{E}\_{nL}^{E} = \frac{-\mathbf{1}}{16b^2} \left[ \frac{\left(2n + h\_{E(L)} + \mathbf{1}\right)^2 + 4b^2 V\_1 \Delta E}{2n + h\_{E(L)} + \mathbf{1}} \right]^2 \tag{73}$$

where

$$h\_{E(L)} = \sqrt{(2L+1)^2 + 4b^2 \Delta EV\_2},\tag{74}$$

and the eigenfunctions are

$$\left(\boldsymbol{\mu}\_{nL}^{E}\left(\boldsymbol{r}\right) = \left(e^{-ar}\right)^{\frac{C\_{E(L)}-1}{2}} (1 - e^{-ar})^{\frac{h\_{E(L)}+1}{2}} {}\_{2}F\_{1}\left(-n, b\_{E(L)}; c\_{E(L)}; e^{-ar}\right) \tag{75}$$

$$b\_{EH(L)} = \frac{\left(h\_{E(L)} + \mathbf{1}\right)\left(h\_{E(L)} + n\right) - 2b^2 \Delta E(V\_2 - V\_1) - 2L\_s}{2n + h\_{E(L)} + \mathbf{1}}\tag{76}$$

and

$$\omega\_{EH(L)} = \frac{-2n\left(h\_{E(L)} + n\right) - 2b^2 \Delta E(V\_2 - V\_1) - 2L\_s}{2n + h\_{E(L)} + 1} \tag{77}$$

It is worth mentioning that in the particular case *D* ¼ 3, the energy spectrum given in Eq. (73) coincides with the results of Akpan et al. [17] by assuming the standard Green and Aldrich approximation (*C*<sup>0</sup> ¼ 0; *C*<sup>1</sup> ¼ *C*<sup>2</sup> ¼ 1).

#### **5.4 The Manning-Rosen potential**

Let us consider now the parameters

$$A = -V\_0/b^2, B = 0, \mathcal{C} = \frac{a(a-1)}{b^2}; k = b \tag{78}$$

from which, one has the Manning-Rosen potential

$$V\_{MR} = \frac{1}{b^2} \left( \frac{a(a-1)e^{-2r/b}}{\left(1 - e^{-r/b}\right)^2} - \frac{V\_0 e^{-r/b}}{1 - e^{-r/b}} \right) + \frac{L\_t}{r^2} \tag{79}$$

with energy spectrum

$$\tilde{E}\_{nL}^{MR} = \frac{-\mathbf{1}}{\mathbf{1}6b^2} \left[ \frac{\left(2n + h\_{MR(L)} + \mathbf{1}\right)^2 + 4\Delta E(V\_0 + a(a-\mathbf{1}))}{2n + h\_{MR(L)} + \mathbf{1}} \right]^2 \tag{80}$$

where

$$h\_{\rm MR(L)} = \sqrt{(2L+1)^2 + 4\Delta E a (a-1)},\tag{81}$$

Besides, the eigenfunctions are

$$\boldsymbol{\psi}\_{n\boldsymbol{L}}^{\rm MR}(\boldsymbol{r}) = \left(\boldsymbol{e}^{-r/b}\right)^{\frac{c\_{\rm MR(L)}-1}{2}} \left(\mathbf{1} - \boldsymbol{e}^{-r/b}\right)^{\frac{b\_{\rm MR(L)}+1}{2}} \,\_2F\_1\left(-n, b\_{\rm MR(L)}; c\_{\rm MR(L)}; \boldsymbol{e}^{-r/b}\right) \tag{82}$$

$$b\_{\rm MR(L)} = \frac{\left(h\_{\rm MR(L)} + \mathbf{1}\right)\left(h\_{\rm MR(L)} + n\right) + 2\Delta EV\_0 - 2\mathbf{L}\_s}{2n + h\_{\rm MR(L)} + \mathbf{1}}\tag{83}$$

and

$$\mathcal{L}\_{\text{MR}(L)} = \frac{-\mathfrak{D}\left(h\_{\text{MR}(L)} + n\right) + \mathfrak{D}\Delta EV\_0 - \mathfrak{L}\_s}{\mathfrak{D}n + h\_{\text{MR}(L)} + \mathfrak{1}} \tag{84}$$

#### **5.5 The improved Manning-Rosen potential**

To get this special case, it becomes necessary the choice *<sup>A</sup>* ¼ �2*De <sup>e</sup>*ð Þ *<sup>α</sup>re* � <sup>1</sup> , *<sup>B</sup>* <sup>¼</sup> 0; *<sup>C</sup>* <sup>¼</sup> *De <sup>e</sup>*ð Þ *<sup>α</sup>re* � <sup>1</sup> <sup>2</sup> , and *<sup>k</sup>* <sup>¼</sup> *<sup>α</sup>*�<sup>1</sup> leading to Improved Manning-Rosen potential

$$V\_{IMR}(r) = V(r) + D\_\epsilon = D\_\epsilon \left(1 - \frac{e^{ar\_\epsilon} - 1}{e^{ar} - 1}\right)^2 + \frac{L\_\epsilon}{r^2} \tag{85}$$

with, according to Eq. (49), energy spectrum given by

$$\bar{E}\_{nL}^{\text{IMR}} = -a^2 \left[ \frac{2n + h\_{\text{IMR}(L)} + \mathbf{1}}{4} - \frac{a^{-2} D\_\epsilon \Delta E (e^{2\sigma r\_\epsilon} - \mathbf{1})}{2n + h\_{\text{IMR}(L)} + \mathbf{1}} \right]^2 + D\_\epsilon \Delta E \tag{86}$$

where

$$h\_{\rm IMR(L)} = \sqrt{\left(2L + 1\right)^2 + 4\alpha^{-2}D\_\epsilon \Delta E \left(e^{\alpha r\_\epsilon} - 1\right)^2},\tag{87}$$

At this point, we want to notice that in the case of *<sup>D</sup>* <sup>¼</sup> 3, the energy spectrum *<sup>E</sup>*~*IMR nL* is in agreement with Eq. (31) of Jia et al. [19] besides, it corrects Eqs. (21) and (24) of the reference [20].

On the other hand, from Eq. (52), the eigenfunctions are in this case

$$\boldsymbol{\nu}\_{n\boldsymbol{L}}^{\text{IMR}}(\boldsymbol{r}) = (\boldsymbol{e}^{-\boldsymbol{a}\boldsymbol{r}})^{\frac{\boldsymbol{c}\_{\text{IMR}(\boldsymbol{L})}-1}{2}} (\mathbf{1} - \boldsymbol{e}^{-\boldsymbol{a}\boldsymbol{r}})^{\frac{\boldsymbol{h}\_{\text{IMR}(\boldsymbol{L})}+1}{2}} \,\_2F\_1(-\boldsymbol{n}, \boldsymbol{b}\_{\text{IMR}(\boldsymbol{L})}; \boldsymbol{c}\_{\text{IMR}(\boldsymbol{L})}; \boldsymbol{e}^{-\boldsymbol{a}\boldsymbol{r}}) \tag{88}$$

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

with

$$b\_{\rm IMR(L)} = \frac{\left(h\_{\rm IMR(L)} + \mathbf{1}\right)\left(h\_{\rm IMR(L)} + n\right) - 4\alpha^{-2}\Delta E(e^{ar\_{\varepsilon}} - \mathbf{1}) - 2L\_{\rm s}}{2n + h\_{\rm IMR(L)} + \mathbf{1}}\tag{89}$$

and

$$\omega\_{\text{IMR}(L)} = \frac{-2n\left(h\_{\text{IMR}(L)} + n\right) + 4a^{-2}\Delta E(e^{m\_\epsilon} - 1) - 2L\_\epsilon}{2n + h\_{\text{IMR}(L)} + 1} \tag{90}$$

#### **5.6 The Hylleraas potential**

Assuming that *<sup>A</sup>* ¼ �*V*0ð Þ <sup>1</sup> � *<sup>a</sup>* , *<sup>B</sup>* <sup>¼</sup> *<sup>C</sup>* <sup>¼</sup> 0 and *<sup>k</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*<sup>α</sup>* �<sup>1</sup> the potential given in Eq. (39) reduces to

$$V(r) = \frac{-V\_0(1-a)e^{-2ar}}{1 - e^{-2ar}}\tag{91}$$

for which the Hylleraas potential in D-dimensions is given by

$$V\_{Hy} = V(r) + V\_0 a = V\_0 \left(\frac{a - e^{-2ar}}{1 - e^{-2ar}}\right) + \frac{L\_t}{r^2} \tag{92}$$

As before, from Eq. (51), the corresponding energy spectrum will be

$$\tilde{E}\_{nL}^{Hy} = \frac{-a^2}{4} \left[ \frac{\left(2n + h\_{Hy(L)} + 1\right)^2 + a^{-2} D\_t \Delta EV\_0(a - 1)}{2n + h\_{Hy(L)} + 1} \right]^2 + aV\_0 \Delta E \tag{93}$$

where

$$h\_{\rm Hy(L)} = \mathfrak{L}L + \mathfrak{1},\tag{94}$$

in agreement with Hassamaadi et al. [21] when considering their parameters *b* ¼ 1 and *V*<sup>1</sup> ¼ *V*<sup>2</sup> ¼ 0.

In relation with wavefunctions, from Eq. (52), these are

$$\Psi\_{nL}^{H\_{\rm p}}(r) = (e^{-ar})^{\frac{c\_{H\rm p(L)}-1}{2}} (1 - e^{-ar})^{\frac{k\_{H\rm p(L)}+1}{2}} {}\_2F\_1(-n, b\_{H\rm p(L)}; c\_{H\rm p(L)}; e^{-ar}) \tag{95}$$

being

$$b\_{Hy(L)} = \frac{\left(h\_{Hy(L)} + \mathbf{1}\right)\left(h\_{Hy(L)} + n\right) - 2a^{-1}\Delta EV\_0(a - \mathbf{1}) - 2\mathbf{L}\_s}{2n + h\_{Hy(L)} + \mathbf{1}}\tag{96}$$

and

$$\omega\_{Hy(L)} = \frac{-2n\left(h\_{Hy(L)} + n\right) - 2\alpha^{-1}\Delta EV\_0(a-1) - 2L\_s}{2n + h\_{Hy(L)} + 1} \tag{97}$$

**85**

where, as in all the above cases, the down index indicates the name of potential, i.e. *Hy* refers to Hylleraas.

#### **5.7 The Deng Fan potential**

In this case, the involved parameters are chosen as

$$A = -2bD\_\epsilon, B = 0, C = D\_\epsilon b^2 \text{and } k = \mathbf{1}/a \tag{98}$$

such that the Deng Fan potential, also called generalized Morse potential will be

$$V\_{DF} = \frac{-2bD\_e e^{-ar}}{1 - e^{-ar}} + \frac{D\_e b^2 e^{-2ar}}{\left(1 - e^{-ar}\right)^2} + \frac{L\_s}{r^2} \tag{99}$$

where *De* is the dissociation energy. The corresponding energy spectrum is obtained from Eq. (51) as

$$\tilde{E}\_{nL}^{DF} = \frac{-a^2}{16} \left[ \frac{\left(2n + h\_{DF(L)} + 1\right)^2 - 4a^{-2} D\_\epsilon \Delta E \left(2b + b^2\right)}{2n + h\_{DF(L)} + 1} \right]^2 \tag{100}$$

which is in agreement with Oluwadare et al. [22] for the three-dimensional case, when considering the Green and Aldrich approximation [12]. Finally, from Eq. (52), the respective wave functions are

$$\Psi\_{nL}^{DF}(r) = (e^{-ar})^{\frac{c\_{DF(L)}-1}{2}} (1 - e^{-ar})^{\frac{k\_{DF(L)}+1}{2}} \,\_2F\_1(-n, b\_{DF(L)}; c\_{DF(L)}; e^{-ar}) \tag{101}$$

where

$$h\_{DF(L)} = \sqrt{\left(2L+1\right)^2 + 4a^{-2}D\_\epsilon \Delta ED\_\epsilon b^2},\tag{102}$$

$$b\_{DF(L)} = \frac{\left(h\_{DF(L)} + \mathbf{1}\right)\left(h\_{DF(L)} + n\right) + 4a^{-2}\Delta E b D\_\varepsilon - 2L\_\varepsilon}{2n + h\_{DF(L)} + \mathbf{1}} \tag{103}$$

and

$$\mathcal{L}\_{\rm DF(L)} = \frac{-2n\left(h\_{\rm DF(L)} + n\right) + 4a^{-2}\Delta E b D\_e - 2L\_s}{2n + h\_{\rm DF(L)} + 1} \tag{104}$$

At this point, it should be noted that the equivalence among the Manning-Rosen potential, the Deng-Fan and Schiöberg models for diatomic molecules have been already shown with detail in references [23, 24]. So, the solutions of the KG equation in arbitrary dimensions derived in this section can also be extended to the Schiöberg potential [25].

Finally, we want to pay attention that in a similar manner to the examples considered in this work, other exponential potentials would be achieved as particular cases from our general proposal of multi-parameter exponential-type potential [26] after a proper selection of the involved parameters.

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

#### **6. Concluding remarks**

Through a direct approach to transform the Schrödinger equation into a hypergeometric differential equation, we have obtained the exact solution of a class of multiparameter exponential-type potentials. Also, we have used the fact that, for equal Lorentz vector and scalar potentials, the Klein-Gordon equation can be written as a Schrödinger-type equation. With these elements, and with a proper redefinition of the involved parameters, we propose an approach to obtain the analytical solutions of the Klein-Gordon equation for exponential-type potentials, in the frame of the Green-Aldrich approximation to the centrifugal term. As a test of the usefulness of the proposed method, by an appropriate selection of parameters, the Klein-Gordon equation has been solved for specific exponential potential models such as Hulthén, Eckart, Manning-Rosen, Improved Manning-Rosen, Hylleraas and generalized Morse or Deng Fan which are derived here as particular cases from the proposal. That is, with this work, we are proposing a unified treatment for solving the Klein-Gordon equation subject to multiparameter exponential-type-potentials, leaving aside the usual methods of solution applied for each one of the aforementioned potentials, for particular parameters, given as examples. So, the displayed method offers an alternative treatment of spinless particles with new exponential-type potentials as well as the possibility to use other schemes of approximations to the centrifugal term.

#### **Acknowledgements**

This work was partially supported by the project UAMA-CBI-2232004-009. One of us (JGR) is indebted to the Instituto Politécnico Nacional-Mexico (IPN) for the financial support given through the COFAA-IPN project SIP-20231470. We are grateful to the SNI-Conacyt-México for the stipend received.

#### **Author details**

José Juan Peña<sup>1</sup> \*, Jesús Morales<sup>1</sup> and Jesús García-Ravelo<sup>2</sup>

1 Metropolitan Autonomous University, Azc. CDMX, Mexico

2 National Polytechnic Institute, ESFM, CDMX, Mexico

\*Address all correspondence to: jjpg@azc.uam.mx

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Nikiforo A, Uvarov V. Special Functions of Mathematical Physics. Bassel: Birkhauser; 1988

[2] Olgar E, Koc R, Tütüncüler H. The exact solution of the s-wave Klein-Gordon equation for the generalized Hulthén potential by the asymptotic iteration method, Phisyca Scripta. 2008; **78**;015011

[3] Ahmadov AI, Nagiyev SM, Qocayeva MV, Uzun K, Tarverdiyeva VA. Bound state solution of the Klein-Fock-Gordon equation with the Hulthén plus a ringshaped-like potential within SUSY quantum mechanics. International Journal of Modern Physics A. 2018; **33**(33):1850203

[4] Yusufoglu E. The variational iteration method for studying the Klein-Gordon equation. Applied Mathematics Letters. 2008;**21**:669

[5] Chatterjee A. Large-N solution of the Klein-Gordon equation. Journal of Mathematical Physics. 1986;**27**:2331

[6] Sun H. Quantization Rule for Relativistic Klein-Gordon Equation. Bulletin of Korean Chemical Society. 2011;**32**:4233

[7] Bülbül B, Sezer M. A New Approach to Numerical Solution of Nonlinear Klein-Gordon Equation. Mathematical Problems in Engineering. 2013;**869749**:7

[8] Klein O. Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik. 1926;**37**:895

[9] Gordon W. Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik. 1926;**40**:117

[10] Okorie US, Ikot AN, Onate CA, Onyeaju MC, Rampho GJ. Bound and scattering states solutions of the Klein-Gordon equation with the attractive radial potential in higher dimensions. Modern Physics Letters A. 2021;**36**(32): 2150230

[11] Peña J, Morales J, García-Ravelo J. Bound state solutions of Dirac equation with radial exponential-type potentials. Journal of Mathematical Physics. 2017; **48**:043501

[12] Nath D, Roy AK. Analytical solution of D dimensional Schrödinger equation for Eckart potential with a new improved approximation in centrifugal term. Chemical Physics Letters. 2021; **780**:138909

[13] Dhahbi A, Landolsi AA. The Klein-Gordon equation with equal scalar and vector Bargmann potentials in D dimensions. Results in Physics. 2022;**33**: 105143

[14] Ikhdair SM. Bound state energies and wave functions of spherical quantum dots in presence of a confining potential model. Journal of Quantum Information Science. 2011;**1**:73

[15] Peña JJ, García-Martínez J, García-Ravelo J, Morales J. Bound state solutions of D-dimensional schrödinger equation with exponential-type potentials. International Journal of Quantum Chemistry. 2015;**115**:158

[16] Jia CS, Diao YF, Yi LZ, Chen T. Arbitrary l-wave Solutions of the Schrödinger Equation with The Hultén Potential Model. International Journal of Modern Physics A. 2009; **24**:4519

[17] Akpan IO, Antia AD, Icot AN. Bound-State Solutions of the

*Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class… DOI: http://dx.doi.org/10.5772/intechopen.112184*

Klein-Gordon Equation with q-Deformed Equal Scalar and Vector Eckart Potential Using a Newly Improved Approximation Scheme. ISRN High Energy Physics. 2012. ID 798209

[18] Saad N. The Klein-Gordon equation with a generalized Hulthén potential in D-dimensions. Physica Scripta. 2007; **76**:623

[19] Jia CS, Chen T, He S. Bound state solutions of the Klein-Gordon equation with the improved expression of the Manning-Rosen potential energy model. Physics Letters A. 2013;**377**:682

[20] Chen XY, Chen T, Jia CS. Solutions of the Klein-Gordon equation with the improved Manning-Rosen potential energy model in D dimensions. The European Physical Journal Plus. 2014; **129**:75

[21] Hassanabadi S, Maghsoodi E, Oudi R, Sarrinkamar S, Rahimov H. Exact solution Dirac equation for an energy-dependent potential. The European Physical Journal Plus. 2012; **127**:120

[22] Oluwadare OJ, Oyewumi KJ, Akoshile CO, Babalola OA. Approximate analytical solutions of the relativistic equations with the Deng-Fan molecular potential including a Pekeris-type approximation to the (pseudo or) centrifugal term. Physica Scripta. 2012; **86**:035002

[23] Wang PQ, Zhang LH, Jia CS, Liu JY. Equivalence of the three empirical potential energy models for diatomic molecules. Journal of Molecular Spectroscopy. 2012;**274**:5

[24] Peña J, Ovando G, Morales J. On the equivalence of radial potential models for diatomic molecules. Theoretical Chemistry Accounts. 2016;**135**:62

[25] Omugbe E, Osafile OE, Okon IB, Enaibe EA, Onyeaju MC. Bound state solutions, Fisher information measures, expectation values, and transmission coefficient of the Varshni potential. Molecular Physics, 2021;**119**:e1909163

[26] Onate CA, Onyeaju MC, Okon IB, Adeoti A. Molecular energies of a modified and deformed exponentialtype potential model. Chemical Physics Impact. 2021;**3**:100045

#### **Chapter 6**

## From a 4-Rank Totally Antisymmetric Field Strength to Two Dual Electromagnetic Fields in Four Time and Four Space Dimensions

*Juan Antonio Nieto*

#### **Abstract**

A 4-rank "electromagnetic" gauge field strength in four-time and four-space dimensions 4 ðð Þ þ 4 -dimensions) is considered. We show that by necessitating such a four-rank gauge field we satisfy the Grassmann–Plücker relations which allows to choose an ansatz (for a basic basis) such that it is broken into two dual electromagnetic fields – one in a 1ð Þ þ 3 -world and the other in a 3ð Þ þ 1 -world. An interesting aspect of this mechanism is that the electromagnetic field 1ð Þ þ 3 -world turns out to be dual to the electromagnetic field in the 3ð Þ þ 1 -world.

**Keywords:** 4-rank field strength, (4 + 4)-dimensions, electromagnetic field, Maxwell equations and higher dimensional theory, antisymmetric field strength

#### **1. Introduction**

It is known that the Grassmann Plücker relations [1–6] of totally antisymmetric forms determine the Plücker coordinates which mean that such a totally antisymmetric form is decomposable (see Ref. [3] and references therein). Physically, the Plücker embedding can be found in a Grassmannian sigma model in *SU*ð Þ2 Yang Mills model [7] and in spherically symmetric instantons of the scale invariant *SU*ð Þ2 gauged Grassmannian model in *d* ¼ 4 [8]. When this developments are applied to 2-rank antisymmetric gauge field in four-dimensions, it is found that the corresponding electromagnetic field strength can be written in terms of the true degrees of freedom [9].

In this work, we make a number of remarks on Plücker coordinates associated with 4-rank totally antisymmetric gauge fields strength *F<sup>μ</sup>*^^*να*^*β*^ (differential 4-form) in ð Þ 4 þ 4 -dimensions. When such a gauge field satisfies the Grassmann–Plücker relations can be, of course, decomposable in terms of more elementary quantities. Surprisingly, for a particular case of ansatz for such elementary basic quantities, the field equations for *<sup>F</sup><sup>μ</sup>*^^*να*^*β*^ lead to both the electromagnetic field equation for *<sup>F</sup>μν* in 1ð Þ <sup>þ</sup> <sup>3</sup> -dimensions

and the electromagnetic field equation for *<sup>G</sup>ij* in 3ð Þ <sup>þ</sup> <sup>1</sup> -dimensions. An interesting aspect of this mechanism is that the source of *Fμν* is determined in part by *Gij* and dually the source of *Gij* is determined in part by *Fμν*.

#### **2. Field equations for a 4-rank field strength**

Let us start considering a totally antisymmetric gauge field strength

$$F^{\vec{\mu}\vec{\nu}\vec{a}\vec{\beta}} = F^{\vec{\mu}\vec{\nu}\vec{a}\vec{\beta}}(\mathfrak{x}^{\sigma}, \mathfrak{y}^{i})$$

in 4ð Þ þ 4 -dimensions, which we shall assume is a function of 1ð Þ þ 3 -coordinates *<sup>x</sup><sup>σ</sup>* and 3ð Þ <sup>þ</sup> <sup>1</sup> -coordinates *<sup>y</sup><sup>i</sup>* . Suppose that *Fμ*^^*να*^*β*^ satisfies the Maxwell type equations:

$$
\partial\_{\hat{\beta}} F^{\hat{\mu}\hat{\nu}\hat{\alpha}\hat{\beta}} = \mathbf{0} \tag{1}
$$

and

$$
\partial\_{\hat{\beta}} \, ^\* F^{\hat{\mu}\hat{\nu}\hat{a}\hat{\beta}} = \mathbf{0}, \tag{2}
$$

where <sup>∗</sup> *Fμ*^^*να*^*β*^ is the dual gauge field defined as

$$\epsilon^\* F^{\hat{\mu}\hat{\nu}\hat{\alpha}\hat{\beta}} = \frac{1}{4!} \epsilon^{\hat{\mu}\hat{\alpha}\hat{\beta}\hat{\sigma}\hat{\rho}\hat{\gamma}\hat{\eta}} F\_{\hat{\sigma}\hat{\rho}\hat{\eta}\hat{\eta}}.\tag{3}$$

In general, we raise and lower indices with a flat Minkowski metric *ημ*^^*<sup>ν</sup>*, which in ð Þ 4 þ 4 -dimensions takes the form

$$\eta\_{\hat{\mu}\hat{\nu}} = \operatorname{diag}(-\mathbf{1}, \mathbf{1}, \mathbf{1}, \mathbf{1}, -\mathbf{1}, -\mathbf{1}, -\mathbf{1}, \mathbf{1}).\tag{4}$$

The ϵ-symbol is a totally antisymmetric symbol (Levi–Civita symbol) defined as

$$
\epsilon^{\hat{\mu}\hat{\alpha}\hat{\beta}\hat{\sigma}\hat{\rho}\hat{\eta}\hat{\eta}} \in \{-1, 0, 1\}.\tag{5}
$$

In fact, the ϵ-symbol has values þ1 or �1 depending on even or odd permutations of ϵ<sup>12</sup>*:::*8, respectively, otherwise the ϵ-symbol is zero. It is verified that the relation

$$
\epsilon^{\hat{\mu}\_1 \dots \hat{\mu}\_8} \epsilon\_{\hat{\nu}\_1 \dots \hat{\nu}\_8} = -\delta^{\hat{\mu}\_1 \dots \hat{\mu}\_8}\_{\hat{\nu}\_1 \dots \hat{\nu}\_8},\tag{6}
$$

where *δ μ*^<sup>1</sup> … *μ*^<sup>8</sup> ^*ν*<sup>1</sup> …^*ν*<sup>8</sup> is a generalized Kronecker delta [10].

Let us assume that the Grassmann–Plücker relations (see Ref. [3] and references therein) hold for *F<sup>μ</sup>*^^*να*^*β*^ , namely

$$F^{\hat{\mu}\hat{\nu}\hat{a}}[^{\hat{\beta}}F^{\hat{\sigma}\hat{\rho}\hat{\eta}\hat{\eta}}] = \mathbf{0}.\tag{7}$$

Here, the bracket *β*^*σ*^^*ρ*^*γ*^*η* … means totally antisymmetric. This implies that *F<sup>μ</sup>*^^*να*^^*<sup>β</sup>* is decomposable. In other words, this means that *F<sup>μ</sup>*^^*να*^*β*^ can be written as

*From a 4-Rank Totally Antisymmetric Field Strength to Two Dual Electromagnetic Fields… DOI: http://dx.doi.org/10.5772/intechopen.112061*

$$F^{\hat{\mu}\hat{\nu}\hat{\alpha}\hat{\beta}} = \frac{1}{4!} \epsilon^{\hat{A}\hat{B}\hat{C}\hat{D}} \upsilon^{\hat{\mu}}\_{\hat{A}} \upsilon^{\hat{\nu}}\_{\hat{B}} \upsilon^{\hat{\alpha}}\_{\hat{C}} \upsilon^{\hat{\beta}}\_{\hat{D}}.\tag{8}$$

Here, the elementary quantities *v μ*^ *<sup>A</sup>*^ ¼ *v μ*^ *<sup>A</sup>*^ *<sup>x</sup><sup>σ</sup>*, *<sup>y</sup><sup>i</sup>* can be considered as basic basis elements and *εA*^*B*^*C*^*D*^ is an 'internal' four-dimensional ϵ-symbol.

Moreover, from (2) one learns that

$$F\_{\hat{\mu}\hat{\nu}\hat{\alpha}\hat{\beta}} = \partial\_{\hat{\mu}} A\_{\hat{\nu}\hat{\alpha}\hat{\beta}]}.\tag{9}$$

So one sees that *A*^*να*^*β*^ is a totally antisymmetric gauge field which under the transformation

$$\mathcal{A}\_{\hat{\nu}\hat{a}\hat{\boldsymbol{\beta}}} \rightarrow \mathcal{A}\_{\hat{\nu}\hat{a}\hat{\boldsymbol{\beta}}} + \partial\_{\hat{\nu}} \mathcal{Q}\_{\hat{a}\hat{\boldsymbol{\beta}}]},\tag{10}$$

the 4-rank tensor *Fμ*^^*να*^*β*^ becomes invariant.

#### **3. Kaluza-Klein type ansatz**

In general, one finds that *Fμ*^^*να*^^*<sup>β</sup>* can be written as

$$\begin{aligned} F^{\hat{\mu}\hat{\alpha}\hat{\beta}} &= F^{\hat{\mu}\hat{\nu}}G^{\hat{\alpha}\hat{\beta}} - F^{\hat{\mu}\hat{\alpha}}G^{\hat{\nu}\hat{\beta}} + F^{\hat{\mu}\hat{\beta}}G^{\hat{\alpha}\hat{\alpha}} \\ &+ G^{\hat{\mu}\hat{\nu}}F^{\hat{\alpha}\hat{\beta}} - G^{\hat{\mu}\hat{\alpha}}F^{\hat{\nu}\hat{\beta}} + G^{\hat{\mu}\hat{\beta}}F^{\hat{\alpha}}, \end{aligned} \tag{11}$$

where

$$F^{\hat{\mu}\hat{\nu}} = \frac{1}{4!} \epsilon^{ab} v\_a^{\hat{\mu}} v\_b^{\hat{\nu}} \tag{12}$$

and

$$\mathbf{G}^{\hat{\mu}\hat{\nu}} = \frac{\mathbf{1}}{4!} \epsilon^{AB} \boldsymbol{v}\_A^{\hat{\mu}} \boldsymbol{v}\_B^{\hat{\nu}}.\tag{13}$$

Our next step is to consider some particular cases. In principle one may assume a kind of Kaluza–Klein ansatz for *v μ*^ *<sup>A</sup>*^ , namely

$$\boldsymbol{\upsilon}\_{\boldsymbol{A}}^{\hat{\boldsymbol{\mu}}} = \begin{pmatrix} \boldsymbol{\upsilon}\_{\boldsymbol{a}}^{\mu} & \boldsymbol{\upsilon}\_{\boldsymbol{A}}^{\mu} \\ \mathbf{0} & \boldsymbol{\upsilon}\_{\boldsymbol{A}}^{i} \end{pmatrix}. \tag{14}$$

However, in this case, one looses the symmetry between the 1ð Þ þ 3 -world and the ð Þ 3 þ 1 -world. So one shall assume that

$$\boldsymbol{\nu}\_{\dot{A}}^{\dot{\mu}} = \begin{pmatrix} \boldsymbol{\nu}\_{a}^{\mu} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\nu}\_{A}^{i} \end{pmatrix}. \tag{15}$$

In this case, one gets

$$F^{\mu i} = \mathbf{0} \tag{16}$$

and

$$F^j = \mathbf{0}.\tag{17}$$

And one also obtains

$$\mathbf{G}^{\mu \dot{\imath}} = \mathbf{0} \tag{18}$$

and

$$\mathbf{G}^{\mu\nu} = \mathbf{0}.\tag{19}$$

Hence, using Eq. (11) one learns that the only nonvanishing component of *Fμ*^^*να*^^*<sup>β</sup>* is

$$F^{\mu\nu\vec{\text{ij}}} = F^{\mu\nu}(\mathfrak{x}^{\sigma}, \mathfrak{y}^{k}) G^{\vec{\text{ij}}}(\mathfrak{x}^{\lambda}, \mathfrak{y}^{k}),\tag{20}$$

where the "electromagnetic" fields *Fμν* and *Gij* corresponds to the 1ð Þ þ 3 -world and 3ð Þ þ 1 -world, respectively.

#### **4. Inhomogeneous Maxwell field equations**

From (1) one knows that

$$
\partial\_{\beta} F^{\hat{\mu}\hat{\nu}\hat{a}\beta} + \partial\_{\hat{\jmath}} F^{\hat{\mu}\hat{\nu}\hat{a}\dagger} = \mathbf{0}.\tag{21}
$$

Thus, the relevant equations that can be obtained from Eq. (21) are

$$
\partial\_{\nu} F^{\mu \neq j} = \mathbf{0} \tag{22}
$$

and

$$
\partial\_{\dot{j}} F^{\mu \dot{\imath} \dot{j}} = 0. \tag{23}
$$

Hence, using Eq. (20) one learns that

$$(\partial\_\nu F^{\mu\nu})G^{\ddagger} + \left(\partial\_\nu G^{\ddagger}\right)F^{\mu\nu} = 0\tag{24}$$

and

$$(\partial\_{\dot{j}}G^{\dot{j}})F^{\mu\nu} + (\partial\_{\dot{j}}F^{\mu\nu})G^{\dddot{j}} = 0.\tag{25}$$

These equations can also be written as

$$
\partial\_{\nu} F^{\mu \nu} = J^{\mu} \tag{26}
$$

and

$$
\partial\_{\vec{j}} \mathcal{G}^{\vec{j}} = \mathcal{J}^{\vec{i}}.\tag{27}
$$

*From a 4-Rank Totally Antisymmetric Field Strength to Two Dual Electromagnetic Fields… DOI: http://dx.doi.org/10.5772/intechopen.112061*

Here,

$$J^{\mu} = -F^{\mu\nu} \partial\_{\nu} \ln \varphi \tag{28}$$

and

$$J^i = -G^{\sharp} \partial\_{\flat} \ln \xi,\tag{29}$$

with

$$\Psi = \left(\frac{1}{2}\mathbf{G}^{\ddagger}\mathbf{G}\_{\ddagger}\right)^{1/2} \tag{30}$$

and

$$\mathfrak{F} = \left(\frac{\mathbf{1}}{2} F^{\mu\nu} F\_{\mu\nu}\right)^{1/2}.\tag{31}$$

Another interesting way to write (26) and (27) is

$$\partial\_{\nu}(\varphi F^{\mu \nu}) = 0 \tag{32}$$

and

$$
\partial\_{\vec{\jmath}} \left( \xi G^{\vec{\jmath}\dagger} \right) = \mathbf{0}, \tag{33}
$$

respectively.

It is evident that (26) (and (27)) or (32) (and (33)) are inhomogeneous Maxwell field type equations. Consequently, *Fμν* can be identified with the electromagnetic field strength in the 1ð Þ <sup>þ</sup> <sup>3</sup> -world and *<sup>G</sup>ij* with the dual-mirror electromagnetic field strength in the 3ð Þ þ 1 -world. However, (26) or (32) establishes something else that the source of the electromagnetic field *<sup>ψ</sup>* arises in part from the 3ð Þ <sup>þ</sup> <sup>1</sup> -world *via Gij* , and the eqs. (27) and (33) indicate that the source of the mirror electromagnetic field emerges from the 1ð Þ <sup>þ</sup> <sup>3</sup> -world *via Fμν*. This process shows a duality between the ð Þ 1 þ 3 -world and the 3ð Þ þ 1 -world. An important point is that both electromagnetic fields *<sup>F</sup>μν* and *<sup>G</sup>ij* are part according to (18) of the 4-rank gauge field strength *<sup>F</sup><sup>μ</sup>*^^*να*^*β*^ which "lives" in a 4ð Þ þ 4 -world.

#### **5. Dual Maxwell field equations**

Moreover, from Eq. (20) one can also show that the only nonvanishing components of the dual gauge field strength <sup>∗</sup> *F<sup>μ</sup>*^^*να*^*β*^ are <sup>∗</sup> *Fμνij* which can be written as

$${}^{\ast}F^{\mu\nu ij} = {}^{\ast}F^{\mu\nu} \left( \mathfrak{x}^{\sigma}, \mathfrak{y}^{k} \right) {}^{\ast}G^{j} \left( \mathfrak{x}^{\lambda}, \mathfrak{y}^{k} \right), \tag{34}$$

where

$${}^{\ast}F^{\mu\nu} = \frac{1}{2!} \epsilon^{\mu\alpha\beta} F\_{\alpha\beta} \tag{35}$$

and

$$\epsilon^\* G^{\dot{j}} = \frac{1}{2!} \epsilon^{\dot{j}kl} G\_{kl}. \tag{36}$$

$$\text{From the field eq. (2), one obtains }$$

$$\partial\_{\nu} \, ^\*F^{\mu \nu} = -\partial\_{\nu} (\ln \, \mu \, ^\*) \, ^\*F^{\mu \nu} \tag{37}$$

and

$$\partial\_{\dot{\jmath}} \, ^\*G^{\ddagger} = -\partial\_{\dot{\jmath}} (\ln \xi \, ^\*) \, ^\*G^{\ddagger}.\tag{38}$$

Here,

$$\boldsymbol{\Psi}^\* = \left(\frac{\mathbf{1}^\*}{2} \, \mathbf{G}^{\circ i} \, \mathbf{G}\_{\circ}\right)^{1/2} \tag{39}$$

and

$$
\xi^\* = \left(\frac{\mathbf{1}}{2}^\* F^{\mu \nu} \,^\* F\_{\mu \nu}\right)^{1/2} . \tag{40}
$$

One finds that (37) and (38) can also be written as

$$
\partial\_{\nu} (\,^\*\mu \,^\*F^{\mu \nu}) = 0 \tag{41}
$$

and

$$
\partial\_{\dot{\jmath}} \left( \,^\*\xi \,^\*G^{\dot{\jmath}} \right) = \mathbf{0}, \tag{42}
$$

respectively.

#### **6. Final remarks**

Let us make some final remarks. Usually, for describing different phenomena in our universe, one time and three-space dimensions ( 1ð Þ þ 3 -dimensions) are the chosen number of real dimensions. But the question emerges, why 1ð Þ þ 3 -dimensions? why not 3ð Þ þ 1 -dimensions? or why not 4ð Þ þ 4 -dimensions? Unfortunately (or fortunately) until now these questions are an open theoretical problem; as far as one knows, nobody knows the answer. It turns out that looking for a possible solution one stumbling with the discovery that there is a triality relation between the signatures ð Þ 1 þ 9 , 5ð Þ þ 5 and 9ð Þ þ 1 [11, 12]. This means that by triality the 5ð Þ þ 5 -dimensional world can always be related to the other basic signatures 1ð Þ þ 9 and 9ð Þ þ 1 . It turns out that 5ð Þ þ 5 -world is a common signature to both type *IIA* strings and type *IIB* strings. From this perspective, one may say that the 4ð Þ þ 4 -world can be considered as the transverse world of the 5ð Þ þ 5 -world (see Refs [11, 12] and references therein). Moreover, it turns out that in 4ð Þ þ 4 -dimensions, there are a number of remarkable mathematical and physical results that are worth mentioning.

*From a 4-Rank Totally Antisymmetric Field Strength to Two Dual Electromagnetic Fields… DOI: http://dx.doi.org/10.5772/intechopen.112061*

Mathematically, it has been suggested that the mathematical structures of oriented matroid theory [13] (see Refs. [14–22] and references therein) and surreal number theory [23, 24] (see also Refs [25, 26] and references therein) may provide interesting routes for a connection with the 4ð Þ þ 4 -world. Physically, the Dirac equation in ð Þ 4 þ 4 -dimensions is consistent with Majorana–Weyl spinors which give exactly the same number of components as the complex spinor of <sup>1</sup> 2 -spin particles such as the electron or the quarks (see Refs. [12, 27]). Second, the most general Kruskal–Szekeres transformation of a black-hole coordinates in 1ð Þ þ 3 -dimensions leads to eightregions (instead of the usual four-regions), which can be better described in 4ð Þ þ 4 dimensions [28]. Third, it also has been shown [29] that duality

$$
\sigma^2 \leftrightarrow \frac{1}{\sigma^2},
\tag{43}
$$

of a Gaussian distribution in terms of the standard deviation *σ* of 4-space coordinates associated with the de Sitter space (anti-de Sitter) and the vacuum zero-point energy yields to a Gaussian of 4-time coordinates of the same vacuum scenario. Moreover, loop quantum gravity in 4ð Þ þ 4 -dimensions [30, 31] admits a self-duality curvature structure analogue to the traditional 1ð Þ þ 3 -dimensions.

In the above sense, the contribution of this work adds to the fact that "electromagnetic" field in a 4ð Þ þ 4 -world described by a 4-rank totally antisymmetric field strength *Fμ*^^*να*^^*<sup>β</sup>* can be broken into two electromagnetic field strengths; the field strength *<sup>F</sup>μν* associated with the 1ð Þ <sup>þ</sup> <sup>3</sup> -world and the field strength *<sup>G</sup>ij* associated with the 3ð Þ þ 1 -world. An interesting aspect of this result is that there is a hidden duality symmetry feature of *Fμν* and *Gij* in the sense that *Gij* contribute to the source of *Fμν* and vice versa.

Finally, it is worth mentioning that 4-rank totally antisymmetric field strength *<sup>F</sup>μ*^^*να*^*β*^ in 1ð Þ <sup>þ</sup> <sup>10</sup> -dimensions are a key mathematical notion in 3-brane theory which, it is known, is an important part in the so-called *M*-theory (see Ref. [15] and references therein). In fact, in Ref. [9] it is shown how totally antisymmetric fields can be related to *p*-brane. For the case of the field strength *Fμ*^^*να*^*β*^ , one uses (6) and writes

$$F^{\hat{\alpha}\hat{\beta}\hat{A}} = \frac{1}{4!} \varepsilon^{\hat{B}\hat{C}\hat{D}\hat{A}} \upsilon\_{\hat{B}}^{\hat{\alpha}} \upsilon\_{\hat{C}}^{\hat{\alpha}} \upsilon\_{\hat{D}}^{\hat{\beta}} \tag{44}$$

and assume that *<sup>F</sup><sup>μ</sup>*^^*να*^^*<sup>β</sup>* <sup>¼</sup> *<sup>F</sup><sup>μ</sup>*^^*να*^^*<sup>β</sup> <sup>x</sup><sup>σ</sup>*, *<sup>y</sup><sup>i</sup>* , *<sup>ξ</sup><sup>A</sup>*^ . Moreover, instead of (1) one considers the field equation:

$$
\partial\_{\dot{A}} F^{\dot{\mu}\ddot{\alpha}\dot{A}} = \mathbf{0}, \tag{45}
$$

where *<sup>∂</sup>A*^ <sup>¼</sup> *<sup>∂</sup> <sup>ξ</sup>A*^ . This expression implies that due to (44) one can write

$$
\boldsymbol{\upsilon}^{\hat{\mu}}\_{\hat{\mathcal{B}}} = \partial\_{\hat{\mathcal{B}}} \boldsymbol{X}^{\hat{\mu}}.\tag{46}
$$

Thus, substituting this result into (8) leads to

$$F^{\hat{\mu}\hat{\alpha}\hat{\beta}} = \frac{1}{4!} \epsilon^{\hat{A}\hat{B}\hat{C}\hat{D}} \partial\_{\hat{A}} X^{\hat{\mu}} \partial\_{\hat{B}} X^{\hat{\nu}} \partial\_{\hat{C}} X^{\hat{\alpha}} \partial\_{\hat{D}} X^{\hat{\beta}}.\tag{47}$$

In turn, this expression can be used to write the Schild type action for the 3-brane in target 4ð Þ þ 4 -dimensions, namely

$$\mathcal{S} = \int \left( F^{\hat{\mu}\hat{\nu}\hat{a}\hat{\beta}} F\_{\hat{\mu}\hat{\nu}\hat{a}\hat{\beta}} \right)^{1/2} d^4 \xi. \tag{48}$$

So it may be interesting for further work to continue relating our present approach with *p*-brane theory and *M*-theory.

#### **Acknowledgements**

I would like to thank the Mathematical, Computational & Modeling Sciences Center of the Arizona State University where part of this work was developed.

#### **Classification**

Pacs numbers: 04.20.Jb, 04.50.-h, 04.60.-m, 11.15.-q.

#### **Author details**

Juan Antonio Nieto1,2

1 Facultad de Ciencias Fsico-Matemáticas Universidad Autónoma de Sinaloa Culiacán, Sinaloa, México

2 Facultad de Ciencias de la Tierra y el Espacio Universidad Autónoma de Sinaloa Culiacán, Sinaloa, México

\*Address all correspondence to: niet@uas.edu.mx; janieto1@asu.edu

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*From a 4-Rank Totally Antisymmetric Field Strength to Two Dual Electromagnetic Fields… DOI: http://dx.doi.org/10.5772/intechopen.112061*

#### **References**

[1] Plücker J. On a new geometry of space. Proceedings. Royal Society of London. 1865;**14**:53

[2] Hodge W, Pedoe D. Methods of Algebraic Geometry, 2. American Branch, New York, Cambridge University Press; 1952

[3] Bokowski JLJ, Sturmfels B. Computational synthetic geometry. In: Dold A, Eckmann B, editors. Lecture Notes in Mathematics. New York: Springer-Verlag; 1980

[4] Kolhatkar R. Grassmann Varieties, Thesis. Montreal, Quebec, Canada: Department of Mathematics and Statistics, McGill University; 2004

[5] Sturmfels B. Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation. Vienna: Springer-Verlag; 1993

[6] Baralic D. How to understand Grassmannians? Teaching Mathamatics. 2011;**XIV**:147

[7] Marsh D. The Grassmannian sigma model in SU (2) Yang-Mills theory. Journal of Physics A. 2007;**40**:9919. hep-th/0702134

[8] Chakrabarti A, Tchrakian DH. Spherically symmetric instantons of the scale invariant SU (2) gauged Grassmannian model in d = 4. Physics Letters B. 1996;**376**:59. e-Print: hep-th/ 9601157

[9] Nieto JA, Nieto-Marín PA, León EA, Garca-Manzanárez E. Remarks on Plücker embedding and totally antisymmetric gauge fields. Modern Physics Letters A. 2020;**35**(22):2050184

[10] Misner CW, Thorne KS, Wheeler JA. Gravitation. San Francisco: W. H. Freeman and Company; 1971

[11] De Andrade MA, Rojas M, Toppan F. Triality of Majorana-Weyl space-times with different signatures. International Journal of Modern Physics A: Particles and Fields; Gravitation; Cosmology; Nuclear Physics. 2001;**16**:4453. hep-th/0005035

[12] Rojas M, De Andrade MA, Colatto LP, Matheus-Valle JL, De Assis LPG and Helayel-Neto JA. Mass Generation and Related Issues from Exotic Higher Dimensions. hep-th/ 1111.2261

[13] Björner A, Las Vergnas M, Sturmfels B, White N, Ziegler GM. Oriented Matroids. Cambridge: Cambridge University Press; 1993

[14] Nieto JA. Advances in Theoretical and Mathematical Physics. 2004;**8**:177. arXiv: hep-th/0310071

[15] Nieto JA. Advances in Theoretical and Mathematical Physics. 2006;**10**:747. arXiv: hep-th/0506106

[16] Nieto JA. Searching for a connection between matroid theory and string theory. Journal of Mathematical Physics. 2004;**45**:285. arXiv: hep-th/0212100

[17] Nieto JA. Phirotopes, super p-branes and qubit theory. Nuclear Physics B. 2014;**883**:350. arXiv:1402.6998 [hep-th]

[18] Nieto JA, Marin MC. Matroid theory and Chern-Simons. Journal of Mathematical Physics. 2000;**41**:7997. hep-th/0005117

[19] Nieto JA. Qubits and oriented matroids in four time and four space dimensions. Physics Letters B. 2013;**718**: 1543. e-Print: arXiv:1210.0928 [hep-th]

[20] Nieto JA. Qubits and chirotopes. Physics Letters B. 2010;**692**:43. e-Print: arXiv:1004.5372 [hep-th]

[21] Nieto JA. Duality, Matroids, Qubits, Twistors, and Surreal Numbers. Frontiers in Physics. 2018;**6**:106

[22] Nieto JA, Leon EA. 2D Gravity with Torsion, Oriented Matroids and 2+2 Dimensions. Brazilian Journal of Physics. 2010;**40**:383. e-Print: 0905.3543 [hep-th]

[23] Conway JH. 24-28 Oval Road. London NW1. England. On Number and Games, London Mathematical Society Monographs. London, England: Academic Press; 1976

[24] Gonshor H. An introduction to the theory of surreal numbers. In: London Mathematical Society Lectures Notes Series. Vol. 110. Cambridge, UK: Cambridge University Press; 1986

[25] Avalos-Ramos C, Felix-Algandar JA, Nieto JA. Dyadic Rationals and Surreal Number Theory. IOSR Journal of Mathematics. 2020;**16**:35

[26] Nieto JA. Some Mathematical and Physical Remarks on Surreal Numbers. Journal of Modern Physics. 2016;**7**:2164

[27] Nieto JA. Dirac equation in four time and four space dimensions. International Journal of Geometric Methods in Modern Physics. 2016;**14**(01):1750014

[28] Nieto JA, Madriz E. Aspects of (4 + 4)-Kaluza-Klein type theory. Physica Scripta. 2019;**94**:115303

[29] Medina M, Nieto JA, Nieto-Marín PA. Cosmological Duality in Four Time and Four Space Dimensions. Journal of Modern Physics. 2021;**12**:1027

[30] Nieto JA. Superfield description of a self-dual supergravity in the context of

MacDowell-Mansouri theory. Classical and Quantum Gravity. 2006;**23**:4387. e-Print: hep-th/0509169

[31] Nieto JA. Towards an Ashtekar formalism in eight dimensions. Classical and Quantum Gravity. 2005;**22**:947. e-Print: hep-th/0410260

#### **Chapter 7**

## Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell Equations and Their Extensions

*George Vahala, Min Soe, Efstratios Koukoutsis, Kyriakos Hizanidis, Linda Vahala and Abhay K. Ram*

#### **Abstract**

It is well known that Maxwell equations can be expressed in a unitary Schrodinger-Dirac representation for homogeneous media. However, difficulties arise when considering inhomogeneous media. A Dyson map points to a unitary field qubit basis, but the standard qubit lattice algorithm of interleaved unitary collision-stream operators must be augmented by some sparse non-unitary potential operators that recover the derivatives on the refractive indices. The effect of the steepness of these derivatives on two-dimensional scattering is examined with simulations showing quite complex wavefronts emitted due to transmissions/reflections within the dielectric objects. Maxwell equations are extended to handle dissipation using Kraus operators. Then, our theoretical algorithms are extended to these open quantum systems. A quantum circuit diagram is presented as well as estimates on the required number of quantum gates for implementation on a quantum computer.

**Keywords:** Schrodinger-Dirac, qubit lattice algorithm, Dyson map, 2D electromagnetic scattering, dissipative systems, Kraus operators, dilation

#### **1. Introduction**

Qubit lattice algorithms (QLA) were first being developed in the late 1990s to solve the Schrodinger equation [1–3] using unitary collision and streaming operators acting on some qubit basis. QLA recovers the Schrodinger equation in the continuum limit to second order in the spatial lattice grid spacing. Because the lattice node qubits are entangled by the unitary collision operator (much like in the formation of Bell states), QLA is encodable onto a quantum computer with an expected exponential speed-up over a classical algorithm run on a supercomputer. Moreover, since QLA is extremely parallelizable on a classical supercomputer, it provides an alternate algorithm for solving difficult problems in computational classical physics.

We then applied these QLA ideas to the study of the nonlinear Schrodinger equation (NLS) [4], by incorporating the cubic nonlinearity in the wave function, j j *ψ* 2 *ψ*, as an external potential operator following the unitary collide-stream operator sequence on the qubits. While the inclusion of such nonlinear terms poses no problem for a hybrid classical-quantum computer, it remains a very important and difficult research topic for their implementation on a quantum computer. The accuracy of the QLA for NLS was tested for soliton-soliton collisions in long-term integration and compared to exact analytic solutions, and while the QLA is second order, it seemed to behave like a symplectic integrator. The QLA was then extended to the totally integrable vector Manakov solitons [5] to handle inelastic soliton scattering. The Manakov solitons are solutions to a coupled set of NLS equations.

Following these successful benchmarking simulations, we moved into QLA for two (2D) and three (3D) dimensional NLS equations—where now there are no exact solutions to these nonlinear equations. In the field of condensed matter, these higher dimensions NLS equations are known as the Gross-Pitaevskii equations and give the mean field representation of the ground state wave function *ψ* of a zero-temperature Bose-Einstein condensate (BEC). For scalar quantum turbulence in 3D, we [6] observed a triple energy cascade on a 57323 grid, with the low-k ("classical") regime exhibiting a Kolmogorov *k*�5*=*<sup>3</sup> cascade in the *compressible* kinetic energy while the incompressible kinetic energy exhibited a long k-range of *k*�<sup>3</sup> spectrum. Similar results were found for both 2D and 3D scalar quantum [7–9], while results for spinor BECs can be found in Refs. [10–12]. A somewhat related, but significantly different, approach is that of the quantum lattice Boltzmann method [13, 14].

Here we will discuss a QLA for the solution of Maxwell equations in a tensor dielectric medium [15–18] and present some simulation results of the scattering of a 1D electromagnetic pulse off 2D localized dielectric objects. This can be viewed as a precursor to examining the scattering of electromagnetic pulses off plasma blobs in the exterior region of a tokamak.

There has been much interest in rewriting the Maxwell equations in operator form and exploit their similarity to the Schrodinger-Dirac equation from the early 1930s (e.g., see the references in [19]). For homogeneous media, the qubit representation of the electric and magnetic fields, **E, H**, leads to a Dirac equation in a fully unitary representation. However, when the media becomes inhomogeneous, a Dyson map [20] is required to yield a unitary Schrodinger-Dirac equation for the evolution of the electromagnetic qubit field representation. In particular, one can use the fields *nxEx*, *nyEy*, *nzEz*, *Bx*, *By*, *Bz* , where *ni* is the refractive index in the *i th*-direction.

A QLA is developed for this representation of the Maxwell equations in Section 3. This particular algorithm is a generalization of that used for the NLS equations. The initial value problem is then solved for the case of an electromagnetic pulse propagating in the *x*-direction and scattering from different 2D localized dielectric objects with refractive index *n x*ð Þ , *y* in Section 4. In particular, we have examined both polarizations of the pulse and ∇ � **B** ¼ 0. In Section 5, we consider the case in which the medium is dissipative. This brings in the field of open quantum systems and interactions with an environment. For illustration, we consider a simplified cold electron-ion dissipative fluid model in an electromagnetic field. Kraus operators are determined by a multidimensional analog of the quantum amplitude damping channel. Some estimates on the quantum gates required are given as well as a quantum circuit diagram illustrating the implementation of Kraus operators on the open Schrodinger equation. We summarize our results in Section 6.

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

Finally, in this introduction, we quickly review the entanglement of qubits and in particular for the 2-qubit Bell state [21].

$$B\_{+} = \frac{1}{\sqrt{2}}(|\mathbf{0}\mathbf{0}\rangle + |\mathbf{1}\mathbf{1}\rangle). \tag{1}$$

The most general 1-qubit states are f g *a*0j0i þ *a*1j1i , and f g *b*0j0i þ *b*1j1i with normalization j j *a*<sup>0</sup> <sup>2</sup> <sup>þ</sup> j j *<sup>a</sup>*<sup>1</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> j j *<sup>b</sup>*<sup>0</sup> <sup>2</sup> <sup>þ</sup> j j *<sup>b</sup>*<sup>1</sup> 2 . The tensor product of these two 1-qubit yields a space of the form

$$a\_0 b\_0|00\rangle + a\_0 b\_1|01\rangle + a\_1 b\_0|10\rangle + a\_1 b\_1|11\rangle. \tag{2}$$

However, the Bell state, Eq. (1) is not part of this tensor product space: to remove the ∣01i state from Eq. (2) either *a*<sup>0</sup> ¼ 0 or *b*<sup>1</sup> ¼ 0. This in turn would remove either the ∣00i or the ∣11i states, respectively. Now consider the unitary collision operator

$$\mathcal{C} = \begin{bmatrix} \cos\theta \sin\theta - \sin\theta \cos\theta \end{bmatrix} \tag{3}$$

acting on the subspace basis f g j00i, j11i . The choice of *θ* ¼ *π=*4 yields the Bell state *B*þ—a maximally entangled state. It is the quantum entanglement of states that will give rise to the exponential speed-up of a quantum algorithm. The QLA is a sequence of interleaved unitary collision-streaming operators that entangle the qubits and then spread that entanglement throughout the lattice.

#### **2. The Dyson map and the generation of a unitary evolution equation for Maxwell equations**

Consider the subset of Maxwell equations

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad , \quad \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} \tag{4}
$$

and treat ∇ � **B** ¼ 0 and ∇ � **D** ¼ 0 as initial constraints that remain satisfied in the continuum limit for all times. (This, of course, follows immediately from taking the divergence of Eq. (4)).

For lossless media, the electric and magnetic fields satisfy the constitutive relations for a tensor dielectric nonmagnetic medium

$$\mathbf{D} = \mathbf{e} \cdot \mathbf{E}, \quad \mathbf{B} = \mu\_0 \mathbf{H}. \tag{5}$$

For Hermitian *ε*, one can transform to a coordinate system in which *ε* is diagonal. Eq. (5) can be rewritten in matrix form

$$\mathbf{d} = \mathbf{W}\mathbf{u} \quad \text{, with} \quad \mathbf{d} \doteq \begin{pmatrix} \mathbf{D}, \mathbf{B} \end{pmatrix}^{\mathrm{T}}, \quad \mathbf{u} \dot{\mathbf{=}} (\mathbf{E}, \mathbf{H})^{\mathrm{T}} \tag{6}$$

where **W** is a 6 � 6 Hermitian block diagonal constitutive matrix

$$\mathbf{W} = \begin{bmatrix} \mathbf{e} \mathbb{I}\_{3 \times 3} & \mathbf{0}\_{3 \times 3} \\ \mathbf{0}\_{3 \times 3} & \mu\_0 \mathbb{I}\_{3 \times 3} \end{bmatrix}. \tag{7}$$

<sup>3</sup>�<sup>3</sup> is the 3 � 3 identity matrix, and the superscript **T** in Eq. (6) is the transpose. In matrix form, the Maxwell equations, Eq. (4) become

$$i\frac{\partial \mathbf{d}}{\partial t} = \mathbf{M}\mathbf{u} \tag{8}$$

where under standard boundary conditions, the curl-matrix operator **M** is Hermitian:

$$\mathbf{M} = \begin{bmatrix} \mathbf{0}\_{3 \times 3} & i\nabla \times \\ -i\nabla \times & \mathbf{0}\_{3 \times 3} \end{bmatrix}. \tag{9}$$

From Eq. (6), since **<sup>W</sup>**�**<sup>1</sup>** exists, **<sup>u</sup>** <sup>¼</sup> **<sup>W</sup>**�<sup>1</sup> **d**, so that Eq. (8) can be written

$$i\frac{\partial \mathbf{u}}{\partial t} = \mathbf{W}^{-1} \mathbf{M} \mathbf{u} \tag{10}$$

If the medium is homogeneous, then **W**�<sup>1</sup> is constant and will commute with the curl-operator **M**. Under these conditions, the product **W**�<sup>1</sup> **M** is Hermitian and Eq. (10) gives unitary evolution for **<sup>u</sup>** <sup>¼</sup> ð Þ **<sup>E</sup>**, **<sup>H</sup>** *<sup>T</sup>*.

However, if the medium is spatially inhomogeneous, then **W**�<sup>1</sup> , **<sup>M</sup>** � � 6¼ 0 and the evolution equation for the **u**-field is not unitary.

#### **2.1 Dyson map**

To determine a unitary evolution of the electromagnetic fields in an inhomogeneous dielectric medium, it [20] has been shown that there exists a Dyson map *ρ*: **u** ! **Q** such that in the new field variables **Q** the resulting evolution equation will be unitary. For the Maxwell equations consider

$$\mathbf{Q} = \rho \mathbf{u} = \mathbf{W}^{1/2} \mathbf{u}. \tag{11}$$

For time-independent media, the evolution equation for the new fields **Q** is

$$i\rho \frac{\partial \mathbf{u}}{\partial t} = \rho \mathbf{W}^{-1} \mathbf{M} \rho^{-1} \rho \mathbf{u} \quad \Rightarrow i \frac{\partial \mathbf{Q}}{\partial t} = \rho \mathbf{W}^{-1} \mathbf{M} \rho^{-1} \mathbf{Q} \tag{12}$$

and is indeed unitary. Explicitly, the new fields, Eq. (11) and the *ρ* are

$$\mathbf{Q} = \begin{bmatrix} q\_0 \\ q\_1 \\ q\_2 \\ q\_3 \\ q\_4 \\ q\_5 \end{bmatrix} = \begin{bmatrix} n\_x E\_x \\ n\_y E\_y \\ n\_z E\_x \\ \mu\_0^{1/2} H\_x \\ \mu\_0^{1/2} H\_y \\ \mu\_0^{1/2} H\_x \end{bmatrix}, \quad \boldsymbol{\rho} = \begin{bmatrix} n\_x & 0 & 0 & 0 & 0 & 0 \\ 0 & n\_y & 0 & 0 & 0 & 0 \\ 0 & 0 & n\_x & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu\_0^{1/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu\_0^{1/2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu\_0^{1/2} \end{bmatrix} \tag{13}$$

The refractive index *ni* ¼ ffiffiffi *εi* <sup>p</sup> . Typically, we will use units where *<sup>μ</sup>*<sup>0</sup> <sup>¼</sup> 1. In component form, Maxwell equations for fields and with constitutive matrix restricted to spatially 2D ð Þ *x*, *y* dependence, Eq. (12) reduces to

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

$$\begin{aligned} \frac{\partial q\_0}{\partial t} &= \frac{1}{n\_x} \frac{\partial q\_5}{\partial \mathbf{y}}, & \frac{\partial q\_1}{\partial t} &= -\frac{1}{n\_\gamma} \frac{\partial q\_5}{\partial \mathbf{x}}, & \frac{\partial q\_2}{\partial t} &= \frac{1}{n\_x} \left[ \frac{\partial q\_4}{\partial \mathbf{x}} - \frac{\partial q\_3}{\partial \mathbf{y}} \right] \\ \frac{\partial q\_3}{\partial t} &= -\frac{\partial (q\_2/n\_x)}{\partial \mathbf{y}}, & \frac{\partial q\_4}{\partial t} &= \frac{\partial (q\_2/n\_x)}{\partial \mathbf{x}}, & \frac{\partial q\_5}{\partial t} &= -\frac{\partial (q\_1/n\_\gamma)}{\partial \mathbf{x}} + \frac{\partial (q\_0/n\_x)}{\partial \mathbf{y}} \end{aligned} \tag{14}$$

#### **3. A qubit lattice representation for 2D tensor dielectric media**

QLA consists of a sequence of unitary collision and streaming operators on a 2D spatial lattice, which will recover the continuum Maxwell equations, Eq. (14) to second order in the spatial grid size, *δ*. In particular, we need to have 6 qubits/lattice sites to represent the field components in Eq. (13). QLA permits us to handle the xand y-dependence separately. Let us first consider the x-dependence and recover the *∂qi <sup>=</sup>∂<sup>x</sup>* - terms. From Eq. (14), we see coupling between *<sup>q</sup>*<sup>1</sup> \$ *<sup>q</sup>*5, *<sup>q</sup>*<sup>2</sup> \$ *<sup>q</sup>*4, Hence, we introduce the local entangling collision operator

$$\mathbf{C}\_{X} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & \cos \theta\_1 & 0 & 0 & 0 & -\sin \theta\_1 \\ 0 & 0 & \cos \theta\_2 & 0 & -\sin \theta\_2 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \sin \theta\_2 & 0 & \cos \theta\_2 & 0 \\ 0 & \sin \theta\_1 & 0 & 0 & 0 & \cos \theta\_1 \end{bmatrix} \tag{15}$$

The collision angles *θ*<sup>1</sup> and *θ*<sup>2</sup> need to be chosen to recover the refractive index factors before the corresponding spatial derivatives,

$$
\theta\_1 = \frac{\delta}{4n\_\mathcal{\mathcal{Y}}} \quad , \quad \theta\_2 = \frac{\delta}{4n\_x} \,. \tag{16}
$$

The first of the unitary streaming operators will stream qubits *q*1, *q*<sup>4</sup> one lattice unit in either direction while leaving the other four qubits fixed: *S*� 14. The other unitary streaming operator will act on qubits *q*2, *q*5: *S*� 25. The final unitary collide-stream sequence, **UX** in the x-direction that leads to a second-order scheme in *δ* can be shown to be

$$\mathbf{U}\_{\mathbf{X}} = \mathbf{S}\_{\mathbf{25}}^{+\mathbf{x}} \mathbf{C}\_{X}^{\dagger} \mathbf{S}\_{\mathbf{25}}^{-\mathbf{x}} \mathbf{C}\_{X} \mathbf{S}\_{\mathbf{14}}^{-\mathbf{x}} \mathbf{C}\_{X}^{\dagger} \mathbf{S}\_{\mathbf{14}}^{+\mathbf{x}} \mathbf{C}\_{X} \mathbf{S}\_{\mathbf{25}}^{-\mathbf{x}} \mathbf{C}\_{X} \mathbf{S}\_{\mathbf{25}}^{+\mathbf{x}} \mathbf{C}\_{X}^{\dagger} \mathbf{S}\_{\mathbf{14}}^{+\mathbf{x}} \mathbf{C}\_{X} \mathbf{S}\_{\mathbf{14}}^{-\mathbf{x}} \mathbf{C}\_{\mathbf{14}} \mathbf{S}\_{\mathbf{14}}^{-\mathbf{x}} \mathbf{C}\_{X}^{\dagger}.\tag{17}$$

It should be noted that if only applies the first 4 collide-stream sequence in Eq. (17) then the algorithm would only be first-order accurate.

Similarly, to recover the *<sup>∂</sup>qi =∂y* terms one would collisionally entangle qubits *q*<sup>0</sup> \$ *q*5, *q*<sup>2</sup> \$ *q*<sup>3</sup> with

$$\mathbf{C}\_{Y} = \begin{bmatrix} \cos\theta\_{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \sin\theta\_{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \cos\theta\_{2} & \sin\theta\_{2} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -\sin\theta\_{2} & \cos\theta\_{2} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\ -\sin\theta\_{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \cos\theta\_{0} \end{bmatrix},\tag{18}$$

with corresponding collision angles *θ*<sup>0</sup> and *θ*2. *θ*<sup>2</sup> is given in Eq. (16), and

$$
\theta\_0 = \frac{\delta}{4n\_\infty}.\tag{19}
$$

The streaming operator *S y* <sup>03</sup> will act on qubits *q*0, *q*<sup>3</sup> only and similarly for the operator *S y* <sup>25</sup>. The final unitary collide-stream second-order accurate or the y-direction for Maxwell equations is

$$\mathbf{U\_{Y}} = \mathbf{S\_{2\circ}^{+y}} \mathbf{C\_{Y}^{\dagger}} \mathbf{S\_{2\circ}^{-y}} \mathbf{C\_{Y}} \mathbf{S\_{0\otimes}^{-y}} \mathbf{C\_{Y}^{\dagger}} \mathbf{S\_{0\otimes}^{+y}} \mathbf{C\_{Y}} \mathbf{S\_{2\circ}^{-y}} \mathbf{C\_{Y}} \mathbf{S\_{2\circ}^{+y}} \mathbf{C\_{Y}^{\dagger}} \mathbf{S\_{0\otimes}^{+y}} \mathbf{C\_{Y}} \mathbf{S\_{0\otimes}^{-y}} \mathbf{C\_{Y}} \mathbf{S\_{0\otimes}^{-y}} \mathbf{C\_{Y}^{\dagger}} \tag{20}$$

We still need to recover the spatial derivatives on the refractive index components in Eq. (14). To obtain the *∂nz=∂x* and *∂ny=∂x* terms, we introduce the (non-unitary) sparse potential matrix

$$V\_X = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -\sin\beta\_2 & 0 & \cos\beta\_2 & 0 \\ 0 & \sin\beta\_0 & 0 & 0 & 0 & \cos\beta\_0 \end{bmatrix} \tag{21}$$

with collision angles

$$
\beta\_0 = \delta^2 \frac{\partial \mathfrak{n}\_{\mathcal{Y}}/\partial \mathfrak{x}}{n\_{\mathcal{Y}}^2} \quad , \quad \beta\_2 = \delta^2 \frac{\partial \mathfrak{n}\_{\mathcal{X}}/\partial \mathfrak{x}}{n\_{\mathcal{z}}^2} , \tag{22}
$$

while the corresponding (non-unitary) sparse potential matrix to recover the *∂=∂y*-derivatives in the refractive index components is

$$V\_Y = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & o \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & \cos \beta\_3 & \sin \beta\_3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ -\sin \beta\_1 & 0 & 0 & 0 & 0 & \cos \beta\_1 \end{bmatrix} \tag{23}$$

with collision angles

$$
\beta\_1 = \delta^2 \frac{\partial n\_x / \partial \mathbf{y}}{n\_x^2} \quad , \quad \beta\_3 = \delta^2 \frac{\partial n\_x / \partial \mathbf{y}}{n\_x^2} . \tag{24}
$$

Thus, the final discrete QLA, that models the 2D Maxwell equations, Eq. (14), to *<sup>O</sup> <sup>δ</sup>*<sup>2</sup> � �, advances the lattice qubit-vector **<sup>Q</sup>**ð Þ*<sup>t</sup>* to **<sup>Q</sup>**ð Þ *<sup>t</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>t</sup>* is

$$\mathbf{Q}(t + \Delta t) = V\_Y. V\_X. \mathbf{U}\_Y. \mathbf{U}\_X. \mathbf{Q}(t) \tag{25}$$

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

Provided, we have diffusion ordering in the space–time lattice, that is, <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup> . It is this ordering that requires us to have the unitary collision angles to be *O*ð Þ*δ* , Eqs. (16) and (19), and the external potential angles *O δ*<sup>2</sup> � �, Eqs. (22). We note that computationally QLA is more accurate if we employ the external potentials twice: once halfway through the collide-stream sequence and then at the end.

#### **3.1 Non-unitary external potential operators**

Recently, considerable effort has been expended into developing more efficient approximation for handling the evolution operator of a complex Hamiltonian system than the standard Suzuki-Trotter expansion Eqs. (21) and (23) [22]. In particular, the idea [23, 24] has been floated of approximating the full unitary operator by a *sum* of unitary operators. The actual implementation onto a quantum computer we will leave to another paper, as one of the outcomes of QLA discussed here will be a quantuminspired highly efficient classical supercomputer algorithm. Moreover, its encoding onto a quantum computer will require error-correcting qubits with long coherence times, something currently out of reach in the noisy qubit regime we are in. Here, we will show the 4 unitary operators needed whose sum yields the sparse non-unitary potential operator *VX*, Eq. (21). Letting <sup>6</sup> be the 6 � 6 identity matrix, then it is easily verified that

$$V\_X = \frac{1}{2} \sum\_{i=1}^{4} LCU\_i \tag{26}$$

where the first two unitaries are diagonal

$$LCU\_1 = \mathbb{I}\_6 \quad , \quad LCU\_2 = diag(-\mathbf{1}, \mathbf{1}, \mathbf{1}, -\mathbf{1}, -\mathbf{1}, -\mathbf{1}) \tag{27}$$

and the remaining two unitaries are

$$LCU\_3 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & o \\ 0 & \cos \beta\_0 & 0 & 0 & 0 & -\sin \beta\_0 \\ 0 & 0 & \cos \beta\_2 & 0 & \sin \beta\_2 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -\sin \beta\_2 & 0 & \cos \beta\_2 & 0 \\ 0 & \sin \beta\_0 & 0 & 0 & 0 & \cos \beta\_0 \end{bmatrix},\tag{28}$$

and

$$LCU\_4 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & o \\ 0 & -\cos\beta\_0 & 0 & 0 & 0 & \sin\beta\_0 \\ 0 & 0 & -\cos\beta\_2 & 0 & -\sin\beta\_2 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -\sin\beta\_2 & 0 & \cos\beta\_2 & 0 \\ 0 & \sin\beta\_0 & 0 & 0 & 0 & \cos\beta\_0 \end{bmatrix}.\tag{29}$$

**107**

#### **3.2 Conservation of energy**

In a fully unitary representation, the norm of **Q** is a constant of the motion. This is simply the conservation of energy in the electromagnetic field. For fields being a function of ð Þ *x*, *y* , we have from Eqs. (11)–(13)

$$\mathcal{R}(t) = \frac{1}{L^2} \int\_0^L dx dy \mathbf{Q} \cdot \mathbf{Q} = \frac{1}{L^2} \int\_0^L dx dy \left[ \varepsilon\_x E\_x^2 + \varepsilon\_y E\_y^2 + \varepsilon\_z E\_z^2 + \mu\_o \left( B\_x^2 + B\_y^2 + B\_z^2 \right) \right] \tag{30}$$

where the (diagonal) tensor dielectric *<sup>ε</sup><sup>x</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> *<sup>x</sup>*, … and we restrict ourselves to nonmagnetic materials, for simplicity.

#### **4. 2D numerical simulations from the QLA for electromagnetic scattering from 2D dielectric objects**

We now present detailed QLA simulations of the initial value problem of the scattering of a 1D electromagnetic pulse from a localized dielectric object. In particular, we consider a 1D Gaussian pulse propagating in the *x*-direction toward a localized dielectric object of refractive index *n x*ð Þ , *y* . The initial pulse has nonzero field components *Ez*, *By*, **Figure 1**, and scatters from either a localized cylindrical dielectric, **Figure 2a**, or a conic dielectric object, **Figure 2b**. These simulations were performed for *δ* ¼ 0*:*1.

It should be noted that QLA is an initial value algorithm. The refractive index profiles are smooth (e.g., hyperbolic tangents for the dielectric cylinder with boundary layer thickness ≈10 lattice units) and so *no* internal boundary conditions are imposed at any time in the simulation.

#### **4.1 Effects of broken symmetry**

When the 1D pulse scatters off the dielectric object with refractive index *n x*ð Þ , *y* the initial electric field spatial dependence *Ez*ð Þ *x* now becomes a function *Ez*ð Þ *x*, *y* , while the initial magnetic field *By*ð Þ *x* will become a function *By*ð Þ *x*, *y* . Now the scattered field

#### **Figure 1.**

*A 1D electromagnetic pulse with initial fields* �*Ez*ð*x*, *t* ¼ *0)*, *By*ð*x*, *t* ¼ *0). 2D simulation grid L* � *L with L* ¼ *8192. Pulse full-width (in lattice units)* ≈ *200. Since the Maxwell equations are linear and homogenous, the initial amplitude of the fields is arbitrary.*

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

**Figure 2.**

*(a) The dielectric cylinder, diameter* ≈ *200, has rapidly increasing boundary dielectric from vacuum n* ¼ 1 *to nmax* ¼ 3*, whereas (b) the conic dielectric, base* ≈ *240, has smoothly increasing dielectric from vacuum to conic peak of nmax* ¼ *3.*

has *<sup>∂</sup>By=∂<sup>y</sup>* 6¼ 0. Thus, for <sup>∇</sup> � **<sup>B</sup>** <sup>¼</sup> 0, the scattered field must develop an appropriate *Bx*ð Þ *x*, *y* . This is seen in our QLA simulations, even though the explicit discrete collidestream algorithm only models asymptotically the Maxwell subset, Eq. (4). ∇ � **D** ¼ 0 and this is exactly conserved in our QLA simulation.

Similarly, for initial *Ey*ð Þ *x* polarization. In this case, the scattered magnetic field *Bz*ð Þ *x*, *y* satisfies, for our 2D scattering in the x-y plane, ∇ � **B** ¼ 0 exactly and no other magnetic field components are generated. Our discrete QLA recovers this ∇ � **B** ¼ 0 exactly. However, in an attempt to preserve ∇ � **D** ¼ 0, the QLA will generate a nonzero *Ex*ð Þ *x*, *y* field.

#### **4.2 Scattering from localized 2D dielectric objects with refractive index** *n x***,** *y*

Consider a 1D pulse with polarization *Ez*ð Þ *x*, *t* propagating in a vacuum toward a 2D dielectric scatterer. In **Figures 3**–**6**, we consider the time evolution of the resultant scattered *Ez*-field. The initial pulse is followed for a short time while it is propagating in the vacuum to verify that the QLA correctly determines its motion. As part of the pulse interacts with the dielectric object, the pulse speed within the dielectric itself is

#### **Figure 3.**

*The scattered Ez field after 15,000 iterations (i.e., t* ¼ *15k). (a) There is an internal reflection at the back of the cylindrical dielectric.*

#### **Figure 4.**

*The scattered Ez field at t* ¼ *23.4k. (a) a reflected circular wavefront occurs as that part of the pulse reaches the back-end of the cylindrical dielectric, along with the initial reflected circular wavefront with its π-changed phase at the front of the vacuum-cylinder boundary. (b) for the conic dielectric, there is an internal reflection from the apex of the cone's nmax, which then propagates out of the weakly varying cone edges.*

#### **Figure 5.**

*The scattered Ez field at t* ¼ *31.2k. (a) There are multiple reflections/transmissions within the boundaries of the dielectric cylinder. (b) There is only one major reflection from the apex of the cone and which then propagates readily out from the cone edge.*

decreased by the inverse of the refractive index profiles, *n x*ð Þ , *y* . The remainder of the 1D pulse propagates undisturbed since it is still propagating within the vacuum.

One sees in **Figure 3a**, a circular-like wavefront reflecting back into the vacuum, with its *Ez* field *π* out of phase as the reflection is occurring from a low to higher refractive index around the vacuum-cylinder interface. One does not find such a reflected wavefront when the pulse interacts with the conic dielectric, **Figure 3b**.

At *t* ¼ 23*:*4*k*, there is a major wavefront emanating from the back of the cylindrical dielectric, **Figure 4a**. For the conic dielectric, there is a major wavefront reflected from the apex of the conic dielectric, and this propagates out of the cone with a little reflection, **Figure 4b**.

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

#### **Figure 6.**

*The scattered Ez field at t* ¼ *49.2k. (a) There are multiple reflections/transmissions within the boundaries of the dielectric cylinder. (b) There is only one major reflection from the apex of the cone and which then propagates readily out from the cone edge.*

One clearly sees at *t* ¼ 31*:*2*k* that more *Ez* wavefronts are being created because of the large refractive index gradients at the vacuum-cylinder dielectric boundary, while such gradients are missing from the vacuum-cone interface which leads to no new wavefronts in the scattering of the dielectric cone, **Figure 5**.

At *t* ¼ 49*:*2*k*, the complex *Ez* wavefronts are due to repeated reflections and transmissions from the cylinder dielectric, **Figure 5a**. However, because of the slowly changing boundaries of the dielectric cone there are no more reflections and one sees only the outgoing wavefront from the pulses' interaction with the region around the *nmax* of the cone. Since the pulse reaches the apex of the cone before the corresponding pulse hits the backend of the dielectric cylinder, the conic wavefront is further advanced than that of the cylindrical wavefront, **Figure 5b**.

### *4.2.1 Auxiliary fields and* ∇ � **B**

For incident *Ez* polarization and with 2D refractive index *n x*ð Þ , *y* , the scattered electromagnetic fields will need to generate a *Bx* field in order to have ∇ � **B** ¼ 0. In **Figure 6a** and **b**, we plot the self-generated *Bx*ð Þ *x*, *y* field at *t* ¼ 23*:*4*k* and *t* ¼ 49*:*2*k* for scattering from the dielectric cylinder. It is also found that ∣∇ � **B**∣*=B*<sup>0</sup> is typically zero everywhere in the spatial lattice except for a very localized region around the vacuumdielectric boundary layer where the normalized *max* ð Þ j∇ � **B**j *=B*<sup>0</sup> reaches around 0.01 at very few isolated grid points (**Figure 7**).

### **4.3 Time dependence of** ð Þ*t* **on perturbation parameter** *δ*

The discrete total electromagnetic energy ℰð Þ*t* . Eq. (30), is not constant since our current QLA is not totally unitary. However, the variations in ℰ decrease significantly as *δ* ! 0. *δ* is a measure of the discrete lattice spacing. The maximal variations occur shortly after the 1D pulse scatters from the 2D dielectric object. For *δ* ¼ 0*:*1, this occurs around *t* ¼ 15*k*, with variations in the 5th decimal, Eq. (31). However, when

#### **Figure 7.**

*The self-consistently generated Bx*ð Þ *x*, *y -field after the 1D incident pulse with By* ¼ *By*ð Þ *x scatters from a local dielectric with refractive index n x*ð Þ , *y at times: (a) t* ¼ *23.4k, corresponding to Figure 4a for Ez, and (b) t* ¼ *49.2k, corresponding to Figure 5a for Ez.*

one reduces *δ* by a factor of 10 on the same lattice grid, then one recovers the same physics a factor of 10 later in time since *δ* controls the speed of propagation in the vacuum. Thus, the wallclock time of a QLA run is also increased by this factor of 10. We find, for *δ* ¼ 0*:*01, that the largest deviation in the total electromagnetic energy is now in the 8th decimal, Eq. (31).


For *<sup>δ</sup>* <sup>¼</sup> <sup>10</sup>�<sup>3</sup> , there is variation in ℰ in the 11th decimal.

#### **4.4 Multiple reflections/transmissions within dielectric cylinder**

We now examine the scattered electromagnetic fields—particularly the polarization *Ez*ð Þ *x*, *y* —within and in the vicinity of the dielectric cylinder. These plots complement the global scattered *Ez*ð Þ *x*, *y* in **Figures 3a, 4a**, and **5a**, but for better resolution we choose *δ* ¼ 0*:*01 and a slightly different ratio of pulse width to dielectric cylinder diameter. In **Figures 8**–**13**, the perspective is looking down from above with the 1D pulse propagating from left to right (!), seen as a dark vertical band. The dielectric cylinder appears as a pink cylinder with the smaller darker pink being the base of the cylinder. The time is expressed in normalized time: *<sup>t</sup>* <sup>¼</sup> *<sup>t</sup>* <sup>∗</sup> *<sup>=</sup>*10, where *<sup>t</sup>* <sup>∗</sup> is the QLA time for *δ* ¼ 0*:*01.

In **Figure 8a**, at time *t* ¼ 7*:*2*k*, a part of the incident pulse has just entered the dielectric cylinder with the transmitted *Ez* field starting to lag behind the main 1D vacuum pulse since 1< *ncyl*. Also, the reflected part of *Ez* emanates from the two boundary points at the sharp vacuum-dielectric boundary and has undergone a

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

#### **Figure 8.**

*A view from the z-axis of the 1D incident Ez wavefront, with x-y the plane of the page. The vacuum pulse is propagating in the x-direction,* ! *(a) the 1D incident pulse has encountered the localized dielectric cylinder, with both transmission and reflection at the thin vacuum-dielectric boundary layer. The reflected Ez circular wavefront undergoes a π-phase change. (b) the transmitted Ez, within the dielectric, has a lower phase speed and so lags the 1D vacuum pulse.*

#### **Figure 9.**

*As the 1D vacuum part of the wavefront moves past the dielectric cylinder, the two vacuum-dielectric boundary "points" move closer together: (a) at t = 18 k, (b) at t = 22.2 K. the vacuum-reflected wavefront keeps radiating out.*

*π*-phase change because the incident 1D pulse is propagating from low to high refractive index. In **Figure 8b**, the slower transmitted *Ez* wavefront within the dielectric is very evident, as is the reflected part of *Ez* back into the vacuum.

By *t* ¼ 18*k*, **Figure 7b**, the 1D pulse has propagated past the dielectric. The *Ez*-field within the dielectric is now being focussed due to its motion toward the backend of the cylinder, with its increasing amplitude but reduced base. As it reaches the backend of the dielectric, part of *Ez* will be transmitted into the vacuum while the other part will be reflected back into the dielectric but now without any phase change since the pulse is propagating from high to low refractive index.

#### **Figure 10.**

*Wavefronts of Ez at times (a) t = 28.2 k, and (b) t = 32 k around and within the dielectric cylinder after the original 1D pulse has moved past the dielectric. (a) the pinching of the two boundary "points" results in a focussing of Ez and its subsequent spiking at t* ¼ 28*:*2*k. This spike now propagates toward the backend of the dielectric cylinder, (b), and "diffuses," one should also note the wavefront emanating from the 1D vacuum pulse.*

#### **Figure 11.**

*Wavefronts of Ez at times around and within the dielectric cylinder. At (a) t = 38.4 k the transmitted Ez within the dielectric is radiating outward, with one part reaching the back of the dielectric and resulting in a complex transmission into the vacuum region at the back end of the dielectric, (b) at t = 43.2 k, and a complex reflection back into the dielectric. There is no phase change in the reflected Ez.*

#### **5. Dissipative classical systems, open quantum systems, and Kraus representation**

So far we have treated Maxwell equations as a closed system based on the energy conservation dictated from the Hermiticity and positive definiteness of the constitutive matrix **W**, Eq. (7) since we have restricted ourselves to perfect materials. However, when we wish to consider actual materials, there is dissipation. This immediately defeats any attempt to pursue a unitary representation in the original Hilbert space. The obvious question is: can we embed our dissipative system into a higher dimension *Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

#### **Figure 12.**

*Wavefronts of Ez at times (a) t = 47.4 k, and (b) t = 52.2 k around and within the dielectric cylinder. The major vacuum wavefront that is transmitted out of the dielectric now radiates out in the xy-plane. The two boundary contact "points" of the wavefront are now propagating back to the front of the dielectric cylinder, as clearly seen in (a) and (b). These localized wavefronts will have their global wavefronts similar to those shown in Figures 3a, 4a, and 5a.*

**Figure 13.** *Wavefronts of Ez at times (a) t = 55.8 k, and (b) t = 60 k around and within the dielectric cylinder.*

closed Hilbert space, and thus recover unitary evolution in this new space and build an appropriate QLA that can be encoded onto quantum computers? To accomplish this, we resort to open quantum system theory [21] to describe classical dissipation as the observable result of interaction between our system of interest and its environment.

For a closed quantum system, the time evolution of a pure state ∣*ψ*ð Þi *t* is given by the unitary evolution from the Schrodinger equation: ∣*ψ*ð Þi ¼ *t U t*ð Þ∣*ψ*ð Þi 0 with *U* ¼ *exp* ½ � �*itH*<sup>0</sup> unitary for the Hermitian Hamiltonian *H*0. The evolution of the density matrix, *ρ* ¼ ∣*ψ*ih*ψ*∣, is governed by the corresponding von Neumann equation: *<sup>ρ</sup>*ðÞ¼ *<sup>t</sup> U t*ð Þ*ρ*ð Þ <sup>0</sup> *<sup>U</sup>*† ð Þ*t* . The density matrix formulation is required when dealing with composite systems. Kraus realized that the density matrix retains its needed properties if one generalized its evolution operator to

*Schrödinger Equation – Fundamentals Aspects and Potential Applications*

$$\rho(t) = \sum\_{k} K\_{k} \rho(0) K\_{k}^{\dagger}, \quad \text{with} \sum\_{k} K\_{k}^{\dagger} K\_{k} = I \tag{32}$$

where the only restriction on the set of so-called Kraus matrices *Kk* is that the sum of *K*† *<sup>k</sup>Kk* is the identity matrix. The evolution of the density matrix, Eq. (32), is no longer unitary for *k*≥ 2.

The Kraus representation [21] is most useful when dealing with quantum noisy operations due to interaction with an environment. For those problems in which this noisy operation translates into a dissipative process, the Hamiltonian for the system in the Schrodinger representation has both a Hermitian part, *H*0, and an anti-Hermitian part, *iH*1, that models the dissipation. A simple but nontrivial example is the 1*D* Maxwell equations (without sources) for a homogeneous scalar medium with electrical losses,

$$i\frac{\partial}{\partial t}\begin{bmatrix}E\_{\mathcal{V}}\\H\_{x}\end{bmatrix}=\begin{bmatrix}\mathbf{0} & \frac{\boldsymbol{\varepsilon}^{\*}}{|\boldsymbol{\varepsilon}|^{2}}\hat{\boldsymbol{p}}\_{x}\\\-\frac{1}{\mu\_{0}}\hat{\boldsymbol{p}}\_{x} & \mathbf{0}\end{bmatrix}\begin{bmatrix}E\_{\mathcal{V}}\\H\_{x}\end{bmatrix}\tag{33}$$

with complex permittivity *<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup><sup>R</sup>* <sup>þ</sup> *<sup>i</sup>εI: <sup>ε</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>ε</sup><sup>R</sup>* � *<sup>i</sup>εI: <sup>p</sup>*^*<sup>x</sup>* ¼ �*i∂<sup>x</sup>* is the momentum operator. Introducing the Dyson map *ρ* ¼ *diag* j*ε*j*=* ffiffiffiffiffi *εR* p , ffiffiffiffiffi *μ*0 � � p into Eq. (33) and after some algebraic manipulations the evolution equation can be written as

$$i\frac{\partial \mathbf{Q}}{\partial t} = \left[v\_{\delta}\left(\sigma\_{\mathbf{x}} + \frac{1}{2}\delta\sigma\_{\mathbf{y}}\right)\hat{p}\_{\mathbf{x}} - \frac{i}{2}\delta v\_{\delta}\sigma\_{\mathbf{x}}\hat{p}\_{\mathbf{x}}\right]\mathbf{Q},\tag{34}$$

where the state vector **Q** ¼ *ρ***u**, where **u** is defined in Eq. (6). *δ* ¼ *εI=ε<sup>R</sup>* is the loss angle, *v<sup>δ</sup>* is the phase velocity *v<sup>δ</sup>* ¼ 1*=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ε</sup>Rμ*<sup>0</sup> <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> <sup>q</sup> � � and *<sup>σ</sup>x*, *<sup>σ</sup>y*, *<sup>σ</sup><sup>z</sup>* are the Pauli matrices.

#### **5.1 Classical dissipation as a quantum amplitude damping channel**

In symbolic form, the Maxwell equations with electric resistive losses, Eq. (34), can be written in the Schrodinger-form

$$i\frac{\partial|\boldsymbol{\mu}\_{\boldsymbol{S}}\rangle}{\partial\boldsymbol{t}} = \left(\hat{H}\_{0}(\mathbf{r}) - i\hat{H}\_{1}(\mathbf{r})\right)|\boldsymbol{\mu}\_{\boldsymbol{S}}\rangle\tag{35}$$

where the Hamiltonians *H*^ <sup>0</sup> and *H*^ <sup>1</sup> are Hermitian, and the dissipative operator *iH*^ <sup>1</sup> is anti-Hermitian and positive definite. The positive definiteness requirement for the specific case of propagation in a lossy medium translates to

$$\operatorname{Im}\left[E\_{\text{y}}^{\*}\,\frac{\partial H\_{x}}{\partial \mathbf{x}} + H\_{x}^{\*}\,\frac{\partial E\_{\text{y}}}{\partial \mathbf{x}}\right] > 0,\quad\text{with}\quad \varepsilon\_{l} > 0.\tag{36}$$

In general, the dissipative operator *H*^ <sup>1</sup> is relatively simple and models the phenomenological or coarse-graining of the underlying microscopic dissipative processes.

We aim to represent the dissipation in the Schrodinger picture, Eq. (35), as an open quantum system S interacting with its environment *Env*. The full system,

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

*<sup>S</sup>* <sup>þ</sup> *Env*, is closed, and hence its time evolution is unitary. Let <sup>U</sup>^ be this unitary operator, and *<sup>ρ</sup>* the total density matrix with <sup>U</sup>^ : *<sup>ρ</sup>*ð Þ! <sup>0</sup> *<sup>ρ</sup>*ð Þ*<sup>t</sup>* . We make the usual assumption that the initial total density matrix is separable into the system and into the environmental Hilbert spaces: *ρ*ð Þ¼ 0 *ρS*ð Þ 0 ⊗ *ρE*ð Þ 0 . A quantum operation *E* on the open system of interest is defined as the map that propagates the open system density in time *t*:

$$
\rho\_S(t) = E(\rho\_S(\mathbf{0})).\tag{37}
$$

But under the conditions of initial separability, the action of the full unitary operators on the total density matrix will yield, after taking the trace over the environment,

$$\rho\_S(t) = Tr\_E(\rho(t)) = Tr\_E\left(\hat{\mathcal{U}}\rho(\mathbf{0})\hat{\mathcal{U}}^\dagger\right) \tag{38}$$

Assuming a stationary environment, *ρE*ð Þ¼ 0 ∣*a*ih*a*∣, Eq. (38) can be written as

$$
\rho\_{\mathcal{S}}(\mathbf{t}) = \sum\_{\mu} \hat{\mathcal{K}}\_{\mu} \rho\_{\mathcal{S}}(\mathbf{0}) \hat{\mathcal{K}}\_{\mu}^{\dagger}. \tag{39}
$$

where *<sup>K</sup>*^ *<sup>μ</sup>* <sup>¼</sup> *<sup>μ</sup>*jU^ <sup>j</sup>*<sup>a</sup>* � �. These operators *<sup>K</sup>*^ *<sup>μ</sup>* will form a Kraus representation for the quantum operation *E* for an open system, Eq. (37), provided the so-called Kraus operators satisfy the extended "unitarity" condition

$$\sum\_{\mu} \mathcal{K}\_{\mu}^{\dagger} \mathcal{K}\_{\mu} = I \tag{40}$$

Note that the individual Kraus operator need not be unitary. Based on this framework for open quantum systems, we proceed to construct a physical unitary dilation for the combined system-environment by identifying dissipation as an amplitudedamping operation, [21].

Let *d* be the dimension of the system Hilbert space, and *r* the dimension of the dissipative Hamiltonian *H*1, Eq. (35). We require *d*≥2*r*, for optimal results but the dilation technique can be also applied to systems with *d* ¼ *r*. If the system was quantum mechanical in nature, then there can be a set of *d*<sup>2</sup> Kraus operators at most. The matrix representation of the total unitary dilation evolution operator consists of listing all the Kraus matrices in the first column block. The remaining columns must then be determined, so that U^ is unitary. This unitary dilation is equivalent to the Stinespring dilation theorem [25]. The advantage of the Kraus approach is that it avoids the need to actually know the physical properties of the environment.

Returning to the Schrodinger representation of the classical system Eq. (35), one can employ the Trotter-Suzuki expansion to *exp* �*iδ<sup>t</sup> <sup>H</sup>*^ <sup>0</sup> � *iH*^ <sup>1</sup> � � � �

$$|\psi(\delta t)\rangle = \left[e^{-i\delta t\hat{H}\_0} \cdot e^{-\delta t\hat{H}\_1} + O\left(\delta t^2\right)\right]|\psi(0)\rangle. \tag{41}$$

Even though *exp* �*δtH*^ <sup>1</sup> � � is not unitary, *H*^ <sup>1</sup> is Hermitian and can be diagonalized by a unitary transformation *U*<sup>1</sup>

$$
\hat{H}\_1 = \hat{U}\_1 \hat{D}\_1 \hat{U}\_1^\dagger \quad \text{with diagonal} \quad \hat{D}\_1 = \text{diag}[\gamma\_1, \dots, \gamma\_r], \tag{42}
$$

where *γ<sup>i</sup>* >0 are the dissipative rate eigenvalues of *H*^ 1. Thus, Eq. (41) becomes

$$|\psi(\delta t)\rangle = \left[e^{-i\delta t\hat{H}\_0}\hat{U}\_1\hat{K}\_0\hat{U}\_1^\dagger + \mathcal{O}\left(\delta t^2\right)\right]|\psi(\mathbf{0})\rangle,\tag{43}$$

where *K*^ <sup>0</sup> is

$$\hat{\mathbf{K}}\_0 = \begin{bmatrix} \hat{\Gamma}\_{r \times r} & \mathbf{0}\_{r \times r} \\ \mathbf{0}\_{(d-r) \times (d-r)} & I\_{(d-r) \times (d-r)} \end{bmatrix}, \quad \text{with diagonal } \hat{\Gamma}\_{r \times r} = \text{diag}\left[e^{-\gamma\_1 \mathcal{H}} \dots e^{-\gamma\_r \mathcal{H}}\right]. \tag{44}$$

The non-unitary *K*^ <sup>0</sup> will be one of our Kraus operators, and it describes the physical dissipation in the open system. We must now introduce a second Kraus operator *<sup>K</sup>*^1, so that *<sup>K</sup>*^ † <sup>0</sup>*K*^ <sup>0</sup> <sup>þ</sup> *<sup>K</sup>*^ † <sup>1</sup>*K*^<sup>1</sup> <sup>¼</sup> <sup>ℐ</sup>:

$$
\hat{\mathbf{K}}\_1 = \begin{bmatrix}
\mathbf{0}\_{(d-r)\times(d-r)} & \mathbf{0}\_{(d-r)\times(d-r)} \\
\sqrt{I\_{r\times r} - \hat{\Gamma}^2} & \mathbf{0}\_{r\times r}
\end{bmatrix}.
\tag{45}
$$

*K*^<sup>1</sup> represents a transition that is not of direct interest. These Kraus operators *K*^ 0,*K*^<sup>1</sup> are the multidimensional analogs of the quantum amplitude damping channel [21]: with *K*^ <sup>0</sup> corresponding to the dissipation processes, while *K*^<sup>1</sup> corresponds to an unwanted quantum transition.

The block structure of the final *unitary* dilation evolution operator U^ *diss*, corresponding to the non-unitary dissipation operator *e*�*δtH*^ <sup>1</sup> , consists of column blocks of the Kraus operators ð Þ *<sup>K</sup>*<sup>0</sup> *<sup>K</sup>*<sup>1</sup> … *<sup>T</sup>*, and the remaining column blocks are of those matrices required to make U^ *diss* unitary [21]:

$$
\hat{\mathcal{U}}\_{\text{dis}} = \begin{bmatrix}
\hat{\Gamma} & \mathbf{0} & \mathbf{0} & -\sqrt{I\_{r \times r} - \hat{\Gamma}^2} \\
\mathbf{0} & I\_{(d-r) \times (d-r)} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & I\_{(d-r) \times (d-r)} & \mathbf{0} \\
\sqrt{I\_{r \times r} - \hat{\Gamma}^2} & \mathbf{0} & \mathbf{0} & \hat{\Gamma}
\end{bmatrix}.\tag{46}
$$

Thus, it can be shown that the evolution of the system ∣*ψ*0i and environment ∣0i is given by

$$|0\rangle|\psi\_0\rangle = \frac{1}{\sqrt{E\_0}}\sum\_q \psi\_{0q}|0\rangle|q\rangle \to |0\rangle \otimes e^{-i\hbar \hat{H}\_0} \hat{U}\_1 \hat{K}\_0 \hat{U}\_1^\dagger |\psi\_0\rangle + |1\rangle \otimes \hat{K}\_1 |\psi\_0\rangle,\tag{47}$$

where measurement of the first qubit-environment by ∣0ih0∣ ⊗ *Id*�*<sup>d</sup>* yields a state analogous to ∣0i∣*ψ δ*ð Þi *t* . Finally, on taking the trace over the environment will yield the desired system state ∣*ψ δ*ð Þi *t* . The corresponding quantum circuit for Eq. (47) is shown in **Figure 14**.

It is important to highlight that the implementation of the dissipative case is directly overlapping with the QLA framework. The QLA can be used to implement the exp �*iδtH*^ <sup>0</sup> � � part in Eq. (43) as proposed in the previous sections. Specifically, for the lossy medium, the exponential operator of the *H*^ <sup>0</sup> term in Eq. (35) can be easily handled with QLA [12, 13].

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

**Figure 14.** *Quantum circuit diagram for Eq. (47).*

#### **6. Conclusions**

The Schrodinger-Dirac equations are the backbone of the work presented here on Maxwell equations both in lossless inhomogeneous and lossy dielectric media. In both cases, straightforward application of unitary algorithms fail, in the first case, somewhat surprisingly one finds that even though a Dyson map points to the required electromagnetic field variables in a tensor dielectric, its implementation has till now defied a fully unitary representation. Our current QLA approach requires some external non-unitary operators that recover the terms involving the spatial derivatives on the refractive indices of the medium. These sparse matrices can be modeled by the sum of a linear combination of unitaries (LCU), which can then be encoded onto a quantum computer [23, 24]. In the second case, handling dissipative systems immediately forces us to consider an open quantum system interacting with its environment. Typically, this forces us into a density matrix formulation and a clever introduction of what is known as Kraus operators [21, 25]. The beauty of the Kraus representation is that even though the system of interest is interacting with the environment, the Kraus operators do not need detailed information on the environment.

We presented detailed 2D scattering of a 1D electromagnetic pulse off localized dielectric objects. QLA is an initial value scheme. No internal boundary conditions are imposed at the vacuum-dielectrix interface. For dielectrics will large spatial gradients in the refractive index, QLA simulations show strong internal reflection/transmission within the dielectric object. These lead to quite complex time evolution of wavefronts from the dielectric objects. On the other hand, for weak spatial gradients in the refractive index, there are negligible reflections from the vacuum-dielectric interface. This is reminiscent of WKB-like effects in the ray tracing approximation.

In considering the dissipative counterpart, one must now include both the system and its environment in order to get a closed system with unitary representation. The Kraus operators are the most general scheme that will retain the properties of the density matrix in time. The probability of obtaining the desired non-unitary evolution of the open system after the measurement operator *<sup>P</sup>*^<sup>0</sup> <sup>¼</sup> <sup>∣</sup>0ih0∣ ⊗ *Id*�*<sup>d</sup>* is

$$p(\mathbf{0}) = \sum\_{i=1}^{r} e^{-2\gamma \mathbf{\hat{x}}t} \left| \boldsymbol{\mu}\_{i0} \right|^2 + \sum\_{i=r+1}^{d} \left| \boldsymbol{\mu}\_{i0} \right|^2 \ge \mathbf{1} + \left( e^{-2\gamma\_{\text{max}} \mathbf{\hat{x}}} - 1 \right) \sum\_{i=1}^{r} e^{-2\gamma\_i \mathbf{\hat{x}}t} \left| \boldsymbol{\mu}\_{i0} \right|^2 \tag{48}$$

The form of the unitary operator U^ *diss*, Eq. (42) implies that it can be decomposed into r two-level unitary rotations *<sup>R</sup>*^*y*ð Þ *<sup>θ</sup><sup>i</sup>* with cosð Þ¼ *<sup>θ</sup>i=*<sup>2</sup> *<sup>e</sup>*�*γiδ<sup>t</sup> :* Then, the quantum circuit implementation of U^ *diss* requires *O rlog*<sup>2</sup> <sup>2</sup>*<sup>d</sup>* � � CNOT - and *<sup>R</sup>*^*y*ð Þ *<sup>θ</sup><sup>i</sup>* quantum gates, so that there is an improvement in the circuit depth of *O r=d*<sup>2</sup> � �. Our multidimensional amplitude damping channel approach is directly related to the Sz. Nagy dilation by a rotation. The Sz. Nagy dilation [26] is the minimal unitary dilation containing the original dissipative (non-unitary) system.

#### **Acknowledgements**

This research was partially supported by Department of Energy grants DE-SC0021647, DE-FG02-91ER-54109, DE-SC0021651, DE-SC0021857, and DE-SC0021653. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 - EUROfusion). Views and opinions expressed, however, are those of the authors only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. E. K. is supported by the Basic Research Program, NTUA, PEVE. K.H is supported by the National Program for Controlled Thermonuclear Fusion, Hellenic Republic. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award FES-ERCAP0020430.

#### **Author details**

George Vahala<sup>1</sup> \*†, Min Soe2†, Efstratios Koukoutsis3†, Kyriakos Hizanidis3†, Linda Vahala4† and Abhay K. Ram5†

1 William & Mary, Williambsurg, USA

2 Rogers State University, Claremore, USA

3 National Technical University of Athens, Zographou, Greece

4 Old Dominion University, Norfolk, USA

5 Massachusetts Institute of Technology, Cambridge, USA

\*Address all correspondence to: gvahala@gmail.com

† These authors contributed equally.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Qubit Lattice Algorithms Based on the Schrödinger-Dirac Representation of Maxwell… DOI: http://dx.doi.org/10.5772/intechopen.112692*

#### **References**

[1] Boghosian BM, Taylor W IV. Quantum lattice gas models for the many-body Schrodinger equation. International Journal of Modern Physics. 1997;**C8**:705-716

[2] Boghosian BM, TaylorW IV. Simulating quantum mechanics on a quantum computer. Physica D. 1998;**120**:30-42

[3] Yepez J, Boghosian BM. An efficient and accurate quantum lattice-gas model for the many-body Schroedinger wave equation. Computer Physics Communications. 2002;**146**:280-294

[4] Vahala G, Yepez J, Vahala L. Quantum lattice gas representation of some classical solitons. Physics Letters A. 2003;**310**:187-196

[5] Vahala G, Vahala L, Yepez J. Inelastic vector soliton collisions: A lattice-based quantum representation. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Royal Society. 2004;**362**:1677-1690

[6] Yepez J, Vahala G, Vahala L, Soe M. Superfluid turbulence from quantum kelvin waves to classical Kolmogorov cascades. Physical Review Letters. 2009; **103**:084501

[7] Vahala G, Yepez J, Vahala L, Soe M, Zhang B, Ziegeler S. Poincare recurrence and spectral cascades in threedimensional quantum turbulence. Physical Review E. 2011;**84**:046713

[8] Zhang B, Vahala G, Vahala L, Soe M. Unitary quantum lattice algorithm for two dimensional quantum turbulence. Physical Review E. 2011;**84**:046701

[9] Vahala G, Zhang B, Yepez J, Vahala L, Soe M. Chpt 11., unitary qubit lattice gas representation of 2D and 3D quantum

turbulence. In: Oh HW, editor. Advanced Fluid Dynamics. London, UK: InTech; 2012. pp. 239-272

[10] Vahala L, Soe M, Vahala G, Yepez J. Unitary qubit lattice algorithms for spin-1 Bose Einstein condensates. Radiation Effects and Defects in Solids. 2019;**174**: 46-55

[11] Vahala G, Vahala L, Soe M. Qubit unitary lattice algorithms for spin-2 Bose-Einstein condensates, I - theory and Pade initial conditions. Radiation Effects and Defects in Solids. 2020;**175**:102-112

[12] Vahala G, Soe M, Vahala L. Qubit unitary lattice algorithms for spin-2 Bose-Einstein condensates, II - vortex reconnection simulations and nonabelian vortices. Radiation Effects and Defects in Solids. 2020;**175**:113-119

[13] Palpacelli S, S. Succi "the quantum lattice Boltzmann equation: Recent developments". Computer Physics Communications. 2008;**4**:980-1007

[14] Succi S, Fillion-Gourdeau F, Palpacelli S. Quantum lattice Boltzmann is a quantum walk. EPJ Quantum Technology. 2015;**2**:12

[15] Vahala G, Vahala L, Soe M, Ram AK. Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media. Journal of Plasma Physics. 2020;**86**:905860518

[16] Vahala G, Hawthorne J, Vahala L, Ram AK, Soe M. Quantum lattice representation for the curl equations of Maxwell equations. Radiation Effects and Defects in Solids. 2022;**177**:85

[17] Vahala G, Soe M, Vahala L, Ram A, Koukoutsis E, Hizanidis K. Qubit lattice algorithm simulations of Maxwell's

equations for scattering from anisotropic dielectric objects. e-print arXiv: 2301.13601. 2023

[18] Oganesov A, Vahala G, Vahala L, Soe M. Effect of Fourier transform on the streaming in quantum lattice gas algorithms. Radiation Effects and Defects in Solids. 2018;**173**:169

[19] Khan SA, Jagannathan R. A new matrix representation of the Maxwell equations based on the Riemann-Silberstein-weber vector for a linear inhomogeneous medium. arXiv: 2205.09907. 2022

[20] Koukoutsis E, Hizanidis K, Ram AK, Vahala G. Dyson maps and unitary evolution for Maxwell equations in tensor dielectric media. Physical Review A. 2023;**107**:042215

[21] Nielsen MA, Chuang IL. Quantum Computation and Quantum Information. 10th ed. New York: Cambridge University Press; 2010

[22] Suzuki M. Generalized Trotter's formula and systematic approximants of exponential operators and inner derivatives with applications to manybody probloems. Communications in Mathematical Physics. 1976;**51**:183

[23] Childs AM, Wiebe N. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information and Computation. 2012;**12**:901

[24] Childs AM, Kothari R, Somma RD. Quantum algorithm for Systems of Linear Equations with exponentially improved dependence on precision. SIAM Journal on Computing. 2017;**46**: 1920

[25] Stinespring WF. Positive functions on *C*<sup>∗</sup> -algebras. Proceedings of the

American Mathematical Society. 1955; **6**:211

[26] Paulsen V. Completely Bounded Maps and Operator Algebras. New York: Cambridge University Press; 2003

#### **Chapter 8**

## Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential

### *Masaru Hamano and Masahiro Ikeda*

#### **Abstract**

This is a sequel to the paper "Characterization of the ground state to the intercritical NLS with a linear potential by the virial functional" by the same authors. We continue to study the Cauchy problem for a nonlinear Schrödinger equation with a potential. In the previous chapter, we investigated some minimization problems and showed global existence of solutions to the equation with initial data, whose action is less than the value of minimization problems and positive virial functional. In particular, we saw that such solutions are bounded. In this chapter, we deal with solutions to the equation with initial data, whose virial functional is negative contrary to the previous paper and show that such solutions are unbounded.

**Keywords:** nonlinear Schrödinger equation, linear potential, standing wave, blow-up, grow-up, global existence

#### **1. Introduction**

In this chapter, we consider the Cauchy problem of the following nonlinear Schrödinger equation with a linear potential:

$$i\partial\_t u + \Delta\_V u = -|u|^{p-1}u, \quad (t, \boldsymbol{\omega}) \in \mathbb{R} \times \mathbb{R}^d,\tag{1}$$

where *<sup>d</sup>*≥1, 1<sup>&</sup>lt; *<sup>p</sup>*<<sup>2</sup> <sup>∗</sup> � 1,

$$\mathcal{Z}^\* \coloneqq \begin{cases} \infty & \text{if } d \in \{1, 2\}, \\ \frac{2d}{d-2} & \text{if } d \ge 3, \end{cases} \tag{2}$$

and <sup>Δ</sup>*<sup>V</sup>* <sup>≔</sup> <sup>Δ</sup> � *<sup>V</sup>* <sup>¼</sup> <sup>P</sup>*<sup>d</sup> j*¼1 *∂*2 *∂x*<sup>2</sup> *j* � *V*. In particular, we consider the Cauchy problem of Eq. (1) with initial condition

$$
u(\mathbf{0}, \cdot) = \boldsymbol{u}\_0 \in H^1(\mathbb{R}^d). \tag{3}$$

Eq. (1) with *V* ∈ *L*<sup>∞</sup> *<sup>d</sup>* � � is a model proposed to describe the local dynamics at a nucleation site (see [1]).

Eq. (1) is locally well-posed in the energy space *H*<sup>1</sup> *<sup>d</sup>* � � under some assumptions, where Eq. (1) is called local well-posedness in *H*<sup>1</sup> *<sup>d</sup>* � � if Eq. (1) satisfies all of the following conditions:


$$\lim\_{t \uparrow T\_{\text{max}}} \|u(t)\|\_{H^1\_x} = \infty \left( \text{resp.} \lim\_{t \downarrow T\_{\text{min}}} \|u(t)\|\_{H^1\_x} = \infty \right). \tag{4}$$

• The solution depends on continuously on the initial condition. That is, if *u*0,*<sup>n</sup>* ! *<sup>u</sup>*<sup>0</sup> in *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � �, then for any closed interval *<sup>I</sup>* <sup>⊂</sup> ð Þ *<sup>T</sup>*min, *<sup>T</sup>*max , there exists *<sup>n</sup>*<sup>0</sup> <sup>∈</sup> such that for any *n* ≥*n*0, the solution *un* to Eq. (1) with *un*ð Þ¼ 0, *x u*0,*<sup>n</sup>*ð Þ *x* is defined on *Ct <sup>I</sup>*; *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � � � � and satisfies *un* ! *<sup>u</sup>* in *Ct <sup>I</sup>*; *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � � � � as *<sup>n</sup>* ! <sup>∞</sup>, where *<sup>u</sup>* is the solution to Eq. (1) with *u*ð Þ¼ 0, *x u*0ð Þ *x* .

To state a local well-posedness result, we define the space

$$\mathcal{K}\_0\left(\mathbb{R}^d\right) \coloneqq \overline{\left\{ f \in L^\infty\left(\mathbb{R}^d\right) : \text{supp} f \text{ is compact} \right\}}^{\|\cdot\|\_{\mathcal{K}}},\tag{5}$$

where

$$\|f\|\_{\mathcal{K}} := \sup\_{\boldsymbol{\chi} \in \mathbb{R}^d} \int\_{\mathbb{R}^d} \frac{|f(\boldsymbol{\chi})|}{|\boldsymbol{\chi} - \boldsymbol{\chi}|^{d-2}} d\boldsymbol{\chi}.\tag{6}$$

We note that

$$L^{\not\preceq -\varepsilon} \left( \mathbb{R}^d \right) \cap L^{\not\preceq +\varepsilon} \left( \mathbb{R}^d \right) \hookrightarrow L^{\not\preceq 1} \left( \mathbb{R}^d \right) \hookrightarrow \mathcal{K} \left( \mathbb{R}^d \right) \coloneqq \{ f : \| f' \|\_{\mathcal{K}} < \infty \} \tag{7}$$

for some *ε*>0, where the space *L<sup>p</sup>*,*<sup>q</sup> <sup>d</sup>* � � denotes the usual Lorentz space.

Theorem 1 (Local well-posedness, [2–4]) Let *<sup>d</sup>*≥1 and 1<*<sup>p</sup>* <sup>&</sup>lt;<sup>2</sup> <sup>∗</sup> � 1. If *<sup>V</sup>* satisfies one of the following, then Eq. (1) is locally well-posed in *H*<sup>1</sup> *<sup>d</sup>* � �.


Moreover, the solution *u* to Eq. (1) conserves its mass and energy with respect to time *t*, where they are defined as

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

$$\begin{aligned} \text{(Mass)} \quad &M[u(t)] := \|u(t)\|\_{L^2\_x}^2, \\ \text{(Energy)} \quad &E\_V[u(t)] := \frac{1}{2} \|u(t)\|\_{\dot{H}^1\_x}^2 + \frac{1}{2} \int\_{\mathbb{R}^d} V(\mathbf{x}) |u(t, \mathbf{x})|^2 - \frac{1}{p+1} \|u(t)\|\_{L^{p+1}\_x}^{p+1}. \end{aligned} \tag{8}$$

We turn to time behaviors of the solution to Eq. (1). A solution to Eq. (1) has various kinds of time behaviors by the choice of initial data. For example, we can consider the following time behaviors.

• (Scattering) We say that the solution *u* to Eq. (1) scatters in positive time (resp. negative time) if *<sup>T</sup>*max <sup>¼</sup> <sup>∞</sup> (resp. *<sup>T</sup>*min ¼ �∞) and there exists *<sup>ψ</sup>*<sup>þ</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � � (resp. *<sup>ψ</sup>*� <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � �) such that

$$\lim\_{t \to +\infty} \|u(t) - e^{it\Delta v} \varphi\_+\|\_{H^1\_x} = 0 \quad \left(\text{resp.} \lim\_{t \to -\infty} \|u(t) - e^{it\Delta v} \varphi\_-\|\_{H^1\_x} = 0\right), \tag{9}$$

where *eit*Δ*<sup>V</sup> f* is a solution to the corresponding linear equation with Eq. (1)

$$i\partial\_t u(t, \mathbf{x}) + \Delta\_V u(t, \mathbf{x}) = \mathbf{0}, \quad u(\mathbf{0}, \mathbf{x}) = f(\mathbf{x}). \tag{10}$$

We say that *u* scatters when *u* scatters in positive and negative time.


$$\limsup\_{t \to \infty} \|u(t)\|\_{H^1\_x} = \infty, \quad \left(\text{resp.} \lim\_{t \to -\infty} \sup \|u(t)\|\_{H^1\_x} = \infty\right). \tag{11}$$

We say that *u* grows up when *u* grows up in positive and negative time.

• (Standing wave) We say that the solution *u* to Eq. (1) is a standing wave if *u* ¼ *e<sup>i</sup>ω<sup>t</sup> Q<sup>ω</sup>*,*<sup>V</sup>* for some *ω*∈ , where *Q<sup>ω</sup>*,*<sup>V</sup>* satisfies the elliptic equation

$$-a\mathbf{Q}\_{a\boldsymbol{o},V} + \Delta\_V \mathbf{Q}\_{a\boldsymbol{o},V} = -\left|\mathbf{Q}\_{a\boldsymbol{o},V}\right|^{p-1} \mathbf{Q}\_{a\boldsymbol{o},V}.\tag{12}$$

In particular, *Q<sup>ω</sup>*,*<sup>V</sup>* is ground state to Eq. (12) if

$$\mathcal{Q}\_{\boldsymbol{w},V} \in \left\{ \phi \in \mathcal{A}\_{\boldsymbol{w},V} : \mathbb{S}\_{\boldsymbol{w},V}(\phi) \le \mathbb{S}\_{\boldsymbol{w},V}(\boldsymbol{\varphi}) \text{ for any } \boldsymbol{\varphi} \in \mathcal{A}\_{\boldsymbol{w},V} \right\} =: \mathcal{G}\_{\boldsymbol{w},V},\tag{13}$$

where *<sup>S</sup><sup>ω</sup>*,*<sup>V</sup>*ð Þ*<sup>f</sup>* <sup>≔</sup> *<sup>ω</sup>* <sup>2</sup> *M f*½ �þ *EV*½ � *f* (and)

$$\mathcal{A}\_{\boldsymbol{\alpha},V} \coloneqq \left\{ \boldsymbol{\psi} \in H^1\left(\mathbb{R}^d\right) \backslash \{\mathbf{0}\} : \mathcal{S}'\_{\boldsymbol{\alpha},V}(\boldsymbol{\psi}) = \mathbf{0} \right\}.\tag{14}$$

We know the following results (Theorems 2 and 3) for time behaviors of the solutions to Eq. (1). For related results, we also list [5–38].

Theorem 2 (Hong, [3]) Let *<sup>d</sup>* <sup>¼</sup> *<sup>p</sup>* <sup>¼</sup> 3, *<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>3</sup> , and *<sup>Q</sup>*1,0 <sup>∈</sup>G1,0. Suppose that *<sup>V</sup>* satisfies *V* ∈ *L* 3 <sup>2</sup> <sup>3</sup> <sup>∩</sup> <sup>K</sup><sup>0</sup> <sup>3</sup> , *<sup>V</sup>* <sup>≥</sup>0, *<sup>x</sup>* � <sup>∇</sup>*<sup>V</sup>* <sup>∈</sup>*<sup>L</sup>* 3 <sup>2</sup> <sup>3</sup> , and *<sup>x</sup>* � <sup>∇</sup>*<sup>V</sup>* <sup>≤</sup> 0. We also assume that

$$\mathcal{M}[\boldsymbol{\mu}\_{0}]E\_{V}[\boldsymbol{\mu}\_{0}] < \mathcal{M}\left[\boldsymbol{Q}\_{1,0}\right]E\_{0}\left[\boldsymbol{Q}\_{1,0}\right] \text{ and} \\ \|\boldsymbol{\mu}\_{0}\|\_{L^{2}}\|\boldsymbol{\mu}\_{0}\|\_{\dot{H}^{1}\_{V}} < \|\boldsymbol{Q}\_{1,0}\|\_{L^{2}}\|\boldsymbol{Q}\_{1,0}\|\_{\dot{H}^{1}}.\tag{15}$$

Then, the solution *u* to Eq. (1) with Eq. (3) scatters.

Theorem 3 (Hamano–Ikeda, [4]) Let *<sup>d</sup>* <sup>¼</sup> 3, <sup>7</sup> <sup>3</sup> <sup>&</sup>lt; *<sup>p</sup>*<5, *<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>3</sup> , and *<sup>Q</sup>*1,0 <sup>∈</sup>G1,0. Suppose that *V* satisfies *V* ≥0 and *x* � ∇*V* ∈*L* 3 <sup>2</sup> <sup>3</sup> . We also assume that

$$\mathcal{M}[\mathfrak{u}\_{0}]^{\frac{1-\iota\_{\rm c}}{\iota\_{\rm c}}}E\_{V}[\mathfrak{u}\_{0}] < \mathcal{M}[Q\_{1,0}]^{\frac{1-\iota\_{\rm c}}{\iota\_{\rm c}}}E\_{0}[Q\_{1,0}],\tag{16}$$

where *sc* ≔ *<sup>d</sup>* <sup>2</sup> � <sup>2</sup> *p*�1 .

1.(Scattering)

$$\text{If } V \in L^{\frac{4}{\epsilon}}(\mathbb{R}^3) \cap \mathcal{K}\_0\left(\mathbb{R}^3\right), \pi \cdot \nabla V \le 0, \text{ and}$$

$$\|u\_0\|\_{L^2}^{\frac{1-\epsilon}{\epsilon}} \|u\_0\|\_{\dot{H}^1} < \|Q\_{1,0}\|\_{L^2}^{\frac{1-\epsilon}{\epsilon}} \|Q\_{1,0}\|\_{\dot{H}^1}, \tag{17}$$

then ð Þ¼ *T*min, *T*max , that is, exists globally in time. Moreover, if *u*<sup>0</sup> and *V* are radially symmetric, then *u* scatters.

2.(Blow-up or grow-up)

If "*V* ∈*L* 3 <sup>2</sup> <sup>3</sup> ∩ K<sup>0</sup> <sup>3</sup> or *V* ∈ *L<sup>σ</sup>* <sup>3</sup> for some <sup>3</sup> <sup>2</sup> <*σ* ≤ ∞," 2*V* þ *x* � ∇*V* ≥ 0, and

$$\|\|u\_0\|\|\_{L^2}^{\frac{1-\iota\_\ell}{\iota\_\ell}} \|u\_0\|\_{\dot{H}^1\_V} > \|Q\_{1,0}\|\_{L^2}^{\frac{1-\iota\_\ell}{\iota\_\ell}} \|Q\_{1,0}\|\_{\dot{H}^1},\tag{18}$$

then *u* blows up or grows up. Furthermore, if one of the following holds:


then *u* blows up.

Remark 1 Mizutani [39] proved that for any *ψ* ∈ *H*<sup>1</sup> , there exists *<sup>ϕ</sup>*� <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>3</sup> such that

$$\lim\_{t \to \pm \infty} \|e^{it\Delta\_V}\varphi - e^{it\Delta}\phi\_{\pm}\|\_{H^1\_x} = \mathbf{0} \tag{19}$$

under the assumptions *V* ∈*L* 3 <sup>2</sup> <sup>3</sup> and *V* ≥0, where the double-sign corresponds. Combining this limit and scattering part in Theorem 3 (or Theorem 2), we can see that the nonlinear solution *<sup>u</sup>* to Eq. (1) approaches to a free solution *<sup>e</sup>it*<sup>Δ</sup>*ϕ*� as *<sup>t</sup>* ! �<sup>∞</sup> for some *<sup>ϕ</sup>*� <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>3</sup> .

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

We realize that there is no potential, which satisfies scattering and blow-up or grow-up parts in Theorem 1 at the same time. Indeed, if *V* satisfies *x* � ∇*V* ≤0 and 2*V* þ *x* � ∇*V* ≥ 0, then *V* ∉ *L* 3 <sup>2</sup> <sup>3</sup> � �. Then, we consider a minimization problem

$$\mathfrak{m}\_{a\circ,V} \coloneqq \inf \left\{ \mathbb{S}\_{a\circ,V}(f) : f \in H^1(\mathbb{R}^d) \backslash \{ \mathbf{0} \}, \ K\_V(f) = \mathbf{0} \right\} \tag{20}$$

to get a potential *V*, which deduces scattering and blow-up or grow-up at the same time. It proved in [40] that the condition Eq. (16) can be rewritten as the following by using *n<sup>ω</sup>*,*V*.

Proposition 1 Let *<sup>d</sup>*≥3, 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt; *<sup>p</sup>*<<sup>2</sup> <sup>∗</sup> � 1, *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � �, and *<sup>Q</sup>*1,0 <sup>∈</sup>G1,0. Assume that *V* satisfies (A2) with ∣a∣ ≤1 and (A6) below. Then, the following two conditions are equivalent.

$$\mathbf{1}. M[f]^{\frac{1-\iota\_c}{\kappa}} E\_V[f] < M\left[Q\_{\mathbf{1},0}\right]^{\frac{1-\iota\_c}{\kappa}} E\_0\left[Q\_{\mathbf{1},0}\right],$$

2.There exists *ω* >0 such that *Sω*,*<sup>V</sup>*ð Þ*f* <*nω*,*<sup>V</sup>*.

Using *nω*,*<sup>V</sup>*, we expect that if *Sω*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> <*nω*,*<sup>V</sup>* and *KV*ð Þ *u*<sup>0</sup> ≥ 0, then the solution *u* scatters and if *Sω*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> <*nω*,*<sup>V</sup>* and *KV*ð Þ *u*<sup>0</sup> < 0, then the solution *u* blows up or grows up, where *KV* is called virial functional and is defined as

$$\begin{split} K\_{V}(f) & \coloneqq \frac{d}{d\lambda}\Big|\_{\lambda=0} S\_{a,V}(e^{d\lambda}f(e^{2\lambda} \cdot)) \\ &= 2\|f\|\_{\dot{H}^{1}}^{2} - \int\_{\mathbb{R}^{d}} (\mathbf{x} \cdot \nabla V) |f(\mathbf{x})|^{2} d\mathbf{x} - \frac{(p-1)d}{p+1} \|f\|\_{\dot{H}^{p+1}}^{p+1} .\end{split} \tag{21}$$

It is well known that *KV*ð Þ *u t*ð Þ denotes variance of the solution and if *xu*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> *<sup>d</sup>* � � then

$$K\_V(u(t)) = \frac{1}{4} \cdot \frac{d^2}{dt^2} \left\| \varkappa u(t) \right\|\_{L^2\_x}^2 \tag{22}$$

for each *t*∈ð Þ *T*min, *T*max . We also consider a minimization problem *r<sup>ω</sup>*,*<sup>V</sup>*, which restricts *n<sup>ω</sup>*,*<sup>V</sup>* to radial functions, that is,

$$r\_{a,V} \coloneqq \inf \left\{ \mathbb{S}\_{o,V}(f) : f \in H^1\_{\text{rad}}(\mathbb{R}^d) \backslash \{0\}, \ K\_V(f) = \mathbf{0} \right\} \tag{23}$$

and expect for radial initial data *u*<sup>0</sup> and radial potential *V* that if *S<sup>ω</sup>*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> < *r<sup>ω</sup>*,*<sup>V</sup>* and *KV*ð Þ *u*<sup>0</sup> ≥0, then the solution *u* scatters and if *S<sup>ω</sup>*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> <*r<sup>ω</sup>*,*<sup>V</sup>* and *KV*ð Þ *u*<sup>0</sup> <0, then the solution *u* blows up. For more general minimization problems

$$\begin{aligned} m\_{a\circ,V}^{a,\theta} &:= \inf \left\{ \mathbb{S}\_{a\circ,V}(f) : f \in H^1(\mathbb{R}^d) \backslash \{ \mathbf{0} \}, \ K\_{a\circ,V}^{a,\theta}(f) = \mathbf{0} \right\}, \\\ r\_{a\circ,V}^{a,\theta} &:= \inf \left\{ \mathbb{S}\_{a\circ,V}(f) : f \in H\_{\mathrm{rad}}^1(\mathbb{R}^d) \backslash \{ \mathbf{0} \}, \ K\_{a\circ,V}^{a,\theta}(f) = \mathbf{0} \right\} \end{aligned} \tag{24}$$

with

$$a > 0, \quad \beta \ge 0, \quad 2a - d\beta \ge 0,\tag{25}$$

the authors showed in [40, 41] the following results (Theorems 4 and 5) Eq. (27), where the functional *K<sup>α</sup>*,*<sup>β</sup> <sup>ω</sup>*,*<sup>V</sup>* is given as

$$K\_{o\circ,V}^{a,\beta}(f) \coloneqq \frac{d}{d\lambda}\Big|\_{\lambda=0} \mathcal{S}\_{o\circ,V}\Big(e^{a\varlambda}f\left(e^{\beta\varlambda}\cdot\right)\Big).\tag{26}$$

Here, we realize *<sup>n</sup><sup>ω</sup>*,*<sup>V</sup>* <sup>¼</sup> *<sup>n</sup>d*,2 *<sup>ω</sup>*,*V*, *r<sup>ω</sup>*,*<sup>V</sup>* ¼ *r d*,2 *<sup>ω</sup>*,*V*, and *KV* <sup>¼</sup> *<sup>K</sup>d*,2 *<sup>ω</sup>*,*V*. To state the results, we give the assumptions of the potential *<sup>V</sup>*: Let <sup>a</sup>∈ð Þ ∪f g<sup>0</sup> *<sup>d</sup>* .

A1. *V* ∈*L* 3 <sup>2</sup> <sup>3</sup> � � ∩ K<sup>0</sup> <sup>3</sup> � � A2. *x*<sup>a</sup>*∂*<sup>a</sup>*V* ∈*L d* <sup>2</sup> *<sup>d</sup>* � � <sup>þ</sup> *<sup>L</sup><sup>σ</sup> <sup>d</sup>* � � for some *<sup>d</sup>* <sup>2</sup> ≤ *σ* < ∞ A3. *x*<sup>a</sup>*∂*<sup>a</sup>*V* ∈*L d* <sup>2</sup> *<sup>d</sup>* � � <sup>þ</sup> *<sup>L</sup>*<sup>∞</sup> *<sup>d</sup>* � � A4. *<sup>x</sup>*<sup>a</sup>*∂*<sup>a</sup>*<sup>V</sup>* <sup>∈</sup>*L<sup>η</sup> <sup>d</sup>* � � <sup>þ</sup> *<sup>L</sup><sup>σ</sup> <sup>d</sup>* � � for some *<sup>d</sup>* <sup>2</sup> < *η*≤*σ* < ∞ A5. *<sup>x</sup>*<sup>a</sup>*∂*<sup>a</sup>*<sup>V</sup>* <sup>∈</sup>*L<sup>η</sup> <sup>d</sup>* � � <sup>þ</sup> *<sup>L</sup>*<sup>∞</sup> *<sup>d</sup>* � � for some *<sup>d</sup>* <sup>2</sup> <*η*< ∞ A6. *V* ≥0, *x* � ∇*V* ≤ 0, 2*V* þ *x* � ∇*V* ≥0 A7. *V* ≥0, *x* � ∇*V* ≤ 0, *ω*≥*ω*<sup>0</sup> for *<sup>ω</sup>*<sup>0</sup> <sup>≔</sup> � <sup>1</sup> ess inf *<sup>x</sup>*<sup>∈</sup> *<sup>d</sup>* ð Þ 2*V* þ *x* � ∇*V :* (27)

We note that the third inequality implies 2*<sup>V</sup>* <sup>þ</sup> *<sup>x</sup>* � <sup>∇</sup>*<sup>V</sup>* <sup>þ</sup> <sup>2</sup>*<sup>ω</sup>* <sup>≥</sup>0 a.e. *<sup>x</sup>*<sup>∈</sup> *<sup>d</sup>*. Theorem 4 Let *<sup>d</sup>*≥3 and 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt;*p*<sup>&</sup>lt; <sup>2</sup> <sup>∗</sup> � 1.

2


$$\begin{split} \mathcal{M}\_{\boldsymbol{\alpha},\boldsymbol{V},\operatorname{rad}}^{\boldsymbol{a},\boldsymbol{\beta}} &:= \left\{ \boldsymbol{\phi} \in H\_{\operatorname{rad}}^{1} \left( \mathbb{R}^{d} \right) : \mathbb{S}\_{\boldsymbol{\alpha},\boldsymbol{V}}(\boldsymbol{\phi}) = r\_{\boldsymbol{\alpha},\boldsymbol{V}}^{\boldsymbol{a},\boldsymbol{\beta}}, \ K\_{\boldsymbol{\alpha},\boldsymbol{V}}^{\boldsymbol{a},\boldsymbol{\beta}}(\boldsymbol{\phi}) = \mathbf{0} \right\}, \\ \mathcal{G}\_{\boldsymbol{\alpha},\boldsymbol{V},\operatorname{rad}} &:= \left\{ \boldsymbol{\phi} \in \mathcal{A}\_{\boldsymbol{\alpha},\boldsymbol{V},\operatorname{rad}} : \mathcal{S}\_{\boldsymbol{\alpha},\boldsymbol{V}}(\boldsymbol{\phi}) \leq \mathcal{S}\_{\boldsymbol{\alpha},\boldsymbol{V}}(\boldsymbol{\varphi}) \text{ for any } \boldsymbol{\varphi} \in \mathcal{A}\_{\boldsymbol{\alpha},\boldsymbol{V},\operatorname{rad}} \right\}, \\ \mathcal{A}\_{\boldsymbol{\alpha},\boldsymbol{V},\operatorname{rad}} &:= \left\{ \boldsymbol{\varphi} \in H\_{\operatorname{rad}}^{1} \left( \mathbb{R}^{d} \right) \backslash \{\mathbf{0} \} : \mathcal{S}\_{\boldsymbol{\alpha},\boldsymbol{V}}^{\boldsymbol{\prime}}(\boldsymbol{\varphi}) = \mathbf{0} \right\}. \end{split} \tag{28}$$

The inequality *n<sup>α</sup>*,*<sup>β</sup> <sup>ω</sup>*,*<sup>V</sup>* ≤*r α*,*β <sup>ω</sup>*,*<sup>V</sup>* holds by their definitions and the attainability of *<sup>n</sup><sup>α</sup>*,*<sup>β</sup> ω*,*V* and *r α*,*β <sup>ω</sup>*,*<sup>V</sup>* deduces the following corollary.

Corollary 1 Under the all assumptions of (Non-radial case) in Theorem 4, we have

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

$$n\_{\alpha,V}^{a,\beta} < r\_{\alpha,V}^{a,\beta}.\tag{29}$$

Remark 2 In the case of *<sup>V</sup>* <sup>¼</sup> 0, it is well known that *<sup>n</sup><sup>α</sup>*,*<sup>β</sup> <sup>ω</sup>*,0 and *r α*,*β <sup>ω</sup>*,0 are attained by *<sup>Q</sup><sup>ω</sup>*,0 <sup>∈</sup>G*ω*,0. That is, *<sup>n</sup><sup>α</sup>*,*<sup>β</sup> <sup>ω</sup>*,0 ¼ *r α*,*β <sup>ω</sup>*,0 <sup>¼</sup> *<sup>S</sup><sup>ω</sup>*,0 *<sup>Q</sup><sup>ω</sup>*,0 holds.

Then, we investigate global existence of a solution to time-dependent Eq. (1). Theorem 5 (Global well-posedness in *H*<sup>1</sup> ) Let *<sup>d</sup>*≥3 and 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt; *<sup>p</sup>*<<sup>2</sup> <sup>∗</sup> � 1.

• (Non-radial case) Let *u*<sup>0</sup> ∈ *H*<sup>1</sup> *<sup>d</sup>* and *Q<sup>ω</sup>*,0 ∈G*ω*,0. Suppose that *V* satisfies "(A1) or (A4) with ∣a∣ ¼ 0," (A2) with ∣a∣ ¼ 1, and (A6). We also assume that there exist ð Þ *α*, *β* satisfying Eq. (25) and *ω* >0 such that

$$\mathcal{S}\_{a,V}(u\_0) < \mathcal{S}\_{w,0}\left(Q\_{w,0}\right) \ \left(= n\_{w,V}^{a,\beta}\right), \quad K\_{a,V}^{a,\beta}(u\_0) \ge 0. \tag{30}$$

Then, the solution *u* to Eq. (1) with Eq. (3) exists globally in time. In particular, it follows that

$$\sup\_{t \in \mathbb{R}} \|u(t)\|\_{H^1\_x} < \infty. \tag{31}$$

• (Radial case) Let *u*<sup>0</sup> ∈ *H*<sup>1</sup> rad *<sup>d</sup>* and *<sup>Q</sup>ω*,*<sup>V</sup>* <sup>∈</sup>G*ω*,*<sup>V</sup>*,rad. Suppose that *<sup>V</sup>* is radially symmetric and satisfies "(A1) or (A5) with ∣a∣ ¼ 0," (A3) with ∣a∣ ¼ 1, 2, (A7), and 3*<sup>x</sup>* � <sup>∇</sup>*<sup>V</sup>* <sup>þ</sup> *<sup>x</sup>*∇<sup>2</sup> *Vx<sup>T</sup>* <sup>≤</sup>0. If there exist ð Þ *<sup>α</sup>*, *<sup>β</sup>* with Eq. (25) and *<sup>ω</sup>*>0 satisfying *ω* ≥*ω*<sup>0</sup> such that

$$\mathcal{S}\_{a,V}(u\_0) < \mathcal{S}\_{a,V}\left(Q\_{a\nu,V}\right) \ \left(= r\_{a\nu,V}^{a,\beta}\right), \quad K\_{a,V}^{a,\beta}(u\_0) \ge 0,\tag{32}$$

then the solution *u* to Eq. (1) with Eq. (3) exists globally in time.

#### **1.1 Main theorem**

In the previous paper, the authors handled the solution *u* to Eq. (1) with initial data *u*<sup>0</sup> satisfying *S<sup>ω</sup>*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> < *m<sup>ω</sup>*,*<sup>V</sup>* and *KV*ð Þ *u*<sup>0</sup> ≥0, where *m<sup>ω</sup>*,*<sup>V</sup>* denotes *n<sup>ω</sup>*,*<sup>V</sup>* or *r<sup>ω</sup>*,*<sup>V</sup>*. We note that *m<sup>ω</sup>*,*<sup>V</sup>* is *m<sup>α</sup>*,*<sup>β</sup> <sup>ω</sup>*,*<sup>V</sup>* with ð Þ¼ *<sup>α</sup>*, *<sup>β</sup>* ð Þ *<sup>d</sup>*, 2 and *<sup>m</sup><sup>α</sup>*,*<sup>β</sup> <sup>ω</sup>*,*<sup>V</sup>* is independent of ð Þ *α*, *β* . In this chapter, we are interested in the solutions to Eq. (1) with initial data satisfying *S<sup>ω</sup>*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> < *m<sup>ω</sup>*,*<sup>V</sup>* and *KV*ð Þ *u*<sup>0</sup> <0. Our main theorem is the following:

Theorem 6 Let *<sup>d</sup>*≥3 and 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt;*<sup>p</sup>* <sup>&</sup>lt;<sup>1</sup> <sup>þ</sup> <sup>4</sup> *d*�2 .

• (Non-radial case) Let *u*<sup>0</sup> ∈ *H*<sup>1</sup> *<sup>d</sup>* and *Q<sup>ω</sup>*,0 ∈G*ω*,0. Suppose that *V* satisfy "(A1) or (A4) with ∣a∣ ¼ 0," (A2) with ∣a∣ ¼ 1, and (A6). We also assume that there exists *ω*>0 such that

$$\mathcal{S}\_{a,V}(\mathfrak{u}\_0) < \mathcal{S}\_{a,0}\left(Q\_{a,0}\right) \ (=\mathfrak{n}\_{a,V}), \quad K\_V(\mathfrak{u}\_0) < 0. \tag{33}$$

Then, the solution *u* to Eq. (1) with Eq. (3) blows up or grows up. Moreover, *u* blows up under the additional assumption *xu*<sup>0</sup> ∈*L*<sup>2</sup> *<sup>d</sup>* .

• (Radial case) Let *u*<sup>0</sup> ∈ *H*<sup>1</sup> rad *<sup>d</sup>* and *<sup>Q</sup><sup>ω</sup>*,*<sup>V</sup>* <sup>∈</sup>G*<sup>ω</sup>*,*<sup>V</sup>*,rad. Suppose that *<sup>V</sup>* is radially symmetric and satisfies "(A1) or (A5) with ∣a∣ ¼ 0," (A3) with ∣a∣ ¼ 1, 2, (A7), and 3*<sup>x</sup>* � <sup>∇</sup>*<sup>V</sup>* <sup>þ</sup> *<sup>x</sup>*∇<sup>2</sup> *Vx<sup>T</sup>* ≤0. We also assume that there exists *ω* >0 satisfying *ω* ≥*ω*<sup>0</sup> such that

$$\mathcal{S}\_{a,V}(\mathfrak{u}\_0) < \mathcal{S}\_{a,V}\left(\mathcal{Q}\_{a\ast,V}\right) \ (= r\_{a\ast,V}), \quad K\_V(\mathfrak{u}\_0) < 0. \tag{34}$$

Then, the solution *u* to Eq. (1) with Eq. (3) blows up.

Remark 3 Let *V* be a potential in Theorem 6. Combining Theorems 5 and 6, we complete bounded and unbounded dichotomy of

*<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* : *<sup>S</sup>ω*,*<sup>V</sup>*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>S</sup><sup>ω</sup>*,0 *<sup>Q</sup><sup>ω</sup>*,0 and global existence and blow-up dichotomy of *u*<sup>0</sup> ∈ *H*<sup>1</sup> rad *<sup>d</sup>* : *<sup>S</sup>ω*,*<sup>V</sup>*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt;*Sω*,*<sup>V</sup> <sup>Q</sup>ω*,*<sup>V</sup>* by using sign of the virial functional of initial data.

Remark 4 The following potential satisfies all of conditions in Theorem 6:

$$V(\boldsymbol{x}) = \frac{\chi\{\log(\mathbf{1} + |\boldsymbol{x}|)\}^{\theta}}{|\boldsymbol{x}|^{\mu}}, \quad (\boldsymbol{\eta} > \mathbf{0}, \mathbf{0} \le \theta \le \mu < \mathbf{2}, \mu > \mathbf{0}). \tag{35}$$

Theorem 6 with the potential Eq. (35) having *θ* ¼ 0 was considered in the previous paper [19] by the authors. As the other example, we put

$$V(\infty) \coloneqq \frac{\gamma}{\langle \infty \rangle^{\mu}}, \quad (\gamma > 0, 0 < \mu < 2), \tag{36}$$

where h i� is called the Japanese bracket and is defined as 1 þ �j j<sup>2</sup> <sup>1</sup> 2 .

#### **1.2 Organization of the paper**

The organization of the rest of this chapter is as follows. In Section 2, we collect some notations and tools used throughout this chapter. In Section 3, we prove nonradial case in Theorem 6 by using an argument in [13]. In Section 4, we show radial case in Theorem 6 by using an argument in [33].

#### **2. Preliminaries**

In this section, we define some notations and collect some tools, which are used throughout this chapter.

#### **2.1 Notation and definition**

For 1<sup>≤</sup> *<sup>p</sup>*<sup>≤</sup> <sup>∞</sup>, *Lp* <sup>¼</sup> *<sup>L</sup><sup>p</sup> <sup>d</sup>* denotes the usual Lebesgue space. For a Banach space *X*, we use *L<sup>q</sup>* ð Þ *<sup>I</sup>*;*<sup>X</sup>* to denote the Banach space of functions *<sup>f</sup>* : *<sup>I</sup>* � *<sup>d</sup>* ! whose norm is

$$\|f\|\_{L^{\mathfrak{g}}(I;X)} := \|\|f(t)\|\_{X}\|\_{L^{\mathfrak{g}}(I)} < \infty. \tag{37}$$

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

We extend our notation as follows: If a time interval is not specified, then the *t*-norm is evaluated over ð Þ �∞, ∞ . To indicate a restriction to a time subinterval *<sup>I</sup>* <sup>⊂</sup> ð Þ �∞, <sup>∞</sup> , we will write as *Lq* ð Þ*<sup>I</sup>* . *<sup>H</sup><sup>s</sup> <sup>d</sup>* � � and *<sup>H</sup>*\_ *<sup>s</sup> <sup>d</sup>* � � are the usual Sobolev spaces, whose norms <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*H<sup>s</sup>* ≔ ∥ð Þ <sup>1</sup> � <sup>Δ</sup> *<sup>s</sup>* <sup>2</sup>*<sup>f</sup>* <sup>∥</sup>*L*<sup>2</sup> and <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*H*\_ *<sup>s</sup>* ≔ ∥ð Þ �<sup>Δ</sup> *<sup>s</sup>* <sup>2</sup>*f* ∥*L*<sup>2</sup> respectively. We also define the Sobolev spaces *H<sup>s</sup> <sup>V</sup> <sup>d</sup>* � � and *<sup>H</sup>*\_ *<sup>s</sup> <sup>V</sup> <sup>d</sup>* � � with the potential *<sup>V</sup>* via norms ∥*f* ∥*H<sup>s</sup> <sup>V</sup>* ≔ ∥ð Þ 1 � Δ*<sup>V</sup> s* <sup>2</sup>*f* ∥*L*<sup>2</sup> and ∥*f* ∥*H*\_ *<sup>s</sup> V* ≔ ∥ð Þ �Δ*<sup>V</sup> s* <sup>2</sup>*f* ∥*L*<sup>2</sup> respectively.

#### **2.2 Some tools**

Proposition 2 Let *p*≥1. For *f* ∈ *H*<sup>1</sup> rad *<sup>d</sup>* � �, we have

$$\|\|f\|\|\_{L^{p+1}(\mathbb{R}\leq|x|)}^{p+1} \leq \frac{\mathsf{C}}{\mathsf{R}^{\frac{(d-1)(p-1)}{2}}} \|\|f\|\|\_{L^{2}(\mathbb{R}\leq|x|)}^{\frac{p+3}{2}} \|\|f\|\|\_{\dot{H}^{1}(\mathbb{R}\leq|x|)}^{\frac{p-1}{2}} \tag{38}$$

for any *R*>0, where the implicit constant *C* is independent of *R* and *f*. To state the next proposition, we define two functions:

$$\mathcal{X}\_R := R^2 \mathcal{X}\left(\frac{|\mathbf{x}|}{R}\right),\tag{39}$$

where X : ½ Þ! 0, ∞ ½ Þ 0, ∞ (forms)

$$\mathcal{X}(r) \coloneqq \begin{cases} r^2 & (0 \le r \le 1), \\ s smooth & (1 \le r \le 3), \\ 0 & (3 \le r) \end{cases} \tag{40}$$

and satisfies X00ð Þ*r* ≤ 2.

$$\mathcal{Y}\_R(\varkappa) \coloneqq \mathcal{Y}\left(\frac{|\varkappa|}{R}\right),\tag{41}$$

where Y : ½ Þ! 0, ∞ ½ Þ 0, ∞ (forms)

$$\mathcal{Y}(r) \coloneqq \begin{cases} 0 & \left(0 \le r \le \frac{1}{2}\right), \\\\ smooth & \left(\frac{1}{2} \le r \le 1\right), \\\ 1 & (1 \le r) \end{cases} \tag{42}$$

and satisfies 0≤Y<sup>0</sup> ð Þ*r* ≤ 3.

Proposition 3 (Localized virial identity, [3]) Let *w* be X*<sup>R</sup>* or Y*<sup>R</sup>* defined as Eqs. (39) and (41) respectively. For the solution *u* to Eq. (1), we define

$$I\_w(t) \coloneqq \int\_{\mathbb{R}^d} w(\boldsymbol{\varkappa}) |u(t, \boldsymbol{\varkappa})|^2 d\boldsymbol{\varkappa}.\tag{43}$$

Then, we have

$$\begin{split} I\_{w'}(t) &= 2\text{Im} \int\_{\mathbb{R}^d} \frac{\mathbf{x} \cdot \nabla u}{|\mathbf{x}|} \overline{u} w' d\mathbf{x}, \\ I\_{w'}(t) &= \int\_{\mathbb{R}^d} F\_1 |\mathbf{x} \cdot \nabla u|^2 d\mathbf{x} + 4 \int\_{\mathbb{R}^d} \frac{w'}{|\mathbf{x}|} |\nabla u|^2 d\mathbf{x} - \int\_{\mathbb{R}^d} F\_2 |u|^{p+1} d\mathbf{x} \\ &\quad - \int\_{\mathbb{R}^d} F\_3 |u|^2 d\mathbf{x} - 2 \int\_{\mathbb{R}^d} \frac{w'}{|\mathbf{x}|} (\mathbf{x} \cdot \nabla V) |u|^2 d\mathbf{x}. \end{split} \tag{44}$$

where

$$\begin{split} F\_1(w, |\mathbf{x}|) &:= \mathbf{4} \left( \frac{w^{\prime\prime}}{|\mathbf{x}|^2} - \frac{w^{\prime}}{|\mathbf{x}|^3} \right), \qquad F\_2(w, |\mathbf{x}|) := \frac{2(p-1)}{p+1} \left( w^{\prime\prime} + \frac{d-1}{|\mathbf{x}|} w^{\prime} \right), \\ F\_3(w, |\mathbf{x}|) &:= w^{(4)} + \frac{2(d-1)}{|\mathbf{x}|} w^{(3)} + \frac{(d-1)(d-3)}{|\mathbf{x}|^2} w^{\prime\prime} + \frac{(d-1)(3-d)}{|\mathbf{x}|^3} w^{\prime}. \end{split} \tag{45}$$

#### **3. Non-radial case of main theorem**

In this section, we prove (Non-radial case) for Theorem 6. First, we recall rewriting of *nω*,*<sup>V</sup>*, which is given in [40].

Lemma 1 Let *<sup>d</sup>*≥3, 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt;*<sup>p</sup>* <sup>&</sup>lt;<sup>1</sup> <sup>þ</sup> <sup>4</sup> *d*�2 , and *Qω*,0 ∈ G*ω*,0. Assume that *V* satisfies (A2) with ∣a∣ ≤ 1 and (A6). Then,

$$\mathcal{S}\_{a,0}(Q\_{a,0}) = \mathfrak{n}\_{a,V} = \inf \left\{ T\_{a,V}(f) : f \in H^1(\mathbb{R}^d) \backslash \{ 0 \}, \ K\_V(f) \le 0 \right\} \tag{46}$$

holds, where the functional *Tω*,*<sup>V</sup>* is defined as

$$T\_{\alpha,V}(f) \coloneqq \mathbb{S}\_{\alpha,V}(f) - \frac{1}{4}K\nu(f). \tag{47}$$

Next, we give uniform estimate of the virial functional *KV*.

Lemma 2 Under the all assumptions of (Non-radial) in Theorem 6, there exists *δ*>0 such that

$$\sup\_{t \in (T\_{\min}, T\_{\max})} K\_V(u(t)) \le -\delta < 0. \tag{48}$$

**Proof:** Let *δ* ≔ 4 *S<sup>ω</sup>*,*<sup>V</sup> Q<sup>ω</sup>*,*<sup>V</sup>* � � � *<sup>S</sup><sup>ω</sup>*,*<sup>V</sup>*ð Þ *<sup>u</sup>*<sup>0</sup> � �>0. Applying Lemma 1, we have

$$\begin{split} \mathcal{S}\_{o,V}(Q\_{o,V}) &\leq T\_{o,V}(u(t)) = \mathcal{S}\_{o,V}(u\_0) - \frac{1}{4}K\_V(u(t)) \\ &= \mathcal{S}\_{o,V}(Q\_{o,V}) - \frac{1}{4}\delta - \frac{1}{4}K\_V(u(t)), \end{split} \tag{49}$$

which implies the desired result.

The blow-up result with *xu*<sup>0</sup> ∈*L*<sup>2</sup> *<sup>d</sup>* � � of (Non-radial case) in Theorem 1.1 follows immediately from Lemma 2.

**Proof of blow-up part in (Non-radial case) for Theorem 6:** We assume that the solution *u* exists globally in time for contradiction. When *xu*<sup>0</sup> ∈*L*<sup>2</sup> *<sup>d</sup>* � �, we have Eq. (22). Combining Eq. (22) and Lemma 2, there exists *δ*>0 such that

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

$$\frac{d^2}{dt^2} \left\| \varkappa u(t) \right\|\_{L^2}^2 = 4K\_V(u(t)) < -4\delta < 0\tag{50}$$

for any *<sup>t</sup>*<sup>∈</sup> . Therefore, we obtain <sup>∥</sup>*xu t*ð Þ∥<sup>2</sup> *<sup>L</sup>*<sup>2</sup> <0 if ∣*t*∣ is sufficiently large. However, this is contradiction.

We consider Lemmas 3 and 4 to prove blow-up or grow-up part in (Non-radial case) for Theorem 6.

Lemma 3 Let *<sup>d</sup>*≥3 and 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt;*p*<sup>&</sup>lt; <sup>1</sup> <sup>þ</sup> <sup>4</sup> *d*�2 . We assume that *<sup>u</sup>* <sup>∈</sup>*<sup>C</sup>* ½ Þ 0, <sup>∞</sup> ; *<sup>H</sup>*<sup>1</sup> � � be a solution to Eq. (1) satisfying *<sup>C</sup>*<sup>0</sup> <sup>≔</sup> sup*t*∈½ Þ 0,<sup>∞</sup> <sup>∥</sup>*u t*ð Þ∥*H*\_ <sup>1</sup> *x* < ∞. Then, it follows that

$$\left\|\boldsymbol{\mu}(t)\right\|\_{L^{2}(|\boldsymbol{x}|\geq R)}^{2} \leq o\_{R}(\mathbf{1}) + \eta \tag{51}$$

for any *η*>0, *R*> 0, and *t* ∈ 0, *<sup>η</sup><sup>R</sup>* 6*C*0∥*u*∥*L*<sup>2</sup> *x* � �, where *oR*ð Þ<sup>1</sup> goes to zero as *<sup>R</sup>* ! <sup>∞</sup> and is independent of *t*.

**Proof:** We consider *I*Y*<sup>R</sup>* given in Eq. (43). Using Proposition 3,

$$\begin{split} I(t) &= I(\mathbf{0}) + \int\_{0}^{t} I'(s) ds \le I(\mathbf{0}) + \int\_{0}^{t} |I'(s)| \, ds \\ &\le I(\mathbf{0}) + \frac{2t}{R} \| \mathbf{J}' \|\_{L^{\infty}} \sup\_{t \in [0, \infty)} \| u(t) \|\_{\dot{H}^{1}\_{x}} \| u \|\_{L^{2}\_{x}} \le I(\mathbf{0}) + \frac{6C\_{0} \| \| u \|\_{L^{2}\_{x}} t}{R} \end{split} \tag{52}$$

for any *t*∈½ Þ 0, ∞ . By the definition of Y*R*, we have

$$I(\mathbf{0}) = \int\_{\mathbb{R}^d} \mathcal{Y}\_R(\mathbf{x}) |u\_0(\mathbf{x})|^2 d\mathbf{x} \le \|u\_0\|\_{L^2\left(|\mathbf{x}| \ge \frac{\mathbf{g}}{2}\right)}^2 = o\_R(\mathbf{1})\tag{53}$$

and hence, we obtain

$$\left\|\boldsymbol{u}(t)\right\|\_{L^{2}(|\boldsymbol{x}|\geq\boldsymbol{R})}^{2} \leq \boldsymbol{I}(t) \leq o\_{R}(\mathbf{1}) + \eta. \tag{54}$$

Lemma 4 Let *<sup>d</sup>*≥3 and 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt; *<sup>p</sup>*<<sup>1</sup> <sup>þ</sup> <sup>4</sup> *d*�2 . Let *<sup>u</sup>* <sup>∈</sup>*<sup>C</sup>* ½ Þ 0, <sup>∞</sup> ; *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* � � � � be a solution to Eq. (1). Then, for *<sup>q</sup>*<sup>∈</sup> *<sup>p</sup>* <sup>þ</sup> 1, 2 <sup>∗</sup> ð Þ, there exist constants *<sup>C</sup>* <sup>¼</sup> *C q*, <sup>∥</sup>*u*0∥*<sup>L</sup>* ð Þ <sup>2</sup> ,*C*<sup>0</sup> <sup>&</sup>gt;0 and *θ<sup>q</sup>* > 0 such that the estimate

$$I\_{X\_R} \prime (t) \le 4K\nu(\boldsymbol{u}(t)) + C \|\boldsymbol{u}(t)\|\_{L^2(\boldsymbol{R} \le |\boldsymbol{x}|)}^{(p+1)\theta\_q} + \frac{C}{R^2} \tag{55}$$

holds for any *<sup>R</sup>*>0 and *<sup>t</sup>*∈½ Þ 0, <sup>∞</sup> , where *<sup>θ</sup><sup>q</sup>* <sup>≔</sup> <sup>2</sup>f g *<sup>q</sup>*�ð Þ *<sup>p</sup>*þ<sup>1</sup> ð Þ *<sup>p</sup>*þ<sup>1</sup> ð Þ *<sup>q</sup>*�<sup>2</sup> <sup>∈</sup> 0, <sup>2</sup> *p*þ1 � �, *<sup>C</sup>*<sup>0</sup> is given in Lemma 3, and *I*<sup>X</sup>*<sup>R</sup>* is defined as Eq. (43).

**Proof:** Using Proposition 3, we have

$$I\_{\mathcal{X}\_{\mathbb{R}}}'(t) = 4K\_V(u(t)) + \mathcal{R}\_1 + \mathcal{R}\_2 + \mathcal{R}\_3 + \mathcal{R}\_4,\tag{56}$$

where R*<sup>k</sup>* ¼ R*k*ð Þ*t* ð Þ *k* ¼ 1,2,3,4 are defined as

$$\begin{split} \mathcal{R}\_1 &= 4 \int\_{\mathbb{R}^d} \left\{ \frac{1}{\left| \boldsymbol{\varkappa} \right|^2} \mathcal{X}'' \left( \frac{\boldsymbol{r}}{R} \right) - \frac{R}{\left| \boldsymbol{\varkappa} \right|^3} \mathcal{X}' \left( \frac{\left| \boldsymbol{\varkappa} \right|}{R} \right) \right\} |\boldsymbol{\varkappa} \cdot \nabla \boldsymbol{u}|^2 d\boldsymbol{x} \\ &+ 4 \int\_{\mathbb{R}^d} \left\{ \frac{R}{\left| \boldsymbol{\varkappa} \right|} \mathcal{X}' \left( \frac{\left| \boldsymbol{\varkappa} \right|}{R} \right) - 2 \right\} |\nabla \boldsymbol{u}(t, \boldsymbol{x})|^2 d\boldsymbol{x}, \end{split} \tag{57}$$

*Schrödinger Equation – Fundamentals Aspects and Potential Applications*

$$\mathcal{R}\_2 \coloneqq -\frac{2(p-1)}{p+1} \int\_{\mathbb{R}^d} \left\{ \mathcal{X}' \left( \frac{|\mathbf{x}|}{R} \right) + \frac{(d-1)R}{|\mathbf{x}|} \mathcal{X}' \left( \frac{|\mathbf{x}|}{R} \right) - 2d \right\} |u(t, \mathbf{x})|^{p+1} d\mathbf{x}, \tag{58}$$

$$\begin{split} \mathcal{R}\_{3} & \coloneqq -\int\_{\mathbb{R}^{d}} \left\{ \frac{1}{R^{2}} \mathcal{X}^{(4)} \left( \frac{|\mathbf{x}|}{R} \right) + \frac{2(d-1)}{R|\mathbf{x}|} \mathcal{X}^{(3)} \left( \frac{|\mathbf{x}|}{R} \right) + \frac{(d-1)(d-3)}{|\mathbf{x}|^{2}} \mathcal{X}^{\prime} \left( \frac{|\mathbf{x}|}{R} \right) \right\} \\ & + \frac{(d-1)(3-d)R}{|\mathbf{x}|^{3}} \mathcal{X}^{\prime} \left( \frac{|\mathbf{x}|}{R} \right) \Bigg) |u(t, \mathbf{x})|^{2} d\mathbf{x}, \end{split} \tag{59}$$

$$|\mathbf{x}|^{3} \qquad \overset{\textstyle \cdot}{\longrightarrow} \begin{Bmatrix} \mathbf{x} \end{Bmatrix} \int \mathbf{x}^{\left\lfloor \mathbf{x} \cdot (\mathbf{x}, \mathbf{x}) \right\rfloor} \cdots$$

$$\mathcal{R}\_{4} \coloneqq 2 \int\_{R \leq |\mathbf{x}|} \left\{ 2 - \frac{R}{|\mathbf{x}|} \mathcal{X}' \left( \frac{|\mathbf{x}|}{R} \right) \right\} (\mathbf{x} \cdot \nabla V) |u(t, \mathbf{x})|^{2} d\mathbf{x}. \tag{60}$$

We set

$$\Delta \Omega := \left\{ \mathbf{x} \in \mathbb{R}^d : \frac{\mathbf{1}}{\left| \mathbf{x} \right|^2} \mathcal{X}'' \left( \frac{|\mathbf{x}|}{R} \right) - \frac{R}{\left| \mathbf{x} \right|^3} \mathcal{X}' \left( \frac{|\mathbf{x}|}{R} \right) \le \mathbf{0} \right\}. \tag{61}$$

By the inequality X<sup>0</sup> <sup>∣</sup>*x*<sup>∣</sup> *R* � �≤ <sup>2</sup>∣*x*<sup>∣</sup> *<sup>R</sup>* , we have

$$\mathcal{R}\_1 \le 4 \int\_{\Omega^\varepsilon} \left\{ \mathcal{X}'' \left( \frac{r}{R} \right) - 2 \right\} |\nabla u(t, \infty)|^2 d\mathbf{x} \le \mathbf{0},\tag{62}$$

where Ω*<sup>c</sup>* denotes a complement of Ω.

Next, we estimate R2. Applying Hölder's inequality and Sobolev's embedding, we have

$$\begin{split} \left\| \mathcal{R}\_{2} \leq & C \| u(t) \| \|\_{L^{p+1}(\mathbb{R} \leq |\mathbf{x}|)}^{p+1} \leq \mathcal{C} \| u(t) \| \|\_{L^{q}(\mathbb{R} \leq |\mathbf{x}|)}^{(p+1)\left(1-\theta\_{q}\right)} \| u(t) \| \|\_{L^{2}(\mathbb{R} \leq |\mathbf{x}|)}^{(p+1)\theta\_{q}} \\ \leq & C \| u(t) \| \|\_{H^{1}}^{(p+1)\left(1-\theta\_{q}\right)} \| u(t) \| \|\_{L^{2}(\mathbb{R} \leq |\mathbf{x}|)}^{(p+1)\theta\_{q}} \leq \mathcal{C} \| u(t) \| \|\_{L^{2}(\mathbb{R} \leq |\mathbf{x}|)}^{(p+1)\theta\_{q}}. \end{split} \tag{63}$$

Next, we estimate R3.

$$\mathcal{R}\_3 \le \frac{C}{R^2} \left\| u(t) \right\|\_{L^2(\mathbb{R} \le |x|)}^2 \le \frac{C}{R^2}.\tag{64}$$

Finally, R<sup>4</sup> is estimated as R<sup>4</sup> ≤0 by X<sup>0</sup> <sup>∣</sup>*x*<sup>∣</sup> *R* � �≤ <sup>2</sup>∣*x*<sup>∣</sup> *<sup>R</sup>* and *x* � ∇*V* ≤0, which completes the proof of the lemma.

**Proof of blow-up or grow-up part in (Non-radial case) for Theorem 6.** We assume that

$$T\_{\max} = \infty \quad \text{and} \quad \sup\_{t \in [0, \infty)} \|u(t)\|\_{\dot{H}^1\_x} < \infty \tag{65}$$

for contradiction. By Lemmas 2, 3, and 4, there exists *δ*>0 such that

$$I\_{\mathcal{X}\_R}''(s) \le -4\delta + \mathcal{C} \|u(s)\|\_{L\_x^2(R \le |x|)}^{(p+1)\theta\_q} + \frac{\mathcal{C}}{R^2} \le -4\delta + \mathcal{C} \eta^{\frac{(p+1)\theta\_q}{2}} + o\_R(1) \tag{66}$$

for any *η*>0, *R*> 0, and *s* ∈ 0, *<sup>η</sup><sup>R</sup>* 6*C*0∥*u*0∥*L*<sup>2</sup> h i. We take *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>0</sup> <sup>&</sup>gt; 0 sufficiently small such as

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

$$C\eta\_0^{\frac{(p+1)\theta\_0}{2}} \le 2\delta. \tag{67}$$

and set

$$T = T(R) := a\_0 R := \frac{\eta\_0 R}{\mathfrak{G}C\_0 \|\|\mu\_0\|\|\_{L^2}}. \tag{68}$$

Applying Eq. (67), integrating Eq. (66) over *s*∈½ � 0, *t* , and integrating over *t*∈½ � 0, *T* , we have

$$\begin{split} I\_{\mathcal{X}\_{\mathbb{R}}}(T) &\leq I\_{\mathcal{X}\_{\mathbb{R}}}(\mathbf{0}) + I\_{\mathcal{X}\_{\mathbb{R}}}'(\mathbf{0})T + \frac{1}{2}(-2\delta + o\_{\mathbb{R}}(\mathbf{1}))T^{2} \\ &= I\_{\mathcal{X}\_{\mathbb{R}}}(\mathbf{0}) + I\_{\mathcal{X}\_{\mathbb{R}}}'(\mathbf{0})a\_{0}\mathbf{R} + \frac{1}{2}(-2\delta + o\_{\mathbb{R}}(\mathbf{1}))a\_{0}^{2}\mathbf{R}^{2}. \end{split} \tag{69}$$

Here, we can see

$$I\_{\mathcal{X}\_{\mathbb{R}}}(\mathbf{0}) = o\_{\mathbb{R}}(\mathbf{1}) \mathsf{R}^2 \quad \text{and} \ I'\_{\mathcal{X}\_{\mathbb{R}}}(\mathbf{0}) = o\_{\mathbb{R}}(\mathbf{1}) \mathsf{R}. \tag{70}$$

Indeed, we get

$$\left\| I\_{\mathcal{X}\_{\mathbb{R}}}(\mathbf{0}) \leq \mathsf{R} \left\| \boldsymbol{\mu}\_{0} \right\|\right\|\_{L^{2}\left(|\boldsymbol{x}| \leq \sqrt{\mathsf{R}}\right)}^{2} + c\mathsf{R}^{2} \left\| \boldsymbol{\mu}\_{0} \right\|\_{L^{2}\left(\sqrt{\mathsf{R}} \leq |\boldsymbol{x}|\right)} = o\_{\mathsf{R}}(\mathsf{1})\mathsf{R}^{2},\tag{71}$$

and

$$\mathcal{I}\_{\mathcal{X}\_{\mathbb{R}}}(\mathbf{0}) \le 4\sqrt{R} \|\boldsymbol{\mu}\_{0}\|\_{\dot{H}^{1}} \|\boldsymbol{\mu}\_{0}\|\_{L^{2}\left(|\mathbf{x}| \le \sqrt{R}\right)} + c\mathcal{R} \|\boldsymbol{\mu}\_{0}\|\_{\dot{H}^{1}} \|\boldsymbol{\mu}\_{0}\|\_{L^{2}\left(\sqrt{R} \le |\mathbf{x}|\right)} = o\_{\mathbb{R}}(\mathbf{1})\mathcal{R}.\tag{72}$$

Combining Eqs. (69) and (70), we get

$$I\_{\mathcal{X}\_R}(T) \le \left(o\_R(\mathbf{1}) - \delta \alpha\_0^2\right) \mathcal{R}^2. \tag{73}$$

We take *<sup>R</sup>*>0 such as *oR*ð Þ� <sup>1</sup> *δα*<sup>2</sup> <sup>0</sup> <0. However, this contradicts *I*X*<sup>R</sup>* ð Þ *T* ≥0.

#### **4. Radial case of main theorem**

In this section, we prove (Radial case) for Theorem 6. First, we introduce another characterization of *r<sup>ω</sup>*,*<sup>V</sup>*.

Lemma 5 Let *<sup>d</sup>*≥3, 1 <sup>þ</sup> <sup>4</sup> *<sup>d</sup>* <sup>&</sup>lt;*p*<sup>&</sup>lt; <sup>1</sup> <sup>þ</sup> <sup>4</sup> *d*�2 , and *Q<sup>ω</sup>*,*<sup>V</sup>* ∈ G*<sup>ω</sup>*,*<sup>V</sup>*,rad. Assume that *V* is radially symmetric and satisfies (A3) with <sup>∣</sup>a∣ ≤2, (A7), and 3*<sup>x</sup>* � <sup>∇</sup>*<sup>V</sup>* <sup>þ</sup> *<sup>x</sup>*∇<sup>2</sup>*VxT* <sup>≤</sup>0. Then,

$$\mathcal{S}\_{a,V}(Q\_{a,V}) = r\_{a,V} = \inf \left\{ U\_{a,V}(f) : f \in H^1\_{\text{rad}}(\mathbb{R}^d) \, | \, \{ 0 \}, \, K\_V(f) \le 0 \right\} \tag{74}$$

holds, where the functional *U<sup>ω</sup>*,*<sup>V</sup>* is defined as

$$U\_{o,V}(f) \coloneqq \mathcal{S}\_{o,V}(f) - \frac{1}{d(p-1)} K\_V(f). \tag{75}$$

**Proof:** The lemma follows from proof of Lemma 1 (see [40], Lemma 4.3) combined 2*ω* þ 2*V* þ *x* � ∇*V* ≥0.

**Proof of (Radial case) for Theorem 6***.* Assume that the solution *u* to Eq. (1) exists globally in time for contradiction. We consider *I*X*<sup>R</sup>* again and recall

$$I\_{\mathcal{X}\_{\mathbb{R}}}^{\prime\prime}(t) = 4\mathcal{K}\_{V}(u(t)) + \mathcal{R}\_1 + \mathcal{R}\_2 + \mathcal{R}\_3 + \mathcal{R}\_4,\tag{76}$$

where R*<sup>k</sup>* ð Þ 1≤*k*≤ 4 are defined as Eqs. (57) � (60). We use same estimates with proof of blow-up or grow-up for R1, R3, and R4. Applying Proposition 2 and the Young's inequality, we have

$$\begin{split} \mathcal{R}\_{2} &\leq \frac{C}{R^{\frac{(d-1)(p-1)}{2}}} \|u(t)\|\_{L^{2}(\mathbb{R}\leq|x|)}^{\frac{p+3}{2}} \|u(t)\|\_{\dot{H}^{1}(\mathbb{R}\leq|x|)}^{\frac{p-1}{2}} \\ &\leq \frac{C}{R^{\frac{2(d-1)(p-1)}{5-p}}} \frac{4}{\varepsilon^{\frac{4}{5-p}}} \|u\|\_{L^{2}}^{\frac{2(p+3)}{5-p}} + 2\{d(p-1)-4\}\varepsilon \|u\|\_{\dot{H}^{1}}^{2} \\ &\leq \frac{C}{R^{\frac{2(d-1)(p-1)}{5-p}}} \frac{4}{\varepsilon^{\frac{4}{5-p}}} \|u\|\_{L^{2}}^{\frac{2(p+3)}{5-p}} + 4d(p-1)\varepsilon U\_{o,V}(u) \end{split} \tag{77}$$

for each positive *ε*> 0, which is chosen later. Collecting these estimates, we have

$$\begin{split} &I\_{X\_{k}}^{\mu}(t) \\ &\leq 4K\_{V}(u) + 4d(p-1)\epsilon U\_{a,V}(u) + \frac{C}{R^{\frac{2(d-1)(p-1)}{5-p}}} + \frac{C}{R^{2}} \\ &= 4d(p-1)\{S\_{a,V}(u) - U\_{a,V}(u)\} + 4d(p-1)\epsilon U\_{a,V}(u) + \frac{C}{R^{\frac{2(d-1)(p-1)}{5-p}}} + \frac{C}{R^{2}} \\ &< 4d(p-1)(1-\delta)S\_{a,V}\left(Q\_{a,V}\right) + 4d(p-1)(e-1)U\_{a,V}(u) + \frac{C}{R^{\frac{2(d-1)(p-1)}{5-p}}} + \frac{C}{R^{2}} \\ &\leq 4d(p-1)(e-\delta)S\_{a,V}\left(Q\_{a,V}\right) + \frac{C}{R^{\frac{2(d-1)(p-1)}{5-p}}} + \frac{C}{R^{2}}, \end{split} \tag{78}$$

where the second inequality is used *Sω*,*<sup>V</sup>*ð Þ *u* <ð Þ 1 � *δ Sω*,*<sup>V</sup> Qω*,*<sup>V</sup>* for some *<sup>δ</sup>*∈ð Þ 0, 1 and the third inequality is used *S<sup>ω</sup>*,*<sup>V</sup> Q<sup>ω</sup>*,*<sup>V</sup>* <sup>≤</sup> *<sup>U</sup><sup>ω</sup>*,*<sup>V</sup>* (see Lemma 5). Taking *<sup>ε</sup>*∈ð Þ 0, *<sup>δ</sup>* and sufficiently large *R*>0, there exists *η*>0 such that *I* 00 <sup>X</sup>*<sup>R</sup>* ð Þ*t* < � *η*< 0 for each *t*∈ . However, this inequality implies that if ∣*t*∣ is sufficiently large, then . This is contradiction and hence, we complete the proof.

#### **5. Conclusions**

In this chapter, our main result is Theorem 6. Combining the main result and a previous result (Theorem 5), we can classify time behavior of solutions to Eq. (1) with initial data below the ground state in the sense of their action *S<sup>ω</sup>*,*<sup>V</sup>* by using sign of the virial functional for the initial data. More precisely, for the solution *u t*ð Þ with *<sup>S</sup><sup>ω</sup>*,*<sup>V</sup>*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt;*S<sup>ω</sup>*,0 *<sup>Q</sup><sup>ω</sup>*,0 , if *KV*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>≥</sup>0 then *<sup>u</sup>* is bounded in *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* and if *KV*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>0</sup> then *<sup>u</sup>* is unbounded in *<sup>H</sup>*<sup>1</sup> *<sup>d</sup>* . In addition, for the radial solution *u t*ð Þ with *S<sup>ω</sup>*,*<sup>V</sup>*ð Þ *u*<sup>0</sup> <*S<sup>ω</sup>*,*<sup>V</sup> Q<sup>ω</sup>*,*<sup>V</sup>* , if *KV*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>≥</sup>0, then *<sup>u</sup>* exists globally in time and if *KV*ð Þ *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt;<sup>0</sup> then *u* blows up.

#### **Acknowledgements**

M.H. is supported by JSPS KAKENHI Grant Number JP22J00787. M.I. is supported by JSPS KAKENHI Grant Number JP19K14581 and JST CREST Grant Number JPMJCR1913.

#### **Conflict of interest**

The authors declare no conflict of interest.

### **Author details**

Masaru Hamano<sup>1</sup> \*† and Masahiro Ikeda2,3†

1 Faculty of Science and Engineering, Waseda University, Tokyo, Japan

2 RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

3 Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan

\*Address all correspondence to: m.hamano3@kurenai.waseda.jp

† These authors contributed equally.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Rose HA, Weinstein MI. On the bound states of the nonlinear Schrödinger equation with a linear potential. Physica D. 1988;**30**(1–2): 207-218

[2] Cazenave T. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics. Vol. **10**. New York; Providence, RI: New York University, Courant Institute of Mathematical Sciences; American Mathematical Society; 2003. p. xiv+323. MR2002047

[3] Hong Y. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis. 2016;**15**(5):1571-1601. MR3538870

[4] Hamano M, Ikeda M. Global dynamics below the ground state for the focusing Schrödinger equation with a potential. Journal of Evolution Equations. 2020;**20**(3):1131-1172. MR4142248

[5] Akahori T, Nawa H. Blowup and scattering problems for the nonlinear Schrödinger equations. Kyoto Journal of Mathematics. 2013;**53**(3):629-672. MR3102564

[6] Arora AK, Dodson B, Murphy J. Scattering below the ground state for the 2d radial nonlinear Schrödinger equation. Proceedings of the American Mathematical Society. 2020;**148**(4): 1653-1663. MR4069202

[7] Dinh VD. On nonlinear Schrödinger equations with attractive inverse-power potentials. Topological Methods in Nonlinear Analysis. 2021;**57**(2):489-523. MR4359723

[8] Dinh VD. On nonlinear Schrödinger equations with repulsive inverse-power potentials. Acta Applicandae Mathematicae. 2021;**171**:52. Paper No. 14, MR4198524

[9] Dodson B. Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state. Advances in Mathematics. 2015;**285**:1589-1618. MR3406535

[10] Dodson B. Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension *d* ¼ 4. Annales Scientifiques de l'École Normale Supérieure. 2019;**52**(1): 139-180. MR3940908

[11] Dodson B, Murphy J. A new proof of scattering below the ground state for the 3D radial focusing cubic NLS. Proceedings of the American Mathematical Society. 2017;**145**(11): 4859-4867. MR3692001

[12] Dodson B, Murphy J. A new proof of scattering below the ground state for the non-radial focusing NLS. Mathematical Research Letters. 2018;**25**(6):1805-1825. MR3934845

[13] Du D, Wu Y, Zhang K. On blow-up criterion for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems. 2016;**36**(7): 3639-3650. MR3485846

[14] Duyckaerts T, Holmer J, Roudenko S. Scattering for the nonradial 3D cubic nonlinear Schrödinger equation. Mathematical Research Letters. 2008;**15**(6):1233-1250. MR2470397

[15] Fang D, Xie J, Cazenave T. Scattering for the focusing energy-subcritical nonlinear Schrödinger equation. Science

*Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential DOI: http://dx.doi.org/10.5772/intechopen.113907*

China Mathematics. 2011;**54**(10): 2037-2062. MR2838120

[16] Farah LG, Pastor A. Scattering for a 3D coupled nonlinear Schrödinger system. Journal of Mathematical Physics. 2017;**58**(7):33, 071502. MR3671163

[17] Glassey RT. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. Journal of Mathematical Physics. 1977;**18**(9): 1794-1797. MR0460850

[18] Hamano M. Global dynamics below the ground state for the quadratic Schödinger system in 5d. Preprint, arXiv: 1805.12245

[19] Hamano M, Ikeda M. Equivalence of conditions on initial data below the ground state to NLS with a repulsive inverse power potential. Journal of Mathematical Physics. 2022;**63**(3):16. Paper No. 031509, MR4393612

[20] Hamano M, Ikeda M. Scattering solutions to nonlinear Schrödinger equation with a long range potential. Journal of Mathematical Analysis and Applications. 2023;**528**(1). Paper No. 127468. MR4602980

[21] Hamano M, Ikeda M, Inui T, Shimizu I. Global dynamics below a threshold for the nonlinear Schrödinger equations with the Kirchhoff boundary and the repulsive Dirac delta boundary on a star graph. Preprint, arXiv: 2212.06411

[22] Hamano M, Inui T, Nishimura K. Scattering for the quadratic nonlinear Schrödinger system in <sup>5</sup> without mass-resonance condition. Fako de l'Funkcialaj Ekvacioj Japana Matematika Societo. 2021;**64**(3):261-291. MR4360610

[23] Holmer J, Roudeko S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Communications in Mathematical Physics. 2008;**282**(2):435-467. MR2421484

[24] Ibrahim S, Masmoudi N, Nakanishi K. Scattering threshold for the focusing nonlinear Klein-Gordon equation. Analysis of PDEs. 2011;**4**(3): 405-460. MR2872122

[25] Ikeda M, Inui T. Global dynamics below the standing waves for the focusing semilinear Schrödinger equation with a repulsive Dirac delta potential. Analysis of PDEs. 2017;**10**(2): 481-512. MR3619878

[26] Inui T, Kishimoto N, Nishimura K. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete and Continuous Dynamical Systems. 2019; **39**(11):6299-6353. MR4026982

[27] Inui T, Kishimoto N, Nishimura K. Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without massresonance. Nonlinear Analysis. 2020; **198**:10, 111895. MR4090442

[28] Kenig CE, Merle F. Global wellposedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Inventiones Mathematicae. 2006;**166**(3): 645-675. MR2257393

[29] Kenig CE, Merle F. Global wellposedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Mathematica. 2008; **201**(2):147-212. MR2461508

[30] Killip R, Murphy J, Visan M, Zheng J. The focusing cubic NLS with inverse-square potential in three space dimensions. Differential and Integral Equations. 2017;**30**(3–4):161-206. MR3611498

[31] Killip R, Visan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 2010; **132**(2):361-424. MR2654778

[32] Lu J, Miao C, Murphy J. Scattering in *H*<sup>1</sup> for the intercritical NLS with an inverse-square potential. Journal of Differential Equations. 2018;**264**(5): 3174-3211. MR3741387

[33] Ogawa T, Tsutsumi Y. Blow-up of *H*<sup>1</sup> solution for the nonlinear Schrödinger equation. Journal of Differential Equations. 1991;**92**(2):317-330. MR1120908

[34] Wang H, Yang Q. Scattering for the 5D quadratic NLS system without massresonance. Journal of Mathematical Physics. 2019;**60**(12):23, 121508. MR4043361

[35] Xu G. Dynamics of some coupled nonlinear Schrödinger systems in <sup>3</sup> . Mathematicsl Methods in the Applied Sciences. 2014;**37**(17):2746-2771. MR3271121

[36] Xu C. Scattering for the non-radial focusing inhomogeneous nonlinear Schrödinger-Choquard equation. Preprint, arXiv: 2104.09756

[37] Zhang J, Zheng J. Scattering theory for nonlinear Schrödinger equations with inverse-square potential. Journal of Functional Analysis. 2014;**267**(8): 2907-2932. MR3255478

[38] Zheng J. Focusing NLS with inverse square potential. Journal of Mathematical Physics. 2018;**59**(11):14, 111502. MR3872306

[39] Mizutani H. Wave operators on Sobolev spaces. Proceedings of the American Mathematical Society. 2020; **148**(4):1645-1652. MR4069201

[40] Hamano M, Ikeda M. Characterization of the ground state to the intercritical NLS with a linear potential by the virial functional. In: Advances in Harmonic Analysis and Partial Differential Equations, Trends Math. Cham: Birkhäuser/Springer; 2020. pp. 279-307. MR4174752

[41] Hamano M, Ikeda M. Global wellposedness below the ground state for the nonlinear Schrödinger equation with a linear potential. Proceedings of the American Mathematical Society. 2020; **148**(12):5193-5207. MR4163832

### *Edited by Muhammad Bilal Tahir, Muhammad Sagir, Muhammad Isa Khan and Muhammad Rafique*

Unlock the secrets of the universe with *Schrödinger Equation - Fundamental Aspects and Potential Applications*. Delve into the heart of quantum mechanics, where matter, energy, and mathematics intertwine in a dance of profound discovery. This essential volume introduces you to the spectral theory of the Schrödinger equation, offering a sturdy foundation to explore its enigmatic depths. Discover the fascinating world of scattering theory, unraveling the intricacies of quantum interactions, while the principles of quantization and Feynman path integrals reveal the mechanics of quantum systems. With a fresh perspective, we explore relative entropy methods and transformation theory, unveiling their significance in crafting singular diffusion processes akin to Schrödinger equations. This well-organized and accessible book caters to a diverse audience, from students and researchers to professionals in functional analysis, probability theory, and quantum dynamics. Within these pages, you'll uncover the profound wonders of the Schrödinger equation and its vast potential in science, engineering, and technology. Embark on a journey through the quantum cosmos and let your understanding of the universe expand as you explore the quantum realm. Welcome to a world where matter and energy dance to the tune of Schrödinger's equation, a world filled with infinite possibilities and extraordinary insights.

Published in London, UK © 2024 IntechOpen © Chayanan / iStock

Schrödinger Equation - Fundamentals Aspects and Potential Applications

Schrödinger Equation

Fundamentals Aspects and Potential

Applications

*Edited by Muhammad Bilal Tahir, Muhammad Sagir, Muhammad Isa Khan* 

*and Muhammad Rafique*