**3. The Landau Hamiltonian** *H<sup>ν</sup>* **on the Poincaré disk**

Let <sup>=</sup>f g *<sup>z</sup>*∈ℂ, *zz*<<sup>1</sup> be the complex unit disk with the Poincaré metric *ds*<sup>2</sup> <sup>¼</sup> 4 1ð Þ � *zz* �<sup>2</sup> *dzdz:* is a complete Riemannian manifold with all sectional curvature equal �1*:* It has an ideal boundary ∂ID identified with the circle f g *ω* ∈ℂ, *ωω* ¼ 1 *:* One refers to points *ω*∈ ∂ID as points at infinity. The geodesic distance between two points *z* and *w* is given by

$$\cosh d(z, w) = 1 + \frac{2(z - w)(\overline{z} - \overline{w})}{(1 - z\overline{z})(1 - w\overline{w})}.\tag{11}$$

By Ref. [5] the Schrödinger operator on with a constant magnetic field of strength proportional to *ν*>0 can be written as:

$$\mathcal{L}\_{\nu} := -\left(\mathbf{1} - \left|\boldsymbol{x}\right|^2\right)^2 \frac{\partial^2}{\partial \boldsymbol{z} \partial \overline{\boldsymbol{z}}} - \nu \overline{\omega} \left(\mathbf{1} - \left|\boldsymbol{z}\right|^2\right) \frac{\partial}{\partial \boldsymbol{z}} + \nu \overline{\omega} \left(\mathbf{1} - \left|\boldsymbol{z}\right|^2\right) \frac{\partial}{\partial \overline{\boldsymbol{z}}} + \nu^2 \left|\boldsymbol{z}\right|^2. \tag{12}$$

which is also called Maass Laplacian on the disk. A slight modification of L*<sup>ν</sup>* is given by the operator

$$H\_{\nu} \coloneqq 4\mathcal{L}\_{\nu} - 4\nu^2 \tag{13}$$

acting in the Hilbert space

$$L^{2,0}(\mathbb{D}) \coloneqq \left\{ \boldsymbol{\varrho} : \mathbb{D} \to \mathbb{C}, \int\_{\mathbb{D}} |\boldsymbol{\varrho}(\boldsymbol{z})|^{2} \left(\mathbb{1} - |\boldsymbol{z}|^{2}\right)^{-2} d\boldsymbol{\mu}(\boldsymbol{z}) < +\infty \right\},\tag{14}$$

For our purpose, we shall consider the unitary equivalent realization *H*~ *<sup>ν</sup>* of the operator *H<sup>ν</sup>* in the Hilbert space

$$L^{2, \nu}(\mathbb{D}) \coloneqq \left\{ \boldsymbol{\varrho} : \mathbb{D} \to \mathbb{C}, \int\_{\mathbb{D}} \left| \boldsymbol{\varrho}(\boldsymbol{z}) \right|^{2} \left( \mathbb{1} - \left| \boldsymbol{z} \right|^{2} \right)^{2\nu - 2} d\mu(\boldsymbol{z}) < +\infty \right\},\tag{15}$$

which is defined by

$$
\tilde{H}\_{\nu} \coloneqq \mathbf{Q}\_{\nu}^{-1} H\_{\nu} \mathbf{Q}\_{\omega}, \tag{16}
$$

*The Singular Values of the Logarithmic Potential Transform on Bound States Spaces DOI: http://dx.doi.org/10.5772/intechopen.107090*

where *Q<sup>ν</sup>* : *L*2,*<sup>ν</sup>* ð Þ! *<sup>L</sup>*2,0ð Þ is the unitary transformation defined by the map *φ* ↦ *Qν*½ � *φ* ≔ 1 � j j *z* <sup>2</sup> �*<sup>ν</sup> φ:* Different aspects of the spectral analysis of the operator *H*~ *<sup>ν</sup>* have been studied by many authors. For instance, note that *H*~ *<sup>ν</sup>* is an elliptic densely defined operator on the Hilbert space *L*2,*<sup>ν</sup>* ð Þ and admits a unique self-adjoint realization that we denote also by *H*~ *<sup>ν</sup>:* The spectrum of *H*~ *<sup>ν</sup>* in *L*2,*<sup>ν</sup>* ð Þ consists of two parts: ð Þ*i* a continuous part 1, ½ ½ þ∞ , which corresponds to *scattering states*, ð Þ *ii* a finite number of eigenvalues (*hyperbolic Landau levels*) of the form

$$\varepsilon\_m^\nu \coloneqq 4(\nu - m)(1 - \nu + m), \ m = 0, 1, 2, \cdots, \left[\nu - \frac{1}{2}\right] \tag{17}$$

with infinite degeneracy, provided that 2*ν*>1. The eigenvalues in (17) correspond eigenfunctions, which are called *bound states* since the particle in such a state cannot leave the system without additional energy. A concrete description of these bound states spaces will be the goal of the next section.

### **4. The bound states spaces** A**<sup>2</sup>** *<sup>ν</sup>***,***m*ð Þ

Here, we consider the eigenspace

$$\mathcal{A}^2\_{\nu,m}(\mathbb{D}) \coloneqq \left\{ \Phi : \mathbb{D} \to \mathbb{C}, \,\Phi \in L^{2,\nu}(\mathbb{D}) \text{ and } \tilde{H}\_{\nu}\Phi = \varepsilon^{\nu}\_{m}\Phi \right\}.\tag{18}$$

See Refs. [6, 7], for the following proposition. **Proposition 4.1.** *Let* <sup>2</sup>*ν*><sup>1</sup> *and m* <sup>¼</sup> 0,1,2,⋯, *<sup>ν</sup>* � <sup>1</sup> 2 *:Then, we have.* ð Þ*<sup>i</sup> an orthogonal basis* of <sup>A</sup><sup>2</sup> *<sup>ν</sup>*,*<sup>m</sup>*ð Þ *is given by the functions*

$$\phi\_k^{\nu, m}(z) \coloneqq |z|^{|m-k|} \left( 1 - |z|^2 \right)^{-m} e^{-i(m-k)\text{arg}z}$$

$$\lambda \times\_2 F\_1 \left( -m + \frac{m-k+|m-k|}{2}, 2\nu - m + \frac{|m-k|-m+k}{2}, 1 + |m-k|; |z|^2 \right) \tag{19}$$

*k* ¼ 0,1,2,⋯, *in terms of a terminating* <sup>2</sup>*F*<sup>1</sup> *Gauss hypergeometric function.* ð Þ *ii the norm square of <sup>ϕ</sup><sup>ν</sup>*,*<sup>m</sup> <sup>k</sup> in L*2,*<sup>ν</sup>* ð Þ *is given by*

$$\left||\phi\_{k}^{\nu,m}\right||^{2} = \frac{\pi(\Gamma(1+|m-k|))^{2}}{(2(\nu-m)-1)} \frac{\Gamma\left(m-\frac{|m-k|+m-k}{2}+1\right)\Gamma\left(2\nu-m-\frac{|m-k|+m-k}{2}\right)}{\Gamma\left(m+\frac{|m-k|-m+k}{2}+1\right)\Gamma\left(2\nu-m+\frac{|m-k|-m+k}{2}\right)}.\tag{20}$$

**Corollary 4.1.** *The functions* Φ*<sup>ν</sup>*,*<sup>m</sup> k* , *<sup>k</sup>* <sup>¼</sup> 0,1,2, … , *given by*

$$\Phi\_k^{\nu, m}(z) := (-1)^k \left( \frac{2(\nu - m) - 1}{\pi} \right)^{\frac{1}{2}} \left( \frac{k! \Gamma(2(\nu - m) + m)}{m! \Gamma(2(\nu - m) + k)} \right)^{\frac{1}{2}} \tag{21}$$

$$\times \left( \mathbf{1} - \left| z \right|^2 \right)^{-m} \overline{z}^{m-k} P\_k^{(m-k, \ 2(\nu-m)-1)} (\mathbf{1} - 2z\overline{z}), \tag{22}$$

*in terms of Jacobi polynomials constitute an orthonormal basis* of A2,*<sup>ν</sup> <sup>m</sup>* ð Þ . **Proof.** Write the connection between the <sup>2</sup>*F*1-sum and the Jacobi polynomial

$$P\_k^{a,\beta}(u) = \frac{(\mathbf{1} + a)\_k}{k!} \,\_2F\_1\left( -k, \, \mathbf{1} + a + \beta + k, \, \mathbf{1} + a; \, \frac{\mathbf{1} - u}{2} \right),$$

then the functions

$$\phi\_k^{\nu,m}(z) = \frac{(-1)^{\min(m,k)}}{\left(1-|z|^2\right)^m} |z|^{|m-k|} e^{-i(m-k)\arg z} P\_{\min(m,k)}^{(|m-k|, \ 2(\nu-m)-1)}(1-2z\overline{z}),\tag{23}$$

constitute an orthonormal basis of A<sup>2</sup> *<sup>ν</sup>*,*<sup>m</sup>:* The norm square of *<sup>ϕ</sup>ν*,*<sup>m</sup> <sup>k</sup>* in *<sup>L</sup>*2,*<sup>ν</sup>* ð Þ is given by

$$\left\| \left| \phi\_k^{\nu, m} \right| \right\|^2 = \frac{\pi}{(2(\nu - m) - 1)} \frac{(m \vee k)! \Gamma(2(\nu - m) + m \wedge k)}{(m \wedge k)! \Gamma(2(\nu - m) + m \vee k)}.\tag{24}$$

Here, *m* ∧ *k* ≔ min ð Þ *m*, *k* and *m* ∨ *k* ≔ max ð Þ *m*, *k :* Thus, the set of functions

$$\Phi\_k^{\nu,m} := \frac{\phi\_k^{\nu,m}}{||\phi\_k^{\nu,m}||}, k = 0, 1, 2, \dots \tag{25}$$

is an orthonormal basis of A<sup>2</sup> *<sup>ν</sup>*,*<sup>m</sup>*ð Þ and can be rewritten as.

$$\Phi\_k^{\nu,m}(z) = (-1)^k \left( \frac{2(\nu - m) - 1}{\pi} \right)^{\frac{1}{2}} \left( \frac{k! \Gamma(2(\nu - m) + m)}{m! \Gamma(2(\nu - m) + k)} \right)^{\frac{1}{2}} \tag{26}$$

$$\times \left(\mathbf{1} - |z|^2\right)^{-m} \overline{z}^{m-k} P\_k^{(m-k, \ 2(\nu-m)-1)} (\mathbf{1} - \mathbf{2}z\overline{z}) \tag{27}$$

by making appeal to the identity ð Þ *S*, *p:*63 :

$$\frac{\Gamma(m+1)}{\Gamma(m-s+1)}P\_{m}^{(-s,-a)}(u) = \frac{\Gamma(m+a+1)}{\Gamma(m-s+a+1)} \left(\frac{u-1}{2}\right)^{s} P\_{m-s}^{(s,a)}(u), \mathbb{1} \le s \le m \tag{28}$$

for *s* ¼ *m* � *k*, *t* ¼ 1 � 2j j *z* 2 , and *<sup>α</sup>* <sup>¼</sup> <sup>2</sup>ð Þ� *<sup>ν</sup>* � *<sup>m</sup>* <sup>1</sup>*:* … □.

**Corollary 4.2.** *The L*<sup>2</sup> �*eigenspace* <sup>A</sup><sup>2</sup> *<sup>ν</sup>*,0ð Þ *, corresponding to m* ¼ 0 *in* ð Þ 3*:*1 *and associated with the bottom energy ε<sup>ν</sup>* <sup>0</sup> ¼ 0 *in* ð Þ 2*:*6 , *reduces further to the weighted Bergman space consisting of holomorphic functions ϕ:* ! *such that*

$$\int\_{\mathbb{D}} |\phi(x)|^2 \left(1 - |x|^2\right)^{2\nu - 2} d\mu(x) < +\infty. \tag{29}$$

*The Singular Values of the Logarithmic Potential Transform on Bound States Spaces DOI: http://dx.doi.org/10.5772/intechopen.107090*
