3. Exact first-order finite differences derivative for the exponential function

Let us consider the exact backward and forward finite differences derivatives of ev x, at xj, given by

$$(D\_\vartheta e^{\boldsymbol{v}\cdot\boldsymbol{x}})\_{\circ} \coloneqq \frac{e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}} - e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ -1}}}{\chi\_2(\boldsymbol{v}, \boldsymbol{j} - 1)} = \boldsymbol{v}\,e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}} \text{ and } \left(D\_{\boldsymbol{f}}e^{\boldsymbol{v}\cdot\boldsymbol{x}}\right)\_{\circ} \coloneqq \frac{e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ} + 1} - e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}}}{\chi\_1(\boldsymbol{v}, \boldsymbol{j})} = \boldsymbol{v}\,e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}},\tag{6}$$

where v∈ C can be a pure real or pure imaginary constant, and the spacing functions χ1ð Þ v; j and χ2ð Þ v; j are defined as

$$\chi\_1(\upsilon, j) \coloneqq \frac{e^{\upsilon \cdot \Lambda\_{\dot{\gamma}}} - 1}{\upsilon} \cong \Lambda\_{\dot{\gamma}} + \frac{\upsilon}{2} \Lambda\_{\dot{\gamma}}^2 + O\left(\Delta\_{\dot{\gamma}}^3\right). \tag{7}$$

and

2. Exact first-order finite differences derivatives of functions

2.1. Backward and forward finite differences derivatives

a partition P ¼ f g x1; ; x2; ⋯; xN of N non-uniformly spaced points xj

� �<sup>≔</sup> g xj

spacing function Δj, is obtained by solving the above equality for χ2ð Þj ,

� � � g xj � <sup>Δ</sup><sup>j</sup>�<sup>1</sup> � �

g<sup>0</sup> xj

Df g � � xj

<sup>χ</sup>1ð Þ<sup>j</sup> <sup>≔</sup> g xj <sup>þ</sup> <sup>Δ</sup><sup>j</sup>

finite differences derivative, as can be seen in Eqs. (2) and (4).

This is an expression which is valid for points xj different from the zeroes of g<sup>0</sup>

� �<sup>≔</sup> g xj <sup>þ</sup> <sup>Δ</sup><sup>j</sup>

� �

� � <sup>¼</sup> <sup>1</sup>

� � � g xj

g<sup>0</sup> xj

ð Þ Dbg xj

<sup>χ</sup>2ð Þ <sup>j</sup> � <sup>1</sup> <sup>≔</sup> g xj

A definition for forward finite differences at xj is

valid for points different from the zeroes of g<sup>0</sup>

differences derivatives.

164 Numerical Simulations in Engineering and Science

requirement that

where

In this section, we intend to introduce a finite differences derivative, which has the same eigenfunction as for the continuous variable case. We start with results valid for any function, but we will concentrate, later in the chapter, on the exponential function because that function is used to perform translations along several directions in the quantum realm. The resulting derivative operator will depend on the point at which it is evaluated as well as on the partition of the interval and on the function of interest. This is the trade-off for having exact finite

An exact, backward, finite differences derivative of an absolutely continuous function g xð Þ (this class of functions is the domain of the momentum operator in Quantum Mechanics), on

> � � � g xj � <sup>Δ</sup><sup>j</sup>�<sup>1</sup> � � <sup>χ</sup>2ð Þ <sup>j</sup> � <sup>1</sup> <sup>¼</sup> <sup>g</sup><sup>0</sup> xj

where Δ<sup>j</sup> ¼ xjþ<sup>1</sup> � xj and the spacing function χ2ð Þj , which is a replacement for the usual

g<sup>0</sup> xj � �X<sup>∞</sup> k¼1

� � � g xj

g<sup>0</sup> xj � �X<sup>∞</sup> k¼1

ð Þx . These definitions coincide with the usual finite differences derivative when the function to which they act on is the linear function g xð Þ¼ a<sup>0</sup> þ a1x, a0, a<sup>1</sup> ∈ C. An exact finite differences derivative of other functions need of more terms than the one found in the usual definition of a

� � <sup>χ</sup>1ð Þ<sup>j</sup> <sup>¼</sup> <sup>g</sup><sup>0</sup> xj

> 1 k! gð Þ<sup>k</sup> xj � � Δ<sup>j</sup> � �<sup>k</sup>

� � <sup>¼</sup> <sup>1</sup>

� �<sup>N</sup>

ð Þ �<sup>1</sup> <sup>k</sup>�<sup>1</sup>

<sup>k</sup>! <sup>g</sup>ð Þ<sup>k</sup> xj

� �Δ<sup>k</sup>

<sup>1</sup> , is defined through the

<sup>j</sup>�<sup>1</sup>: (2)

ð Þx .

� �, (3)

, (4)

� �, (1)

$$\chi\_2(v,j) := \frac{1 - e^{-v \cdot \Delta\_{\dot{j}}}}{v} \cong \Delta\_{\dot{j}} - \frac{v}{2}\Delta\_{\dot{j}}^2 + O\left(\Delta\_{\dot{j}}^3\right). \tag{8}$$

Xm j¼n

<sup>j</sup> <sup>¼</sup> ghxjþ<sup>1</sup>

v hx ð Þ <sup>j</sup>þ<sup>1</sup> �h x ð Þ ð Þ<sup>j</sup> �<sup>1</sup>

� � <sup>¼</sup> <sup>1</sup> � <sup>e</sup>

where 1 ≤ n < m < N, and

where 1 < n < m ≤ N.

Df ghx ð Þ ð Þ � �

where

χ<sup>1</sup> v; Δh,j � � <sup>¼</sup> <sup>e</sup>

where χ<sup>2</sup> v; Δh,j�<sup>1</sup>

functions is

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> v ev xj <sup>¼</sup> <sup>X</sup><sup>m</sup>

Xm j¼n

Chain rule. The finite differences versions of the chain rule are

� � � � � ghxj

Df g � �

<sup>v</sup> and Δh,j ¼ h xjþ<sup>1</sup>

�v hx ð Þ ð Þ<sup>j</sup> �h xð Þ <sup>j</sup>�<sup>1</sup> � �=v,

ð Þ Dbg <sup>j</sup> ≔v

<sup>j</sup> <sup>¼</sup> gjþ<sup>1</sup>hjþ<sup>1</sup> � gj

<sup>¼</sup> gjþ<sup>1</sup> Df <sup>h</sup> � �

where 1 ≤ j < N. Also, for the backwards derivative, we have

and h xð Þ is any absolutely continuous complex function on ½ � a; b .

Df gh � �

<sup>j</sup> ≔v

ð Þ Dbghx ð Þ ð Þ <sup>j</sup> ¼ ð Þ Dbg <sup>j</sup>

ghxj

1 � e

hj <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> <sup>¼</sup> gjþ<sup>1</sup>

<sup>j</sup> <sup>þ</sup> hj Df <sup>g</sup> � �

The derivative of a product of functions. The exact finite differences derivative of a product of

� � � � � ghxj�<sup>1</sup>

�v hx ð Þ ð Þ<sup>j</sup> �h xð Þ <sup>j</sup>�<sup>1</sup>

hjþ<sup>1</sup> � hj <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> <sup>þ</sup>

<sup>j</sup> <sup>¼</sup> <sup>e</sup>�<sup>v</sup> <sup>Δ</sup><sup>j</sup>

gjþ<sup>1</sup> � gj <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> hj

gjþ<sup>1</sup>ð Þ Dbh <sup>j</sup>þ<sup>1</sup> <sup>þ</sup> hj Df <sup>g</sup> � �

� � � � <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> <sup>¼</sup> <sup>v</sup>

j¼n

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> v ev xj <sup>¼</sup> <sup>e</sup>

χ1ð Þ v; j Dfe

ghxjþ<sup>1</sup>

� � � � � ghxj

� �, and

χ<sup>2</sup> v; Δh,j�<sup>1</sup> � �

� � � �

e

<sup>¼</sup> Df <sup>g</sup> � � j

ghxjþ<sup>1</sup>

� � � h xj

e

v x � �

v xm � <sup>e</sup>

� � � � � ghxj

χ<sup>1</sup> v; Δh,j � � <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> ,

v hx ð Þ ð Þ <sup>j</sup>þ<sup>1</sup> �h xð Þ<sup>j</sup> � <sup>1</sup>

� � � �

� � � �

e

v hx ð Þ ð Þ <sup>j</sup>þ<sup>1</sup> �h xð Þ<sup>j</sup> � <sup>1</sup> , (15)

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> , (16)

: (17)

j , (18)

<sup>j</sup> ¼ e

v xmþ<sup>1</sup> � <sup>e</sup>

v xn , (12)

v xn�<sup>1</sup> , (13)

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956

> v hx ð Þ ð Þ <sup>j</sup>þ<sup>1</sup> �h xð Þ<sup>j</sup> � <sup>1</sup> v χ1ð Þ v; j

(14)

167

Note that we recover the usual definitions of a finite differences derivative in the limit Δ<sup>j</sup> ! 0 (N ! ∞) in which case χ1ð Þ¼ v; j χ2ð Þ! v; j Δj. Hereafter, the exact finite differences derivatives that we will consider are

$$(D\_b g)\_j \coloneqq \frac{g\_j - g\_{j-1}}{\chi\_2(v, j-1)} \text{ and } \left(D\_f g\right)\_j \coloneqq \frac{g\_{j+1} - g\_j}{\chi\_1(v, j)},\tag{9}$$

with χ1ð Þ v; j and χ2ð Þ v; j given in Eqs. (7) and (8), and some properties of these definitions follow. There is a plot of χ1ð Þ v; j in Figure 2. The spacing function χ1ð Þ v; j is defined for finite values of v and Δj.

The summation of a derivative. As is the case for continuous systems, the summation is the inverse operation to the derivative,

$$\sum\_{j=n}^{m} \chi\_1(v,j) \left( D\_f \mathbf{g} \right)\_j = \sum\_{j=n}^{m} \left( \mathbf{g}\_{j+1} - \mathbf{g}\_j \right) = \mathbf{g}\_{m+1} - \mathbf{g}\_{n'} \tag{10}$$

where 1 ≤ n < m < N, and

$$\sum\_{j=n}^{m} \chi\_2(v, j-1)(D\_b \mathbf{g})\_j = \mathbf{g}\_m - \mathbf{g}\_{n-1'} \tag{11}$$

where 1 < n < m ≤ N.

The exponential function is also an eigenfunction of the summation operation. The usual integral of the exponential function also has its equivalent expression in exact finite differences terms

Figure 2. Three-dimensional plot of <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> for the exponential function ev x.

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 167

$$\sum\_{j=n}^{m} \chi\_1(v,j)v \, e^{v \cdot x\_j} = \sum\_{j=n}^{m} \chi\_1(v,j) \left( D\_f e^{v \cdot x} \right)\_j = e^{v \cdot x\_{n+1}} - e^{v \cdot x\_n} \,. \tag{12}$$

where 1 ≤ n < m < N, and

$$\sum\_{j=n}^{m} \chi\_2(v, j-1)v \; e^{v \; \ge \; v \; \ge} = e^{v \; \ge n} - e^{v \; \ge n-1},\tag{13}$$

where 1 < n < m ≤ N.

Chain rule. The finite differences versions of the chain rule are

$$\begin{split} \left( D\_{f} \mathbf{g} (h(\mathbf{x})) \right)\_{j} &= \frac{\mathbf{g} \left( h(\mathbf{x}\_{j+1}) \right) - \mathbf{g} \left( h(\mathbf{x}\_{j}) \right)}{\chi\_{1}(\mathbf{v}, j)} = \upsilon \frac{\mathbf{g} \left( h(\mathbf{x}\_{j+1}) \right) - \mathbf{g} \left( h(\mathbf{x}\_{j}) \right)}{\mathbf{e}^{\upsilon} \left( h(\mathbf{x}\_{j+1}) - h(\mathbf{x}\_{j}) \right)} \frac{\mathbf{e}^{\upsilon} \left( h(\mathbf{x}\_{j+1}) - h(\mathbf{x}\_{j}) \right)}{\upsilon \chi\_{1}(\mathbf{v}, j)} - 1}{\upsilon \chi\_{1}(\mathbf{v}, j)} \\ &= \left( D\_{f} \mathbf{g} \right)\_{j} \frac{\chi\_{1}(\mathbf{v}, \Delta\_{h,j})}{\chi\_{1}(\mathbf{v}, j)} \end{split} \tag{14}$$

where

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> <sup>≔</sup> <sup>1</sup> � <sup>e</sup>�<sup>v</sup> <sup>Δ</sup><sup>j</sup>

ð Þ Dbg <sup>j</sup> <sup>≔</sup> gj � gj�<sup>1</sup>

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> Df <sup>g</sup> � �

Xm j¼n

Figure 2. Three-dimensional plot of <sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> for the exponential function ev x.

that we will consider are

166 Numerical Simulations in Engineering and Science

values of v and Δj.

inverse operation to the derivative,

where 1 ≤ n < m < N, and

where 1 < n < m ≤ N.

Xm j¼n

v

ffi <sup>Δ</sup><sup>j</sup> � <sup>v</sup> 2 Δ2

Note that we recover the usual definitions of a finite differences derivative in the limit Δ<sup>j</sup> ! 0 (N ! ∞) in which case χ1ð Þ¼ v; j χ2ð Þ! v; j Δj. Hereafter, the exact finite differences derivatives

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> and Df <sup>g</sup> � �

with χ1ð Þ v; j and χ2ð Þ v; j given in Eqs. (7) and (8), and some properties of these definitions follow. There is a plot of χ1ð Þ v; j in Figure 2. The spacing function χ1ð Þ v; j is defined for finite

The summation of a derivative. As is the case for continuous systems, the summation is the

The exponential function is also an eigenfunction of the summation operation. The usual integral of the exponential function also has its equivalent expression in exact finite differences terms

gjþ<sup>1</sup> � gj � �

<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup> j¼n

<sup>j</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>3</sup> j � �

<sup>j</sup> <sup>≔</sup> gjþ<sup>1</sup> � gj

<sup>χ</sup>2ð Þ <sup>v</sup>; <sup>j</sup> � <sup>1</sup> ð Þ Dbg <sup>j</sup> <sup>¼</sup> gm � gn�<sup>1</sup>, (11)

: (8)

<sup>χ</sup>1ð Þ <sup>v</sup>; <sup>j</sup> , (9)

<sup>¼</sup> gmþ<sup>1</sup> � gn, (10)

$$\mathbf{g}\left(\mathbf{D}\_{\mathbf{f}}\mathbf{g}\right)\_{\mathbf{j}} := \upsilon \frac{\mathbf{g}\left(\mathbf{h}\left(\mathbf{x}\_{\mathbf{j}+1}\right)\right) - \mathbf{g}\left(\mathbf{h}\left(\mathbf{x}\_{\mathbf{j}}\right)\right)}{\mathbf{e}^{\upsilon}\left(\mathbf{h}\left(\mathbf{x}\_{\mathbf{j}+1}\right) - \mathbf{h}\left(\mathbf{x}\_{\mathbf{j}}\right)\right)},\tag{15}$$

$$\chi\_1(\mathbf{v}, \Delta\_{b,j}) = \frac{\varepsilon^{\frac{r}{k}\left(\mathbf{z}\_{j+1}\right) - \mathbf{z}\left(\mathbf{z}\_j\right)} - 1}{v} \text{and } \Delta\_{b,j} = h\left(\mathbf{x}\_{j+1}\right) - h\left(\mathbf{x}\_j\right), \text{ and}$$

$$(\mathbf{D}\_b \mathbf{g}(h(\mathbf{x})))\_j = (\mathbf{D}\_b \mathbf{g})\_j \frac{\chi\_2\left(\mathbf{v}, \Delta\_{b,j-1}\right)}{\chi\_2\left(\mathbf{v}, j-1\right)}, \tag{16}$$

where χ<sup>2</sup> v; Δh,j�<sup>1</sup> � � <sup>¼</sup> <sup>1</sup> � <sup>e</sup> �v hx ð Þ ð Þ<sup>j</sup> �h xð Þ <sup>j</sup>�<sup>1</sup> � �=v,

$$g(D\_b g)\_j := \upsilon \frac{g\left(h\left(\mathbf{x}\_j\right)\right) - g\left(h\left(\mathbf{x}\_{j-1}\right)\right)}{1 - e^{-\upsilon \left(h\left(\mathbf{x}\_j\right) - h\left(\mathbf{x}\_{j-1}\right)\right)}}.\tag{17}$$

and h xð Þ is any absolutely continuous complex function on ½ � a; b .

The derivative of a product of functions. The exact finite differences derivative of a product of functions is

$$\begin{split} \left( \left( D\_{f} \text{g} h \right)\_{j} = \frac{\mathbf{g}\_{j+1} h\_{\hat{f}+1} - \mathbf{g}\_{j} h\_{\hat{f}}}{\chi\_{1}(\mathbf{v}, \boldsymbol{j})} = \mathbf{g}\_{j+1} \frac{h\_{\hat{f}+1} - h\_{\hat{f}}}{\chi\_{1}(\mathbf{v}, \boldsymbol{j})} + \frac{\mathbf{g}\_{j+1} - \mathbf{g}\_{j}}{\chi\_{1}(\mathbf{v}, \boldsymbol{j})} h\_{\hat{f}} \\ = \mathbf{g}\_{j+1} \left( D\_{\hat{f}} h \right)\_{j} + h\_{\hat{f}} \left( D\_{\hat{f}} \mathbf{g} \right)\_{j} = e^{-v \cdot \Delta\_{\hat{f}}} g\_{j+1} (D\_{b} h)\_{j+1} + h\_{\hat{f}} \left( D\_{\hat{f}} \mathbf{g} \right)\_{j'} \end{split} \tag{18}$$

where 1 ≤ j < N. Also, for the backwards derivative, we have

$$(D\_b \text{gh})\_j = \mathcal{g}\_j (D\_b \text{h})\_j + h\_{j-1} (D\_b \text{g})\_j = \mathcal{g}\_j (D\_b \text{h})\_j + e^{\nu\_j \Delta\_{j-1}} h\_{j-1} \left( D\_f \text{g} \right)\_{j-1'} \tag{19}$$

where 1 < j ≤ N.

The derivative of the ratio of two functions. For the finite differences, backward derivative of the ratio of two functions we have

$$\begin{split} \left( D\_{b} \mathfrak{z}\_{\overline{h}}^{\mathbb{C}} \right)\_{j} = \frac{1}{\chi\_{2} \left( \upsilon, j-1 \right)} \left( \frac{\mathfrak{z}\_{j}}{h\_{\overline{j}}} - \frac{\mathfrak{z}\_{j-1}}{h\_{\overline{j}-1}} \right) = \frac{1}{\chi\_{2} \left( \upsilon, j-1 \right)} \left( -\frac{\mathfrak{z}\_{j} \left( h\_{\overline{j}} - h\_{\overline{j-1}} \right)}{h\_{\overline{j}} h\_{\overline{j-1}}} + \frac{\left( \mathfrak{z}\_{\overline{j}} - \mathfrak{z}\_{j-1} \right) h\_{\overline{j}}}{h\_{\overline{j}} h\_{\overline{j-1}}} \right) \\ = \frac{(D\_{b} \mathfrak{g})\_{j}}{h\_{\overline{j-1}}} - \mathfrak{g}\_{j} \frac{(D\_{b} h)\_{\overline{j}}}{h\_{\overline{j}} h\_{\overline{j-1}}}, \end{split} \tag{20}$$

$$\left(D\_b \frac{\mathcal{g}}{h}\right)\_j = \frac{(D\_b \mathcal{g})\_j}{h\_j} - \mathcal{g}\_{j-1} \frac{(D\_b h)\_j}{h\_j h\_{j-1}}.\tag{21}$$

summation by parts results can be used in the finding of an appropriate momentum operator for discrete quantum systems. The summation by parts relates two operators between them-

It is advantageous to use a matrix to represent the finite differences derivative on the whole interval so that we can consider the whole set of derivatives on the partition at once. Let us consider the backward and forward exact finite differences derivative matrices Db,f given by

We have used the definition for the backward derivative ð Þ Dbg <sup>j</sup> for all the rows of the backward derivative matrix Db but not for the first line in which we have instead used the forward

The matrix formulation of the derivative operators allows the derivation of some useful results

Many properties can be obtained with the help of the derivative matrices Db,f . Expressions for the exact second finite differences derivative associated to the exponential function are

obtained through the square of the derivative matrices Db,f . These expressions are

<sup>T</sup>

<sup>1</sup>. A similar thing was done for the forward derivative matrix Df . These

∈ C<sup>N</sup>.

ð28Þ

169

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956

ð29Þ

4. The matrix associated to the exact finite differences derivative

selves and with boundary conditions on the functions.

and

derivative Df g

for the derivative itself.

4.1. Higher order derivatives

matrices act on bounded vectors g ¼ g1; g2; ⋯; gN

$$\left(D\_{\hat{f}}\frac{\mathcal{g}}{h}\right)\_{\hat{f}} = \frac{\left(D\_{\hat{f}}\mathcal{g}\right)\_{\hat{f}}}{h\_{\hat{f}+1}} - \mathbf{g}\_{\hat{f}}\frac{\left(D\_{\hat{f}}h\right)\_{\hat{f}}}{h\_{\hat{f}}h\_{\hat{f}+1}},\tag{22}$$

$$\left(D\_f \frac{\mathcal{S}}{h}\right)\_j = \frac{\left(D\_f g\right)\_j}{h\_j} - g\_{j+1} \frac{\left(D\_f h\right)\_j}{h\_j h\_{j+1}}.\tag{23}$$

Additional properties. A couple of equalities that will be needed below are

$$\frac{1}{\chi\_2(v,\vec{j})} - \frac{1}{\chi\_1(v,\vec{j})} = v,\text{ and }\frac{\chi\_1(v,\vec{j})}{\chi\_2(v,\vec{j})} = e^{v\cdot\Delta\_{\vec{j}}}.\tag{24}$$

For instance, these equalities imply that

$$(D\_{\!\!\!/} \mathbf{g})\_{\!\!\!/} = e^{-v \cdot \Delta\_{\!\!/}} (D\_{\!\!\!/} \mathbf{g})\_{\!\!\!/ + 1}. \tag{25}$$

Summation by parts. An important result is the summation by parts. The sum of equalities (18) and (19) combined with equalities (10) and (11) provide the exact finite differences summation by parts results,

$$\sum\_{j=n}^{m} \chi\_2(\mathbf{v}, \mathbf{j}) \mathbf{g}\_{j+1}(D\_b h)\_{j+1} + \sum\_{j=n}^{m} \chi\_1(\mathbf{v}, \mathbf{j}) h\_j(D\_f \mathbf{g})\_j = \mathbf{g}\_{m+1} h\_{m+1} - \mathbf{g}\_n h\_n \tag{26}$$

where 1 ≤ j < N, and

$$\sum\_{j=n}^{m} \chi\_2(\mathbf{v}, j-1) \mathbf{g}\_j(D\_b h)\_j + \sum\_{j=n}^{m} \chi\_1(\mathbf{v}, j-1) h\_{j-1} \{D\_f \mathbf{g}\}\_{j-1} = \mathbf{g}\_m h\_m - \mathbf{g}\_{n-1} h\_{n-1} \tag{27}$$

where 1 < j ≤ N.

The integration by parts theorem of continuous functions is the basis that allows to define adjoint, symmetric and self-adjoint operations for continuous variables [8, 9]. Therefore, the summation by parts results can be used in the finding of an appropriate momentum operator for discrete quantum systems. The summation by parts relates two operators between themselves and with boundary conditions on the functions.
