Introduction

The \ssubject of finite differences is an old, but  useful method with wide application in science and engineion{introduction}

Fering. The story starts with Newton and Leibnitz themselves with the very  definition of the derivative of a function; the limit when the differences become zero, as is well known. The fom calculus of a single variable, the forrmal definition of the derivative \( f'(x_0) \$f'(x)$) of some function \( f(x) \$f(x)$) of a single variable \( x \$x$), at \( x_0 \$x_0$),requ is obtained fromres of taking the limit \( x\to x_0 \)$x\t of any of the following ratiox_0$ of a ratios of differences

      \( \displaystyle f'(x_0) :=\lim_{x\to x^-_0} \frac{f(x_0)-f(x)}{x_0-x}, \quad f'(x_0) :=\lim_{x\to x^+_0} \frac{f(x)-f(x_0)}{x-x_0}. \

%

\[)f'(x)

These r:=\lim_{x\atios o x_0}

\fare known as backward and c{f(x)-f(x_0)}{x-x_0}.

\]

%

Hforward differences, respectively.

Tever, there are at lhaste limit of vanishing finite differences involves variablestwo situations in which are continuous. However, many timewe need to compute the derivative whens it is not possible to perform suchtake the mentioned limit  at all. $x\to x_0$. OneA situation like that appears when doing is nnumerical simulations in a computer due to the limit in how small a number can be represented in the computer. Another situation in which computation. When doing numerical calculations, iit is not possible to take the infinitesimal differencperform ae limit is wheliken the independent variable is a discrete variable by nature, as is the case found in Quantum Mechanics theory regarding the spectrum of quantum operators.

finitesimal separation between the discrete set of poFinite differences are necessary in numerical computations tos of a mesh, and then an approximate several quantities likeion to the derivatives and i is used.

Onhtegrals of functions, as well as, differential and integral equations. Their use as a numerical tool is a well developed subject, see for instance the series of books Numerical Recipes.^[1] An ir situation is in the theory of operators. there are operators with a discrete spectrum. The values of nteresting development is the use of a complex finite differences to evaluate the real derivative of a function.^[2]

On tspectrum of the operator are fixhde other hand, the reader can learn about the calculus of finite differences  from the classical works of Kopal^[3], Boole^[4number and there is no meaning in taking a limit like $x\to x_0$^]. Jordan^[5], RiTcharudson^[6], for instance. 

Another ban exranch of thte finite differences tree, is known as the exact, a finite differences technique. That scheme was developed by researchers like Potts,^[7][8which will give the exact derivative even if $\Delta$ has^] Raonald E. Mickens^[9] y finwith the purpose of obtaining evalue, is desirable.

Usuxaclt finite differences representations of continuous differential equations and of their solutions.

Ancalculus, for equally spaced poinother use of finite differences wa in a mesh, is wells develope^[1][2][3][4],d by Armando MartíHnez-Pérez and Gabino Torres-Vega^[10]re with the intention of obtaining discrete operators for use in Quantum Mechanics theory. It is common that a quantum operator has a discrete spectrum and a roduce finite differences calculus which is exact for the first dderivative with respect to the spectrum is necessary some times. The aim is to obtain discrete operators which comply with discrete versions of the properties that a quantum operator must have.

The exp. We specialize the results to finite intervals, flicrsit form of the exact finorder finite difference derivative depends on the functions that we want to consider. In this article, we discuss the use of an exact finite differences derivative when its and to the real exponential function, the reals eigenfunction is the real expoof the derivative.

\snential function. This will provide with a momentum operator to be used in discrete Quantum Mechanics.

{The partition} 

The partition

Let \( \{q_{i}\}_{i=-N}^{N} \$\{q_{i}\}_{i=-N}^{N}$) be a \( N+1 \) points partition of the real interval \( [-a,a] \$[-a,a]$), that is

%

\begin{align*}

a=q_{-N} < q_{-N+1} <  \dots      < q_{N-1} <  \( \displaystyle -a=q_{-N} < q_{-N+1} < \dots < q_{N-1} < q_{N}=a. \q_{N}=b.

\end{align*})%

The separations between mesh points are

          \( \displaystyle \Delta_j = q_{j+1}-q_{j}, \quad j = -N, \dots, N-1. \)

T

%

\bhese separagtions can or cannot be equal and are finin{align*}

\Dte. The discrlete variable $q_j$ is the independent variable with respect to we will calculate ta_j = q_{j+1}-q_{j}, \quad j = -N, \dots, N-1.

\hen derivative of a function.

The exact finite differences derivative

{align*}

%

For a given \( v\in\mathbb{R} \),$v\in\mathb backward an{R}$, ad forward finite differences , first order, first  derivatives, of a vector \( g=(g_{-N}, g_{-N+1},\dots,g_{N-1},g_N)^T \$g=(g_{-N}, g_{-N-1},\dots,g_{N-1},g_N)^T$), at \( q_j \$q_j$), defined on the partition, are

is 

%

\[       \( \displaystyle (D_bg)_j=\frac{g_{j}-g_{j-1}}{\eta_{j}}, \)

          \( \displaystyle (D_fg)_j=\frac{g_{j+1}-g_j}{\chi_{j}}, \)

wh(D_fg)_j=\fere ac{g_{j+1}-g_j}{\cthe denominatorsi_{f,j}},

\qu are d

\chi_{f,j}=\efinedr as

          \( \displaystyle \eta_{j}=\frac{2}{v}\,e^{-v\Delta_{j-1}/2} \sinh\left(\frac{v}{2}\Delta_{j-1}\right) =\Delta_{j-1}+\frac{v}{2}\Delta^2_{j-1}+{\cal O}(\Delta^3_j), \)

          \( \displaystyle \chi_{j}=\frac{2}{v}\,e^{v\Delta_{j}/2} \sinh\left(\frac{v}{2}\Delta_{j}\right) =\Delta_j+\frac{v}{2}\Delta^2_j+{\cal O}(\Delta^3_j). \)

The disc{rete variable \( q_j \) is^{v\Del ta_j}-1}{v}

=\Dhe independelnt variable with respect to which we will calculate the derivative of a function.a_j+\frac{v}{2}\Delta^2_j+{\cal O}(\Delta^3_j). \]

%

The functions in the denominators of thesise expressions, the functions \( \eta_{j} \) and \( \chi_{j} \$\chi_{f,j}$), makes sure that the real exponential function \( e^{vq} \$e^{vq}$) be, exactly, an eigenfunction of the discrete derivative operation with real eigenvalue \( v \$v$,

%

\begin{align*}

(D_fe^{vq})_j=v\,e^{vq_j}),

          \( \displaystyle (D_be^{vq})_j=(D_fe^{vq})_j=v\,e^{vq_j}, \

\end{align*})%

which is the same property afs the continuous variable derivative has.

When \( |\eta_{j}-\Delta_j| \)$|\chi_{f,j}-\Delt an_j|$d \( |\chi_{j}-\Delta_j| \is) are less than some small \( \epsilon \$\epsilon$), our method and the usual finite-differences derivativ wille give similar results for any vector defined on the mesh

A backwards version. Note that the denominatof the finite differences derivative, at $q_j$, ir

%

\[s \( \eta_{j} \)(D_bg)_j=\frac{g_j-g_{j-1}}{\chi_{b,j}},

\qu an

\chi_{b,j}=\frac{1-e^{-v\Delta_{j-1}}}{v}

=\Delta_{j-1}-\frac{v}{2}\Delta^2_{j-1}

d \( \chi_{j} \+{\cal) O}(\Dblta^3_{j-1}), \]

%

whieh alscom compliese \( \Delta_j \) itw

%

\bhgin{aligen \( v=0 \*}

(D_be^{vq})_j=v\,e^{vq_j})

\end{align*}.

Properties of the finite-differences derivativ

\subsection{Properties}

e

Some properties of the exact finite-differences derivative are

T{\hme connection between forward and backward discrete derivatives. The equalit\/}

%

$y \( \displaystyle \eta_{j+1}=e^{-v\Delta_j} \chi_{j} \\frac{\chi_{b,j+1}}{\chi_{f,j}}

=e^{-v\Delta_j}

$) implies tht

%

\[ (D_fg)_j

=e^{-v\Delaa_j}\,(D_bg)_{j+1}

. \]t

{\em Two discrete versions of the        summation of the derivative.\/}  \( (D_fg)_j=e^{-v\Delta_j}\,(D_bg)_{j+1}. \)

The summation of the derivative. The finite differences versions of \( \displaystyle \int_a^xdy\,g'(y)=g(x)-g(a) \$\int_a^xdy\,g'(y)=g(x)-g(a)$ are

%

\begin{align*}

&\sum_{j=-N}^n\Delta_j(D_fg)_j

%

&=

-\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}

+\sum_{j=-N+1}^n\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

)-\fr ac{\Delta_j}{\chi_{f,j}}\right)g_j

+\frac{\Dre

\( \displaystyle \sum_{j=-N}^n\Delta_j(D_fg)_j =-\frac{\Delta_{-N}}{\chi_{-N}}g_{-N} +\sum_{j=-N+1}^n\left( \frac{\Delta_{j-1}}{\chi_{j-1}} -\frac{\Delta_j}{\chi_{j}}\right)g_j +\frac{\Delta_n}{\chi_{n}}g_{n+1}, \lta_n}{\chi_{f,n}}g_{n+1}

,

\end{align*}

%

and

%

\begin{align*}

&\Delta_{-N}(D_fg)_{-N}

+\sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_j

%

&=

-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)\Delta_{-N}g_{-N}

+\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}

+\sum_{j=-N+1}^{n-1}\left(

\frac{\Delta_{j-1}}{\chi_{b,j}}) -\frac{\Delta_j}{\chi_{b,{j+1}}}\right)g_j

+\fr

ac{\Delta_{n-1}}{\chi_{b,n}}g_n

%,

\ennd\( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_j =-\frac{\Delta_{-N}}{\eta_{-N+1}}g_{-N} +\sum_{j=-N+1}^{n-1}\left( \frac{\Delta_{j-1}}{\eta_{j}} -\frac{\Delta_j}{\eta_{{j+1}}}\right)g_j +\frac{\Delta_{n-1}}{\eta_{n}}g_n, \{align*})%

where \( \)\( n \$n)<\( N \N$). The summation term at the right hand side of thiese equalitieys involve asymmetrinterferencey terms that vanish when the separation between mesh points is the same For the whole interval, we have.%

\begin{align*}

&

\sum_{j=-N}^{N-1}\Delta_j(D_fg)_j

+\Delta_{N-1}(D_bg)_N

\\

%

&=

-\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}

+\sum_{j=-N+1}^{N-1}\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

-\frac{\Delta_j}{\chi_{f,j}}\right)g_j

-\frac{\Delta_{N-1}}{\chi_{b,N}}g_{N-1}+\left(\frac{1}

{\chi_{f,N-1}}

+\frac{1}{\chi_{b,N}}\right)\Delta_{N-1}g_N

,

\end{align*}

%

\begin{align*}

&

\Delta_{-N}(D_fg)_{-N}

+\sum_{j=-N+1}^N\Delta_{j-1}(D_bg)_j\\

&=-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)

\Delta_{-N}g_{-N}

+\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}

+\sum_{j=-N+1}^{N-1}\left(

\frac{\Delta_{j-1}}{\chi_{b,j}}

-\frac{\Delta_j}{\chi_{b,j+1}}\right)g_j

+\frac{\Delta_{N-1}}{\chi_{b,N}}g_N

,

\end{align*}

The eigenfunction of the summation.{\em The eigenfunction of the summation.\/} The finite differences versions of \( \displaystyle \int_a^xdx\,v\,e^{vx}=e^{vx}-e^{va} \)$\int_a^xdx\,v\,e^{vx}=e^{va}-e^{vx}$ are

%

\begin{align*}

\sum_{j=-N}^n\Delta_jv\,e^{vq_j}

&=\sum_{j=-N}^n\Delta_j(D_f\,e^{vq})_j\\

%

&=-\frac{\Delta_{-N}}{\chi_{f,-N}}e^{vq_{-N}}

+\sum_{-N+1}^n\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

-\frac{\Delta_j}{\chi_{f,j}}\right)e^{vq_j}

+\frac{\Delta_n}{\chi_{f,n}}e^{vq_{n+1}},

\end{align*}

%

and

%

\begin{align*}

&\Delta_{-N}v\,e^{vq_{-N}}

+\sum_{j=-N+1}^n\Delta_{j-1}v\,e^{vq_j}

%=\Delta_{-N}(D_f\,e^{vq})_{-N}

% +\sum_{j=-N+1}^n\Delta_{j-1}(D_b\,e^{vq})_j\\

%

&=-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)

\Delta_{-N}e^{vq_{-N}}

+\frac{\Delt _{-N}}{\chi_{f,-N}}e^{vq_{-N+1}}

+\sum_{j=-N+1}^{n-1}\left(

\faac{\Dre\( \displaystyle \sum_{j=-N}^n\Delta_jv\,e^{vq_j} =\sum_{j=-N}^n\Delta_j(D_f\,e^{vq})_j =-\frac{\Delta_{-N}}{\chi_{-N}}e^{vq_{-N}} +\sum_{-N+1}^n\left( \frac{\Delta_{j-1}}{\chi_{j-1}} -\frac{\Delta_j}{\chi_{j}}\right)e^{vq_j} +\frac{\Delta_n}{\chi_{n}}e^{vq_{n+1}}, \lta_{j-1}}{\chi_{b,j}}) -\frac{\Delta_j}{\chi_{b,j+1}}\right)e^{vq_j}

+\fr

ac{\Delta_{n-1}}{\chi_{b,n}}e^{vq_n},

\ennd\( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}v\,e^{vq_j} =\sum_{j=-N+1}^n\Delta_{j-1}(D_b\,e^{vq})_j =-\frac{\Delta_{-N}}{\eta_{-N+1}}e^{vq_{-N}} +\sum_{j=-N+1}^{n-1}\left( \frac{\Delta_{j-1}}{\eta_{j}} -\frac{\Delta_j}{\eta_{j+1}}\right)e^{vq_j} +\frac{\Delta_{n-1}}{\eta_{n}}e^{vq_n}, \{align*}

%

)where \( -N \$-N)<\( n \n)<\( N \N$).

The derivative of a constant function \( c \{\em The derivative of one).\/}

%

\begin{align*}

(D_f\,1)_j=0,

\quad\text{and}\quad

(D_b\,1)_j=0,

\end{align*} 

{\em          \( (D_f\,c)_j=0, \quad\text{and}\quad (D_b\,c)_j=0. \The derivatives of $q$.\/}

%

\begin{align*}

(D_f\,q)_j

=\frac{q_{j+1}-q_j}{\chi_{f,j}}

=\frac{\Delta_j}{\chi_{f,j}},

\quad\text{and}\quad

(D_b\,q)_j

=\frac{q_j-q_{j-1}}{\chi_{b,j}})

The derivatives of \( q \=\frac{\Delta_{j-1}}{\chi_{b,j}}).

\( \displaystyle (D_f\,q)_j =\frac{q_{j+1}-q_j}{\chi_{j}} =\frac{\Delta_j}{\chi_{j}}, \quad\text{and}\quad (D_b\,q)_j =\frac{q_j-q_{j-1}}{\eta_{j}} =\frac{\Delta_{j-1}}{\eta_{j}}. \

\end{align*})%

These derivatives will approach to one in the limit of small \( \Delta_j \$\Delta_j$)

{\em The. chaiIn rule.\/}

%

\begin{parltgn*}

(D_fg(h(q)))_j

=({\icau D}_fg)_j\,(D_fh)_j,

\end{llign*}

%

wheare

%

\begin{align*}

({\cal,D}_fg)_j

=\frac{g(h(q_{j+1}))-g(h(q_j))}{h(q_{j+1})-h(q_j)}.

\end{align*}

%

and

%

\ boegin{align*}

(D_bg(t(q)))_j

=({\calh \( (D_f\,q)_j \) D}_bg)_j\,(D_bh)_j,

\eand \( (D_b\,q)_j \){ lign*}

%

whear

%

\begin{align*}

({\caleD}_bg)_j

=\frac{g(h(q_j))-g(h(q_{j-1}))}{\chi_{b,j}}.

\ eqnd{ual to oign*}

{\nme wheThen \( v=0 \).

The chain rule. Fdeoriva the forward sive of a product of vectors.\/}

%

\begin{align*}

(D_f\,gh)_j

=g_{j+1}(D_fh)_j+(D_fg)_jc_j

.

\hend{align*}

%

\bme gin{align*}

(D_f\,gh)_j

=(D_fg)_jh_{j+1}+g_j(D_ft)_j

.

\end{alhis rgn*}

%

\begin{align*}

(D_b\,gh)_j

=g_j(D_bh)_j+(D_bg)_jh_{j-1}

.

\end{auign*}

%

\ble tugin{aligrns to*}

(D_b\,gh)_j

=(D_bg)_jh_j+g_{j-1}(D_ h)_j

.

\bnd{align*}e

{\em        The derivative of the  \( \displaystyle (D_fg(h(q)))_j=({\cal D}_fg(h))_j\,(D_fh)_j, \)

winvhesre

  of        \( \displaystyle ({\cal D}_fg(h))_j =\frac{g(h(q_{j+1}))-g(h(q_j))}{h(q_{j+1})-h(q_j)}. \)

a vectFor.\/}

%

\begin{align*}

\lef (D_f\,\frac{1}{the }\right)_j

=-\frback{(D_fh)_j}{h_jh_{j+1}},

%

\quward finite\text{and}\qua

%

\left(D_b\,\frac{1}{h}\rdght)_j

=-\iffeac{(D_bh)_j}{h_jh_{j-1}}.

\rend{align*}

{\cems we hThe derivatiave

of  a    ratio of  \( \displaystyle (D_bg(h(q)))_j=({\cal D}_bg(h))_j\,(D_bh)_j, \vectors.\/}

%

\begin{align*}

\left(D_f\,\frac{g}{h}\right)_j

&=\frac{(D_f\,g)_j}{h_j}) -\frac{(D_fh)_j}{h_jh_{j+1}}g_{j+1}

%=\frac{h_{j+1}(D_f\,g)_j-(D_fh)_jg_{j+1}}{h_jw_{j+1}}

.

%

\quad\thxt{oe}\quad

%

\lrft(D_f\,\frac{g}{h}\right)_je

=\frac{(D_f\,g)_j}{h_{j+1}}

          \( \displaystyle ({\cal D}_bg(h))_j =\frac{g(h(q_j))-g(h(q_{j-1}))}{h(q_{j})-h(q_{j-1})}. \)

The derivative of a product of vectors. -\frac{(D_fh)_j}{h_jT_{j+1}}g_j\\

.

\end{align*}

%

and

%

\begin{align*}

\lhft(D_b\,\fere arac{g}{h}\right)_j

=\frac{(D_b\,g)_j}{h_{j-1}}

e-\ rac{(D_bh)_j}{h_jh_{j-1}}g_j

,

%

\quad\text{four equ}\quad

%

\aeft(D_b\,\frac{g}{h}\rlight)_j

=\frac{(D_b\,g)_j}{h_j}

-\frac{(D_bh)_j}{h_jh_{j-1}}g_{j-1}

.

\end{altgn*}

{\ies

m  Summation       \( (D_f\,gh)_j=g_{j+1}(D_fh)_j+(D_fg)_jh_j., \by parts.\/}

%

\begin{align*})

         \( \displaystyle (D_f\,gh)_j=(D_fg)_jh_{j+1}+g_j(D_fh)_j, \&\sum_{j=-N}^{N-1}\Delta_jg_{j+1}(D_fh)_j)

         \( \displaystyle (D_b\,gh)_j=g_j(D_bh)_j+(D_bg)_jh_{j-1}, \+\Delta_{N-1}g_N(D_bh)_N)

          \( \displaystyle (D_b\,gh)_j=(D_bg)_jh_j+g_{j-1}(D_bh)_j. \)

The derivative of a vector of inverses.

         \( \displaystyle \left(D_f\,\frac{1}{h}\right)_j=-\frac{(D_fh)_j}{h_jh_{j+1}}, \quad\text{and}\quad \left(D_b\,\frac{1}{h}\right)_j=-\frac{(D_bh)_j}{h_jh_{j-1}}, \+\sum_{j=-N}^{N-1}\Delta_j(D_fg)_jh_j)+\Delta_{N-1}(D_bg)_Nh_{N-1}

pro=\text{b.vi.t},

\dned \( h_j,h_{j\pm 1} \neq 0 \).

The derivative of a vector of ratios. {align*}

%

wThere

%

\begin{ align*}

&\text{b.i.t.}

=\sum_{j=-N}^{N-1}\Drlta_j(D_f\,gh)_j

e+\Delta_{N-1}(D_b\,gh)_N\\

&=-\ four versioac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}h_{-N}

+\num_{-N+1}^{N-1}\left(

s \of thirac{\Delta_{j-1}}{\chi_{f,j-1}}

s -\fproperty

          \( \displaystyle \left(D_f\,\frac{g}{h}\right)_j=\frac{(D_f\,g)_j}{h_j} -\frac{(D_fh)_j}{h_jh_{j+1}}g_{j+1}, \)

          \( \displaystyle \left(D_f\,\frac{g}{h}\right)_j=\frac{(D_f\,g)_j}{h_{j+1}} -\frac{(D_fh)_j}{h_jh_{j+1}}g_j, \)

          \( \displaystyle \left(D_b\,\frac{g}{h}\right)_j=\frac{(D_b\,g)_j}{h_{j-1}} -\frac{(D_bh)_j}{h_jh_{j-1}}g_j, \)

        \( \displaystyle \left(D_b\,\frac{g}{h}\right)_j=\frac{(D_b\,g)_j}{h_j} -\frac{(D_bh)_j}{h_jh_{j-1}}g_{j-1}. \)

Summation by parts.

          \( \displaystyle \sum_{j=-N}^{N-1}\Delta_jg_{j+1}(D_fh)_j +\sum_{j=-N}^{N-1}\Delta_j(D_fg)_jh_j=\cal{I}_1, \)

ac{\Delta_j}{\chi_{f,j}}\right)h_jg_j

-\frac{\Delta_{N-1}}{\chi_{b,N}}w_{N-1}g_{N-1}\\

&\quad+\lhft(\feac{1}{\chi_{b,{N-1}}}

+\frac{1}{\chi_{b,N}}\right)\Delta_{N-1}h_Ng_N.

\re        \( \displaystyle {\cal I}_1 =\sum_{j=-N}^{N-1}\Delta_j(D_f\,gh)_j =-\frac{\Delta_{-N}}{\chi_{-N}}g_{-N}h_{-N} +\sum_{-N+1}^{N-1}\left( \frac{\Delta_{j-1}}{\chi_{j-1}} -\frac{\Delta_j}{\chi_{j}}\right)h_jg_j +\frac{\Delta_{N-1}}{\chi_{{N-1}}}h_Ng_N. \nd{align*}

%

)The terms in the summation vanish when \( \Delta_j\to0 \)$\Delta_j\to0$, i.eor. when the separation between the mesh points is the same. B

Also

%

\begin{align*}

&\Delut a_{-N}g_{-N+1}(D_fh)_{-N}

+\Dwe ltc_{-N}(D_fg)_{-N}h_{-N}

+\sum_{j=-N+1}^an \Delta_{j-1}g_j(D_bch)_j

+\oum_{j=-N+1}^n\Dse tlta_{j-1}(D_bg)_jh_{j-1}

\\

\\

%

&=

-\lhe vefctors \( \mathbf{h} \)(\fr and \( \mathbf{g} \)c{1}{\ch is s_{f,-N}}

+\frac{1}{\uchi_{b,-N+1}}\right)\Delt a _{-N}g_{-N}h_{-N}

+\frwayc{\Del tha_{-N}}{\chi_{f,-N}}g_{-N+1}h_{-N+1}

+\sum_{j=-N+1}^{N-1}\lefa(

t\frac{\Del ta_{j-1}}{\chi_{b,j}}

-\frac{\Dhe last ta_j}{\chi_{b,{j+1}}}\right)g_jh_j\\

&\qsum ad+\frac{\Deltvanis_{N-1}}{\chi_{b,N}}g_Nh_N

,

\hes. nd{aAlsign*}o

          \( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}g_j(D_bh)_j +\sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_jh_{j-1} =\cal{I}_2, \)

{\em        The commutator between $q$  \( \displaystyle \cal{I}_2 =-\frac{\Delta_{-N}}{\chi_{b,-N+1}}g_{-N}h_{-N} +\sum_{j=-N+1}^{N-1}\left( \frac{\Delta_{j-1}}{\chi_{b,j}} -\frac{\Delta_j}{\chi_{b,{j+1}}}\right)g_jh_j +\frac{\Delta_{N-1}}{\chi_{b,N}}g_Nh_N. \)

The commutator between \( q \) and \( D \$D$)\/}. The discrete version of the relationshi$\frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q)$p \( \displaystyle \frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q) \) becomes

          \( \displaystyle (D_f\,qh)_j-q_{j+1}(D_f\,h)_j=\frac{\Delta_j}{\chi_{j}}q_j, \)

          \( \displaystyle (D_b\,qh)_j-q_j(D_b\,h)_j=\frac{\Delta_j}{\eta_{j}}q_{j-1}. \%)\begin{align*}

Translation of the exponential function. (D_f\,qh)_j-q_{j+1}(D_f\,The di)_j

=\frasc{\Drelta_j}{\chi_{f,j}}q_j,

\ten {align*}

%

\bdegrivatin{align*}

(D_b\,qh)_j-q_j(D_b\,h)_j

=\frac{\Dve isl ta_j}{\chi_{b,j}}q_{j-1}

\hend{ali gen*}

{\neratomr of Ttranslations of the exponential functio.\/}n

          \( \displaystyle (e^{s\,D_{b,f}}e^{vq})_j=e^{v(q_j+s)}. \)

%

\bWe gwin{align*}

(e^{s\,D_{b,f}}e^{vq})_j

=e^{v(q_j+s)}

\end{align*}

{\em Thelbou need oded Fourier transform\/}f the bounded Fourier transform of a given function \( g(v) \$g(v)$) which is defined as

          \( \displaystyle \tilde g_j=(Fg)_j=\frac{1}{\sqrt{{\cal V}}} \int_{-{\cal V}/2}^{{\cal V}/2}e^{ivq_j}g(v)dv, \)%

\begin{aligon*}

\ tildhe mg_j=(Fg)_j

=\frac{1}{\esh.

Wqrt{2a}}

\int_{-{\caleV}/2}^{{\c also V}/2}e^{ivq_j}g(v)dv,

\e d{align*}

{\nm Theed of th discrete Fourier transform\/}e discrete Fourier transform of a vector \( h_j \) $whic_j$h is defined a

%

\begin{align*}s

\tilde    h(v)=({\cal F}h)(v)

=\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal      \( \displaystyle \tilde h(v)=({\cal F}h)(v) =\sum_{j=-N}^{N-1}\sqrt{\frac{\Delta_j}{\cal V}}\,e^{-ivq_j}h_j. \V}}\,e^{-ivq_j}h_j.

\end{align*}

%)

These transformations preserve the norm of vectors and functions

{\em. The conjugate of $D_f$}. CoIsidern the equiality

%

\begdistant case in wh{align*}

p({\icalh \( q_{j+1}-q_j=2\pi/(N-1) \) F}th)_j

&=\sum_{j=-N}^{N-1}\eqrt{\frac{\Dse transforms calta_j}{\cal V}}\,

n be ide^{-ipq_{j+1}}(-iD_fh)_j

+\sqrnt{\ified witrac{\Delta_{N-1}}{\cal V}}e^{-ipq_{N-1}}(-iD_bh)_N

h the u\\

&\qsuadl Fourier transform on \( [-{\cal V},{\cal V}] \+i\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\,)

and with the Fou(D_fe^{-ipq}h)_j

+rier\ series, respectively.

The adjoint of \( D \)qrt{\frac{\Delta_{N-1}}{\cal. ConV}}\,(D_be^{-sider tpq}h)_N.

\he eqnd{uality          \( \displaystyle v({\cal F}h)_j=\sum_{j=-N}^{N-1}\sqrt{\frac{\Delta_j}{\cal V}}\, e^{-ivq_{j+1}}(-iD_fh)_j +i\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\, (D_fe^{-ivq}h)_j. \gn*}

%)

Then, there is the relationship

%

\begin{align*}

p\leftrightarrow 

\begin{cases}

e^{-ip\Delta_j}(-iD_fh)_j,&  j=-N,\dots,N-1,\\

e^{ip\Delta_{N-1}}(-iD_bh)_N,&      \( \displaystyle p\leftrightarrow e^{-iv\Delta_j}(-iD_fh)_j,\quad j=-N,\dots,N-1, \j=N.

\end{cases}

\end{align*}

%)

providedin that the asymmeerference and boundatry term

s          \( \displaystyle \sum_{j=-N}^{N-1}\sqrt{\frac{\Delta_j}{\cal V}}\, (D_fe^{-ivq}h)_j, \)

vanishes.

The conjugate of \( q_j \{\em The conjugate of $q_j$)}. The relationship

%

\begin{align*}

q_j\tilde g_j

&=i\frac{1}{\sqrt{2a}} 

\int_{-{\cal V}/2}^{{\cal  V}/2}

  e^{ivq_j}\frac{d\,g(v)}{dv}dv

-i\frac{1}{\sqrt{2a}}

  e^{ivq_j}g(v)\Big|_{v=-{\cal  V}/2}^{{\cal  \( \displaystyle q_j\tilde g_j =i\frac{1}{\sqrt{{\cal V}}} \int_{-{\cal V}/2}^{{\cal V}/2} e^{ivq_j}\frac{d\,g(v)}{dv}dv -i\frac{1}{\sqrt{{\cal V}}} e^{ivq_j}g(v)\Big|_{v=-{\cal V}/2}^{{\cal V}/2} \V}/2}

\end{align*}

%)

indicates that

%

\begin{align*}

q_j\leftrightarrow          \( \displaystyle q_j\tilde g_j =i\frac{1}{\sqrt{{\cal V}}} \int_{-{\cal V}/2}^{{\cal V}/2} e^{ivq_j}\frac{d\,g(v)}{dv}dv -i\frac{1}{\sqrt{{\cal V}}} e^{ivq_j}g(v)\Big|_{v=-{\cal V}/2}^{{\cal V}/2} \

i\,\frac{d}{dv}

\end{align*}

%)

provided boundary term

      s    \( \displaystyle \frac{i}{\sqrt{{\cal V}}} e^{ivq_j}g(v)\Big|_{v=-{\cal V}/2}^{{\cal V}/2} \)

 vanish.

Eigenvetors of the coordinate operator

{\em

The normalized eigenvector of the coordinate operatoreigenvector\/} of the coordinate operator with eigenvalue \( q_n \$q_n$) in the \( v \$v$) representation is

          \( \displaystyle e_{q_n}(v)=\frac{e^{-ivq_n}}{\sqrt{\cal V}}. \)

T$he_{q_n}(v)=se fun^{-ivq_n}/\sqrt{\cal V}$.

Orcthions arormalitye orthonormal$\ it_{-{\caln a discV}/2}^{{\cal V}/2}dv\,

ret^*_{q_m}(v)

e se_{q_n}(v)

=\text{nse, i.e.nc} 

          \( \displaystyle \int_{-{\cal V}/2}^{{\cal V}/2}dv\,e^*_{q_m}(v) e_{q_n}(v)=\text{sinc} \left[\frac{\cal V}{2}(q_m-q_n)\right] \)

Now,\lef th[\frac{\cale conjugate to the coordinate eigenvector \( e_{q_n}(v) \)V}{2}(q_m-q_n)\r ight]$s

\( \displaystyle \)\( \displaystyle e_{q_n}(m) =\frac{1}{\sqrt{{\cal V}}}\int_{-{\cal V}/2}^{{\cal V}/2} e^{ivq_m}\frac{e^{-ivq_n}}{\sqrt{\cal V}}dv =\sqrt{\frac{{\cal V}}{2a}}\, \text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right], \)

and t{\hme orthonormality between theConjugate vectors.} The conjugatse vectors rea to $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$ ids

\( \displaystyle \sum_{m=-N}^N\Delta_me^*_{q_j}(m)e_{q_n}(m)=\sum_{m=-N}^N \text{sinc}\left[\frac{\cal V}{2}(q_m-q_j)\right]\, \text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right], \

%)\begin{

align*}

e_{q_n}(m)

=\f rac{1}{\esult whicqrt{2a}}\int_{-{\cal V}/2}^{{\cal V}/2}

h becomes^{ivq_m}\frac{e^{-ivq_n}}{\sqrt{\cal V}}dv

=\sqr the {\fKroneckeac{{\cal V}}{2a}}\,

r \tdeltxt{sinc}\left[\frac{\cala \( \delta_{jn} \) V}{2}(q_m-q_n)\rigwt],

\hen \( {\cal V}\to\infty \)d{align*}.

Eigenvectors of the derivative operator

Now{\em, Tthe normalize}d eigenvector of the discrete derivative with eigenector, conjugvaltue \( v \to)$e_{q_n}(v)=e^{- in tvq_n}/\sqrt{\cal V}$ is

%

\bhe coorgdina{align*}

te_{q_n}(m)

=\f reac{1}{\sqpresentt{\Delta_m}}\,

\texa{stion inc}\left[\frac{\cal V}{2}(q_m-q_n)\right],s

          \( \displaystyle e_{v}(n)=\frac{e^{ivq_n}}{\sqrt{\Delta_n(2N+1)}}, \)\end{

ligan*}

{\emd Tthe orthogonality fonormality\/} betweenr these statvectores rea

%

\begin{align*}

\dsum_{m=-N}^N\Delta_me^*_{q_j}

(3m1)          \( \displaystyle \sum_{n=-N}^N\Delta_ne^*_{v'}(n)e_{v}(n) =\frac{1}{2N+1}\sum_{n=-N}^Ne^{i(v-v')q_n} \mathop{\longrightarrow}_{N\to\infty} \delta(v-v'). \)

The conjugate function to \( e_{v}(n)=e^{ivq_n}/\sqrt{2N+1} \) i_{q_n}(m)s

          \( \displaystyle e_{v}(v') =\sum_{j=-N}^N\frac{e^{i(v'-v)q_j}} {\sqrt{{\cal V}(2N+1)}} , \)

Th&=\isum_{m=-N}^N\Delta_m\ funcrac{1}{\Delta_m}

\text{stion approximc}\left(\frac{\cal V}{2}(q_m-q_j)\right)

\atxt{es  a delta funinc}\left(\frac{\cal V}{2}(q_m-q_n)\right)\\

%

&\macthop{\lion with small noigrightarrow}_{{\cal V}\to\infty}

\dse lta_{jin it}.,

\end{align*}

Concluding remarks

{\em

This is a step more in the theory of normalized eigenvector\/} of the discrete operators. It shows that it is possible to have discrete operators very similderivative with eigenvalue $v$ in the coordinate representation is $e_{v}(n)=e^{ivq_n}/\sqrt{\Delta_n(2N+1)}$

{\em Oar to thogonality}

%

\bhe gin{align*}

\usm_{n=-N}^N\Deltual op_ne^*_{v'}(n)e_{v}(n)

=\ferator of contin{1}{2N+1}\sum_{n=-N}^Ne^{i(v-v')q_n}

\mathop{\luous vngariable theory. ghtarrow}_{N\to\infty}

\dWe willta(v-v')

.

\ exnd{aploign*}

{\rme more things about this operator,  TThe conjugate\/} function to $e_{v}(n)=e^{ivq_n}/\sqrt{2N+1}$ is

%

\beghi{alinn*}

e_{v}(v')

=\gsum_{j=-N}^N\sqrt{\frac{\Delta_j}{\ca like it V}}

se^{- inverse, ivq_j}\frac{e^{iv'q_j}}{\sqrt{2N+1}}

=\ts usem_{j=-N}^N\sqrt{\frac{\Delta_j}{{\cal V}(2N+1)}}

e^{ (v'-v)q_j},

\ein obd{taininlign*}

{\emg Thself-adjoint ex normalized vector is}

%

\btgiensio{align*}

e_{v}(v')

=\ns,um_{j=-N}^N\ forac{e^{i(v'-v)q_j}}

r i{\nstqrt{{\cal V}(2N+1)}}

,

\end{aligance*}.

WeT have discusseis isd a local approach to the finite differences first derivative. Another point of view is obtained by collecting the finite differences at each point of the mesh in a single matrix, the subject of another rticle.

Anotheraapproach to the derivative on a mesh make use o future wor second order finite differences. This subject is deal with in another articlesk.

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