introduction

Introduction

FrThe subject om calculus of a single variable, tf finite differences is an old, but  useful method with wide application in science and engineering. 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 formal definition of the derivative $f'(x)$\( f'(x_0) \) of some function $f(x)$\( f(x) \) of a single variable $x$\( x \), at $x_0$\( x_0 \), requires of takings obtained from the limit $x\t\( x\to x_0 \) of x_0$ of aany of the following 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)

:=\lThese ratim_{x\to x_0}

\fs are known ac{f(x)-f(x_0)}{x-x_0}.

\]

Hs backward and forwever, there are atard differences, respectively.

The leasimit two situations in of vanishing finite differences involves variables which we need to compute the derivative when are continuous. However, many times it is not possible to take the mentioned perform such limit $x\to x_0$. One at all. A situation islike that appears when doing numerical simulations in a computation. When doing numerical calculations,er due to the limit in how small a number can be represented in the computer. Another situation in which it is not possible to perform atake the infinitesimal difference limit likeis when the infinitesimal separation between the discrete set of podependent variable is a discrete variable by nature, as is the case found in Quantum Mechanics theory regarding the spectrum of quantum operators.

Finits of a mesh, and then an e differences are necessary in numerical computations to approximation to the several quantities like derivative is used.

Os and inther situation is in the theory of operators. there are operators with a discrete spectrum. The values of the spectrum of the operatorgrals 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 areNumerical Recipes.^[1] fixed Anumber and there is no meaning in taking a limit like $x\to x_0$. Thus, an exact interesting development is the use of a complex finite differences, a finite differences which will giv to evaluate the exactreal derivative even if of a function.^[2]

On $\Dthelta$ has any finite value, is desirable.

Us other hand, the reader can learn about the calcual us of finite differences calculus, for equ from the classical works of Kopal^[3], Booly spaced^[4]. pJoirdants^[5], Rin a meshchardson^[6], for is well develnstance. 

Anopthed^[1][2][3][4],r Hebre we introducanch of the finite differences calculus which istree, is known as the exact for the first derivative. We specialize the resultsinite differences technique. That scheme was developed by researchers like Potts,^[7][8] tRo finnald E. Mickens^[9] witeh intervals, first order finthe purpose of obtaining exact finite differences and to the real exponrepresentations of continuous differential function, the real eigenfunctionequations and of their solutions.

Another use of the derivative.

The Partition

Lfinite differences was developetd $\{q_{i}\}_{i=-N}^{N}$ be a partition of the real interval $[-by Armando Martínez-Pérez and Gabino Torres-Vega,a]$,^[10] withat is

\b the integin{align*}

a=q_{-N}tion <of q_{-N+1} < \dots < q_{N-1} < q_{N}=b.

\eobtaind{align*}

Theng diseparations between mesh points are

\begicrete operators for use in Quantum Mechanics theory. It is common{ thalign*}

\Delt a_j = q_{j+1}-q_{j}, \quad j = -N, \dots, N-1.

\end{ quantum operator has align*}

\[ (D_fg)_j=\frac{g_{j+1}-g_j}{\chdi_{f,j}},

\quad

\schi_{f,j}=\frac{e^{v\Delta_j}-1}{v}

=\Dete spelcta_j+\frac{v}{2}\Delta^2_j+{\cal O}(\Delta^3_j). \]

\brum and a degrin{align*}

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

\end{al wign*}

Wth ren $|\chi_{f,j}-\Delta_j|$ is less thanspect to the spectrum is necessary some small $\epsilon$, our method and the usual finite-differences derivative will give similar results for any vector defined on the mesh.

Atimes. The aim is to obtain discrete operators which comply with discrete versions of the properties that a bquackwards versntum operator must have.

The explicit fonrm of the exact finite differences derivative, at $q_j$, is

%

\[ depends on the (D_bg)_j=\frac{g_j-g_{j-1}}{\chi_{b,j}},

\quad

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

=\Delons thata_{j-1}-\frac{v}{2}\D welta^2_{j-1}

+{\cwal O}(\Delta^3_{j-1}), \]

%

which alsnt to complnsies wider. In th

%

\begin{s align*}

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

\en we d{align*}

Properties

Somescuss properties of thehe use of an exact finite- differences derivative arwhen its e

{\igemnfunction is The equality\/}

%

$the real exponential \frac{\chi_{b,j+1}}{\chi_{f,j}}

=e^{-v\Deunction. This willta_j}

$ implrovies thade with a moment

%

\[um (D_fg)_j

=ope^{-v\Delrator ta_j}\,(D_bg)_{j+1}

.o be \]

{\usemd Twoin discrete versionsQuantum Mechanics.

The partition

Let of\( \{q_{i}\}_{i=-N}^{N} \) thbe a \( N+1 \) pointsumm patirtition of the derivareal interval \( [-a,a] \), that ive.\/}s

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

The fseparatinite diffeons between mesh points arenc

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

These versions of $\int_a^xdy\,g'(y)=g(x)-g(a)$separations can or cannot be equal and are

%

\beg fin{align*}

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

%

&=

-\frac{\Delta_{-N}}{\c. Thi_{f,-N}}g_{-N}

+\e disum_{j=-N+1}^n\lcreft(

te \fvarac{\Delta_{j-1}}{\chi_{f,j-1}}

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

s +\frac{\Delta_n}{\chi_{f,n}}g_{n+1}

,

\hend{al ign*}

%

annd

%

\bepegin{align*}

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

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

%

&=

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

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

+\fle with rac{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}

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

ct to \frac{\Dwelta_{j-1}}{\chi_{b,j}}

-\fr will calc{\Delta_j}{\chi_{b,{j+1}}}\right)g_j

+\frac{\Dulatel ta_{n-1}}{\chi_{b,n}}g_n

%,

\hen d{align*}

%

wherivativere $n<N$. The summation teof a function.

The exact finite differences derivative

Form at the given \( v\in\mathbb{R} \), backwaright hand side of this equality involve interd and forward finite difference terms ths  derivatives, of a vector \( g=(g_{-N}, g_{-N+1},\dots,g_{N-1},g_N)^T \), at va\( q_j \), definished wheon the separation bpartition, aret

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

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

wheen mesh points ire the denominators are defined as

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

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

Thole interval,discrete variable \( q_j \) weis havthe

%

\beg in{aligdepen*}

&

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

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

\\

%

&=

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

ble with +\resum_{j=-N+1}^{N-1}\left(

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

o which we will -\frcac{\Delta_j}{\chi_{f,j}}\right)g_j

lculate the derivative -\ofrac{\Delta_{N-1}}{\chi_{b,N}}g_{N-1}+\left(\frac{1}

a function. {\cThi_{f,N-1}}

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

,

\unctions in the dend{alomign*}

%

\begin{nalign*}

&

\Delta_{-N}(D_ors ofg)_{-N}

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

&=-\l eft(\fxprac{1}{\chi_{f,-N}}

essions, the +\frac{1}{\chi_{bunctions \( \eta_{j} \) and \( \chi_{j} \),-N+1}}\ makes suright)

e that \Dthelta_{-N}g_{-N}

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

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

\frac{\Delta_{j-1}}{\chi_{unction \( e^{vq} \) be,j}}

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

+\frac{\Dexactlta_{N-1}}{\chi_{by,N}}g_N

,

\end{ align*}

{\em The eigenfunction of the summation.\/} The finite differences versions of $\int_a^xdx\,v\,e^{vx}=e^{va}-e^{vx}$ adiscrete derivative operation with re

%

\begin{al eign*}

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

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

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

%

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

+\h is the suam_{-N+1}^n\left(

e \fprac{\Delta_{j-1}}{\chi_{f,j-1}}

operty as the continuous -\fvarac{\Delta_j}{\chi_{f,j}}\right)e^{vq_j}

+\frac{\Diable derivativelt ha_n}{\cs.

Whi_{f,n}}e^{vq_{n+1}},

\en \( |\eta_{j}-\Delta_j| \) and{ \( |\chi_{j}-\Delta_j| \) are lign*}

%

ess thand

%

\b somegin{ smalign*}

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

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

%=\Delr method and ta_{-N}(D_f\,he^{vq})_{-N}

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

%

&=al finite-\ledift(\frac{1}{\chi_{f,-N}}

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

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

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

+\ive give suim_{j=-N+1}^{n-1}\left(

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

esults -\frac{\Delta_j}{\chi_{b,j+1}}\right)e^{vq_j}

+\or any vector defrac{\Delta_{n-1}}{\chi_{b,n}}e^{vq_n},

\enned{alig on*}

%

w ther me $-N<n<N$.

{\sh. Notem The derivative of one.\/}

%

\begithat the den{alomign*}

(D_f\,1)_j=0,

\qunad\text{ors \( \eta_{j} \) and}\quad

(D_ \( \chi_{j} \) b\,1)_j=0,

\ecome \( \Delta_j \) whend{align*} \( v=0 \).

Properties of the finite-differences derivative

{\eSom The derivative properties of $q$.\/}

%

\bthe egin{xalign*}

(D_f\,q)_j

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

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

\quad\text{an-d}\quad

(D_b\,q)_j

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

=\ferac{\Delta_{j-1}}{\chi_{b,j}}.

\encend{align*}

%

These derivatives will approac are

Th to one inconnection between forward and backward discrete derivatives. tThe limitequality of\( \displaystyle \eta_{j+1}=e^{-v\Delta_j} \chi_{j} \) simall $\Delta_j$.plies that

{\em          \( (D_fg)_j=e^{-v\Delta_j}\,(D_bg)_{j+1}. \)

The summation of the derivative. The chafin rule.\/}

%

\begite din{align*}

(D_fg(h(q)))_j

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

\end{align*}

%

where

%

\begin{align*}

({\cales D}_fg)_j

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

\end{alsigon*}

%

s of \( \displaystyle \int_a^xdy\,g'(y)=g(x)-g(a) \) and

%

\bregin{align*}

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

(D_bg(h(q)))_j

=({\cal D}_bg)_j\,(D_bh)_j,

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

%

where

%

\begin{align*}

({\cal \( \)\( n \)<\( N \). D}_bg)_j

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

\end{ summalign*}

{\tion term The derivative of a product of vectors.\/}

%

\bat the right hand side of these egin{qualign*}

(D_f\,gh)_j

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

.

\end{altign*}

%

\begs in{align*}

(D_f\,gh)_j

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

.

\volvend{ align*}

%

\bsymmetry tegin{rms thalign*}

(D_b\,gh)_j

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

.

\end{t valign*}

%

\begin{align*}

(D_b\,gsh)_j

=(D_bg)_j wh_j+g_{j-1}(D_bh)_j

.

\end{align*}

{\em Tthe derivative of the inverse of a vectorseparation between mesh points is the same.\/}

%

\b

The eigenfunction of the summation. Theg fin{align*}

\lite dift(D_f\,\frac{1}{h}\right)_j

=-\ferences versions of \( \displaystyle \int_a^xdx\,v\,e^{vx}=e^{vx}-e^{va} \) arac{(D_fh)_j}{h_jh_{j+1}},e\( \displaystyle \sum_{j=-N}^n\Delta_jv\,e^{vq_j} =\sum_{j=-N}^n\Delta_j(D_f\,e^{vq})_j =-\frac{\Delta_{-N}}{\chi_{-N}}e^{vq_{-N}} +\sum_{-N+1}^n\left( \frac{\Delta_{j-1}}{\chi_{j-1}} -\frac{\Delta_j}{\chi_{j}}\right)e^{vq_j} +\frac{\Delta_n}{\chi_{n}}e^{vq_{n+1}}, \)

%

\quad\text{and}\quad

%

\l\( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}v\,e^{vq_j} =\sum_{j=-N+1}^n\Delta_{j-1}(D_b\,e^{vq})_j =-\frac{\Delta_{-N}}{\eta_{-N+1}}e^{vq_{-N}} +\sum_{j=-N+1}^{n-1}\left( \frac{\Delta_{j-1}}{\eta_{j}} -\frac{\Delta_j}{\eta_{j+1}}\right)e^{vq_j} +\frac{\Delta_{n-1}}{\eta_{n}}e^{vq_n}, \)wheft(D_b\,\frac{1}{h}\right)_je \( -N \)<\( n \)<\( N \).

The derivative of a constant function \( c \). 

          \( (D_f\,c)_j=0, \quad\text{and}\quad (D_b\,c)_j=0. \)

=-\frac{(D_bh)_j}{h_jh_{j-1}}The derivatives of \( q \).

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

\end{align*}

{\em These derivative of a ratio of vectors.\/}

%

\begin{s will approach to one in the limit of smallig \( \Delta_j \). In*}

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

&=\fratic{(D_f\,g)_j}{h_j}

-\frulac{(D_fh)_j}{h_jh_{j+1}}g_{j+1}

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

.

%

\qu both \( (D_f\,q)_j \) and\t \( (D_b\,q)_j \) are ext{or}\quad

%

\l to oneft(D_f\,\frac{g}{h}\ when \( v=0 \).

The chain rule. Forigh t)_j

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

e -\frac{(D_fh)_j}{h_jh_{j+1}}g_j\\

.

\orward schend{align*}

%

and

%

\bmeg thin{align*}

\ls rulef t(D_b\,\frac{g}{h}\right)_j

=\frac{(D_urns to b\,g)_j}{h_{j-1}}

e

          -\frac{(D_bh)_j}{h_jh_{j-1}}g_j\( \displaystyle (D_fg(h(q)))_j=({\cal D}_fg(h))_j\,(D_fh)_j, \)

,

%

\quad\twhext{or}\quade

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

\lFor theft(D_b\,\frac{g}{h}\right)_j

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

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

.

\eind{align*}

{\tem Summation by parts.\/}

%

\bdifferegin{align*}

&\sum_{j=-N}^{N-1}\Dcelta_jg_{j+1}(D_fh)_j

+\Delta_{N-1}g_N(D_bh)_N

+\sum_{j=-N}^{N-1}\D welta_j(D_fg)_jh_j

+\Delt ha_{N-1}(D_bg)_Nh_{N-1}

=\tvext{b.i.t},

        \( \displaystyle (D_bg(h(q)))_j=({\cal D}_bg(h))_j\,(D_bh)_j, \)

\end{align*}

%

where

%

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

The derivative of a product of vectors. Theregin{ align*}

&\trext{b.i.t.}

=\s foum_{j=-N}^{N-1}\Delta_j(D_f\,gh)_j

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

&=-\frac{\Dequalita_{-N}}{\chi_{f,-N}}g_{-N}h_{-N}ies

         \( (D_f\,gh)_j=g_{j+1}(D_fh)_j+(D_fg)_jh_j., \)

         \( \displaystyle (D_f\,gh)_j=(D_fg)_jh_{j+1}+g_j(D_fh)_j, \)

  +\sum_{-N+1}^{N-1}\left(       \( \displaystyle (D_b\,gh)_j=g_j(D_bh)_j+(D_bg)_jh_{j-1}, \)

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

The derivative of a vector of inverses.

         \( \displaystyle \left(D_f\,\frac{1}{h}\right)_j=-\frac{(D_fh)_j}{h_jh_{j+1}}, \quad\text{and}\quad \left(D_b\,\frac{1}{h}\right)_j=-\frac{(D_bh)_j}{h_jh_{j-1}}, \)

prac{\Dovidelta_{j-1}}{\cd \( h_j,h_{j\pm 1} \neq 0 \).

The derivative of a vector of ratios. Thi_{f,j-1}}

-\ferac{\Delta_j}{\chi_{f,j}}\right)h_jg_j

-\ are fourac{\Delta_{N-1}}{\chi_{b,N}}h_{N-1}g_{N-1}\\

&\quad+\le versions of t(\frac{1}{\chi_{b,{N-1}}}

+\fhis prac{1}{\chi_{b,N}}\operigty

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

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

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

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

Summation by parts.

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

wht)\Delta_{N-1}h_Ng_N.

\rend{align*}

%

        \( \displaystyle {\cal I}_1 =\sum_{j=-N}^{N-1}\Delta_j(D_f\,gh)_j =-\frac{\Delta_{-N}}{\chi_{-N}}g_{-N}h_{-N} +\sum_{-N+1}^{N-1}\left( \frac{\Delta_{j-1}}{\chi_{j-1}} -\frac{\Delta_j}{\chi_{j}}\right)h_jg_j +\frac{\Delta_{N-1}}{\chi_{{N-1}}}h_Ng_N. \)The terms in the summation vanish when $\D\( \Delta_j\to0 \), i.elta_j\to0$. or when the separation between the mesh points is the same.

Also

%

\b But wegin{ calign*}

&\Delta_{-N}g_{-N+1}(D_fn ch)_{-N}

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

+\osum_{j=-N+1}^n\Delta_{j-1}g_j(D_bh)_j

+\sum_{j=-N+1}^n\D thelta_{j-1}(D_bg)_jh_{j-1}

\\

\\

%

&=

-\l vefct(\frors \( \mathbf{h} \) ac{1}{\chnd \( \mathbf{g} \) i_{f,-N}}

+\fras suc{1}{\chi_{b,-N+1}}\right)\Delta_{-N}g_{-N}h_{-N}

+\fr a wac{\Dely ta_{-N}}{\chi_{f,-N}}g_{-N+1}h_{-N+1}

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

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

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

&\qast sum vad+\frac{\Delta_{N-1}}{\chi_{b,N}}g_Nh_N

,

\nishend{as. Align*}so

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

{\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 $\frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q)$the relationship \( \displaystyle \frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q) \) becomes

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

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

\b

Translation of the exponential function. Theg din{align*}

(D_f\,qh)_j-q_{j+1}(D_f\,h)_j

=\fscrac{\Delta_j}{\chi_{f,j}}q_j,

\ten d{align*}

%

\begrin{align*}

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

=\frac{\Dvativel is ta_j}{\chi_{b,j}}q_{j-1}

\hend{ali gn*}

{\enemrator of Ttranslations of the exponential function.\/}

%

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

Weg win{align*}

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

=e^{v(q_j+s)}

\end{allig n*}

{\em The bounded Fourier transform\/}d of the bounded Fourier transform of a given function $g(v)$\( g(v) \) which is defined as

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

%

\begion{align*}

\ tildhe g_j=(Fg)_j

=\frac{1}{\mesqrt{2a}}

h.

We \int_{-{\cal V}/2}^{{\cal V}/2}e^{ivq_j}g(v)dv,

\eso nd{align*}

{\em The discrete Fourier transform\/}d of the discrete Fourier transform of a vector \( h_j \) $which_j$ is defined as

%

\begin{align*}

\tilde      h(v)=({\cal  F}h)(v)

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

\end{align*}

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These transformations preserve the norm of vectors and functions.

{\em The coInjugate of $D_f$}. Consider the equality

%

\bidistant caseg in{align*}

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

&=\esum_{j=-N}^{N-1}\e transqrt{\frac{\Delta_j}{\cal V}}\,

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

+\ntified with the usqual Fouriert{\frac{\Delta_{N-1}}{\cal V}}e^{-ipq_{N-1}}(-iD_bh)_N

transform on \\\( [-{\cal V},{\cal V}] \)

&\quad +i\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\nd with the Fourier series,

(D_fre^{-ipq}h)_j

+specti\vely.

The adjoint of \( D \). Consqidert{\frac{\Delta_{N-1}}{\cal V}}\,(D_be^{-ipq}h)_N.

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

%

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

\end{cases}

\end{align*}

%

provided inthat terference and boundahe asymmetry terms

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

vanishes.

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

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

e^{ivq_j}g(v)\Big|_{v=-{\cal V}/2}^{{\cal 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 terms

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

 vanish.

{\em

Eigenvetors of the coordinate operator

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

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

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

Or functhionormalitys are $\orthonormal int_{-{\cal V}/2}^{{\cal V}/2}dv\,

a discrete^*_{q_m}(v)

se_{q_n}(v)nse, i.e. 

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

Now, thext{ conjugate to the coordinate eigenvector \( e_{q_n}(v) \) isi

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

anc}

d \ltheft[\frac{\cal V}{2}(q_m-q_n)\right]$

{\ orthonormality bem Conjugate vectors.} The conjugatween these vector to $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$ is reads

%

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

\begin{align*}

re_{q_n}(m)

=\frac{1}{\sqrult{2a}}\int_{-{\cal V}/2}^{{\cal V}/2}

which becomes the Krone^{ivq_m}\frac{e^{-ivq_n}}{\sqrt{\cal V}}dv

=\sqrt{\frac{{\cal V}}{2a}}\,

\tcker dext{sinc}\left[\frac{\calta \( \delta_{jn} \) V}{2}(q_m-q_n)\rigwht],

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

Eigenvectors of the derivative operator

{\emNow, Tthe normalized} eigenvector of the discrete derivative with eigenvaluec \( v \) in tor,he conjugate to $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$ is

%

\bordinate represegin{align*}

e_{q_n}(m)

=\frac{1}{\sqrt{\Delta_m}}\,

\text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right],on is

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

and{align*}

{\em Tthe orthonormality\/} betweengonality for these vectorstates reads

%

\b(31)          \( \displaystyle \sum_{n=-N}^N\Delta_ne^*_{v'}(n)e_{v}(n) =\frac{1}{2N+1}\sum_{n=-N}^Ne^{i(v-v')q_n} \mathop{\longrightarrow}_{N\to\infty} \delta(v-v'). \)

Theg conjugate function to \( e_{v}(n)=e^{ivq_n}/\sqrt{2N+1} \) in{als

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

Thign*}

\s fum_{m=-N}^N\Delta_me^*_{q_j}(m)e_{q_n}(m)

&=\nction approximatesum_{m=-N}^N\D  a delta_m\frac{1}{\Delta_m}

\text{s functionc}\left(\frac{\cal V}{2}(q_m-q_j)\right)

\t with small noisext{s inc}\left(\frac{\cal V}{2}(q_m-q_n)\right)\\ it.

%

&\mat

Concluding remarks

Thop{\longrightarrow}_{{\cal V}\to\infty}

\delts is a_{jn}

,

\end{align*}

{\em Thste normalized eigenvector\/} of the p more in the theory of discrete derivative with eigenvalue $v$ in the coordinate representation is $e_{v}(n)=e^{ivq_n}/\sqrt{\Delta_n(2N+1)}$

{\emoperators. It shows that it is possible to have discrete operators very similar Orthogonality}

%

\bo thegin{align*}

\ usum_{n=-N}^N\Dealta_ne^*_{v'}(n)e_{v}(n)

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

\montinuous vathop{\longrightarrow}_{N\to\infty}

\dable theory. We willta(v-v')

.

\ end{axplign*}

{\ore m The conjugate\/} function to $e_{v}(n)=e^{ivq_n}/\sqrt{2N+1}$ is

%

\bore things about this operator,  Things likeg in{align*}

ts inve_{v}(v')

=\rsum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}

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

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

e^{se in obtai(v'-v)q_j},

\end{align*}

{\emng Thse normalized vector is}

%

\blf-adjoint extegin{align*}

e_{v}(v')

=\sionsum_{j=-N}^N\, frac{or instance.

We^{i(v'-v)q_j}}

{\sqrt{{\chal V}(2N+1)}}

,

\enve d{align*}

Thiscus ised 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 article.

Another approach futo the derivative on a mesh make use of second order finite differences. This subject is deal with in another articlesure work.^[5][6]