Encyclopedia
We introduce finite-differences derivatives intended to be exact, when applied to the real exponential functionith a finite number of terms, for a particular function. For other functions, these finite differences are also an approximation to the actual derivative. We want to recover the known results of continuous calculus with our finite differences derivatives but in a discrete form. The purpose of this work is to have a discrete momentum operator suitable for use as an operator in discrete quantum mechanics theory. In this article we consider the first derivative on a partition. Our definitions are based on the requirements for the derivative such as that the exponential function is its eigefunction.
- discrete derivative
- discrete symmetric operator,
- discrete quantum mechanics
- Exact finite differenes, derivative, first order
Introduction
introduction
The subject Frof 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 m calculus of a single variable, the formal definition of the derivative \( f'(x_0) \)$f'(x)$ of some function \( f(x) \)$f(x)$ of a single variable \( x \)$x$, at \( x_0 \)$x_0$, requis obtained from res of taking the limit \( x\to x_0 \) $x\tof any of the following ratiosx_0$ of a ratio of differences
\( \displaystyle f'(x_0)
:=\lim_{x\to x^-_0}
\frac{f(x_0)-f(x)}{x_0-x},
\quad
f'(x_0)
:=\lim_{x\to x^+_0}
\frac{f(x)-f(x_0)}{x-x_0}. \)
\[
f'(x)
These ra:=\lim_{x\tios ao x_0}
\fre known as backward and fc{f(x)-f(x_0)}{x-x_0}.
\]
Horward differences, respectively.
Thever, there are at least limit of vanishing finite differences involves variables two situations in which are continuous. However, many timeswe need to compute the derivative when it is not possible to perform such take the mentioned limit at all. A$x\to x_0$. One situation like that appears when doingis numerical simulations in a computer due to the limit in how small a number can be represented in the computer. Another situation in which icomputation. When doing numerical calculations, it is not possible to ta perform a limit like the infinitesimal difference limit is whseparation between the independent variable is a discrete variable by nature, as is discrete set of points of a mesh, and then an approximation to the derivative is used.
Other case found in Quantum Mechanics theory regarding the spectrum of quantum situation is in the theory of operators.
Fini the differences are necessary in numerical computations to approximate several quantities like derivatives and integrals 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] Are are operators with a discrete spectrum. The values of the spectrum of the operator are fixed number and there is no meaning interesting development is the use of a complex taking a limit like $x\to x_0$. Thus, an exact finite differences to evaluat, a finite differences which will give the realexact derivative of a function.[2]
Oneven if $\Delthe other hand, the reader can learn aboa$ has any finite value, is desirable.
Usut the calculus of finite differences from the classical works of Kopalcalculus, for equally spaced points in a mesh, is well developed[31], Boole[42]. Jordan[53], Richardson[64], foHer instance.
Ane we introther branduch of the finite differences tree, is known as thecalculus which is exact finite differences technique. That scheme was developed by researchers like Potts,[7][8]or the first derivative. We specialize the results Rtonald E. Mickens[9] w finithe the purpose of obtaining exact finiintervals, first order finte differences representations of continuous differand to the real exponential equations and of their solutions.
Anotherfunction, the real eigenfunction use of finite differences was developthe derivative.
The Partition
Ledt $\{q_{i}\}_{i=-N}^{N}$ by Armando Martínez-Pérez and Gabino Torres-Vege a partition of the real interval $[-a,a[10]]$, with the intat is When respect to the spectrum is necessary$|\chi_{f,j}-\Delta_j|$ is less than some times. The aim is to obtain discrete operators which comply with discrete versions of the properties thatsmall $\epsilon$, our method and the usual finite-differences derivative will give similar results for any vector defined on the mesh. A ba quantum operator must have. The explckwards versicit formn of the exact finite differences derivative , at $q_j$, is Some properthe use of anies of the exact finite -differences derivative ware {\em Then itsequality\/} {\edm inTwo discrete Quantum Mechanics. Letversions \( \{q_{i}\}_{i=-N}^{N} \)of bthe a \( N+1 \) points pummartitition of the real interval \( [-a,a] \), derivathat is ve.\/} \( \displaystyle -a=q_{-N} < q_{-N+1} < \dots < q_{N-1} < q_{N}=a. \) \( \displaystyle \Delta_j = q_{j+1}-q_{j}, \quad j = -N, \dots, N-1. \) Th Fo$n<N$. The summation term a given \( v\in\mathbb{R} \),t the backward and forward finite diffight hand side of this equality involve interferences derivatives, of a vector \( g=(g_{-N}, g_{-N+1},\dots,g_{N-1},g_N)^T \), terms that vatnish \( q_j \), dwhefined on the partition, ar the separation be \( \displaystyle (D_bg)_j=\frac{g_{j}-g_{j-1}}{\eta_{j}}, \) \( \displaystyle (D_fg)_j=\frac{g_{j+1}-g_j}{\chi_{j}}, \) en mesh points is the same. For \( \displaystyle \eta_{j}=\frac{2}{v}\,e^{-v\Delta_{j-1}/2}
\sinh\left(\frac{v}{2}\Delta_{j-1}\right)
=\Delta_{j-1}+\frac{v}{2}\Delta^2_{j-1}+{\cal O}(\Delta^3_j), \) \( \displaystyle \chi_{j}=\frac{2}{v}\,e^{v\Delta_{j}/2} \sinh\left(\frac{v}{2}\Delta_{j}\right)
=\Delta_j+\frac{v}{2}\Delta^2_j+{\cal O}(\Delta^3_j). \) Tthe whole discrete variablinterval, we \( q_j \) is thave {\em anThe eigenfunction of the discrete derivative operation with summation.\/} The finite differences versions of $\int_a^xdx\,v\,e^{vx}=e^{va}-e^{vx}$ are \( \displaystyle (D_be^{vq})_j=(D_fe^{vq})_j=v\,e^{vq_j}, \)e^{vq})_j\\ Whelign \( |\eta_{j}-\Delta_j| \) *} {\em that the dThe derivative of one.\/} {\eme \( \Delta_j \) wThen \( v=0 \). Somde propertiivatives of t$q$.\/} Ts will approach to one connection between forward and backward discrete derivatives.in Tthe equalitylimit \( \displaystyle \eta_{j+1}=e^{-v\Delta_j} \chi_{j} \)of ismplies thatall $\Delta_j$. {\em \( (D_fg)_j=e^{-v\Delta_j}\,(D_bg)_{j+1}. \) The chain rule.\/} \( \displaystyle \sum_{j=-N}^n\Delta_j(D_fg)_j
=-\frac{\Delta_{-N}}{\chi_{-N}}g_{-N}
+\sum_{j=-N+1}^n\left(
\frac{\Delta_{j-1}}{\chi_{j-1}}
-\frac{\Delta_j}{\chi_{j}}\right)g_j
+\frac{\Delta_n}{\chi_{n}}g_{n+1}, \)gin{align*} {\erm at the right hand side of thesThe derivative of a product of vectors.\/} {\em tThe separation between mesh points is the samederivative of the inverse of a vector.\/} The eigenfunction of the summation. Th\be fgini{align*} The derivative of a constant function \( c \). {\em \( (D_f\,c)_j=0,
\quad\text{and}\quad
(D_b\,c)_j=0. \) \( \displaystyle (D_f\,q)_j
=\frac{q_{j+1}-q_j}{\chi_{j}}
=\frac{\Delta_j}{\chi_{j}},
\quad\text{and}\quad
(D_b\,q)_j
=\frac{q_j-q_{j-1}}{\eta_{j}}
=\frac{\Delta_{j-1}}{\eta_{j}}. \) The chain rule. Fo}\r ight)_j {\em turns to be Summation by \( \displaystyle (D_fg(h(q)))_j=({\cal D}_fg(h))_j\,(D_fh)_j, \)parts.\/} lta_{-N}}{\chi_{f,-N}}g_{-N}h_{-N} The derivative of a product of vectors. The\frac{\De arelta_{j-1}}{\chi_{f,j-1}} \( (D_f\,gh)_j=g_{j+1}(D_fh)_j+(D_fg)_jh_j., \) The derivative of a vector of inverses. \( \displaystyle \left(D_f\,\frac{1}{h}\right)_j=-\frac{(D_fh)_j}{h_jh_{j+1}},
\quad\text{and}\quad
\left(D_b\,\frac{1}{h}\right)_j=-\frac{(D_bh)_j}{h_jh_{j-1}}, \) The derivative of a vector of ratios. Tght)he_jg_j \( \displaystyle \left(D_f\,\frac{g}{h}\right)_j=\frac{(D_f\,g)_j}{h_j} -\frac{(D_fh)_j}{h_jh_{j+1}}g_{j+1}, \) \( \displaystyle \left(D_f\,\frac{g}{h}\right)_j=\frac{(D_f\,g)_j}{h_{j+1}} -\frac{(D_fh)_j}{h_jh_{j+1}}g_j, \) \( \displaystyle \left(D_b\,\frac{g}{h}\right)_j=\frac{(D_b\,g)_j}{h_{j-1}} -\frac{(D_bh)_j}{h_jh_{j-1}}g_j, \) \( \displaystyle \left(D_b\,\frac{g}{h}\right)_j=\frac{(D_b\,g)_j}{h_j} -\frac{(D_bh)_j}{h_jh_{j-1}}g_{j-1}. \) Summation by parts. \( \displaystyle \sum_{j=-N}^{N-1}\Delta_jg_{j+1}(D_fh)_j
+\sum_{j=-N}^{N-1}\Delta_j(D_fg)_jh_j=\cal{I}_1, \) w_{b,N}}\right)\Delta_{N-1}h_Ng_N. Also \( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}g_j(D_bh)_j
+\sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_jh_{j-1}
=\cal{I}_2, \) The commutator between \( q \) and \( D \)$D$.\/} The discrete version of the relationship$\frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q)$ \( \displaystyle \frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q) \) becomes \( \displaystyle (D_f\,qh)_j-q_{j+1}(D_f\,h)_j=\frac{\Delta_j}{\chi_{j}}q_j, \) Translation of the exponential function. T(D_f\,qh)_j-q_{j+1}(D_f\,he dis)_j {\eratorm of tTranslations of the exponential function.\/} W\be wgiln{align*} \( \displaystyle \tilde g_j=(Fg)_j=\frac{1}{\sqrt{{\cal V}}}
\int_{-{\cal V}/2}^{{\cal V}/2}e^{ivq_j}g(v)dv, \) Weqrt{2a}} {\em Thed of the discrete Fourier transform\/} discrete Fourier transform of a vector \( h_j \) w$hich_j$ is defined as {\em IThe conjugate of $D_f$}. Consider the equidality The adjoint of \( D \).qrt{\frac{\Delta_{N-1}}{\cal ConsV}}\,(D_be^{-ider thpq}h)_N. The conjugate of \( q_j \){\em The conjugate of $q_j$.} The relationship {\em The normalized eigenvector of the coordinate operator eigenvector\/} of the coordinate operator with eigenvalue \( q_n \)$q_n$ in the \( v \)$v$ representation is Th$es_{q_n}(v)=e func^{-ivq_n}/\sqrt{\cal V}$. Ortihons areormality orthonormal $\int_{-{\cal a discrV}/2}^{{\cal V}/2}dv\, Now, \lefthe[\frac{\cal conjugate to the coordinate eigenvector \( e_{q_n}(v) \) V}{2}(q_m-q_n)\risght]$ \( \displaystyle \)\( \displaystyle e_{q_n}(m)
=\frac{1}{\sqrt{{\cal V}}}\int_{-{\cal V}/2}^{{\cal V}/2}
e^{ivq_m}\frac{e^{-ivq_n}}{\sqrt{\cal V}}dv
=\sqrt{\frac{{\cal V}}{2a}}\,
\text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right], \) and th{\em orthonormality between theseConjugate vectors.} The conjugate vectors read to $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$ is \( \displaystyle \sum_{m=-N}^N\Delta_me^*_{q_j}(m)e_{q_n}(m)=\sum_{m=-N}^N
\text{sinc}\left[\frac{\cal V}{2}(q_m-q_j)\right]\,
\text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right], \) Now,{\em tThe normalized} eigenvector of the discrete derivative with eigenvector, conjugalute \( v \)to $e_{q_n}(v)=e^{-in thvq_n}/\sqrt{\cal V}$ is \end{alignd*} {\em tThe orthogonality fornormality\/} between these statevectors read The conjugate function to \( e_{v}(n)=e^{ivq_n}/\sqrt{2N+1} \) is_{q_n}(m) Thi&=\s um_{m=-N}^N\Delta_m\functrac{1}{\Delta_m} {\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 similaderivative with eigenvalue $v$ in the coordinate representation is $e_{v}(n)=e^{ivq_n}/\sqrt{\Delta_n(2N+1)}$ {\em Or to thhogonality} {\em more things about this operator, ThThe conjugate\/} function to $e_{v}(n)=e^{ivq_n}/\sqrt{2N+1}$ is We Thave discussedis is 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 to the derivative on a mesh make use ofuture work second order finite differences. This subject is deal with in another articles.
\begintion{align*}
a=q_{-N} < q_{-N+1} < \dof obtts < q_{N-1} < q_{N}=b.
\end{alininggn*}
The discrete operators for use in Quantum Mechanics theparations between mesh points are
\beory. It gin{alis common gn*}
\Delthat a quantum op_j = q_{j+1}-q_{j}, \quad j = -N, \dots, N-1.
\end{align*}
\[ (D_fg)_j=\frator c{g_{j+1}-g_j}{\chas i_{f,j}},
\qua d
\chis_{f,j}=\fracret{e^{v\Delta_j}-1}{v}
=\De splta_j+\frac{v}{2}\Dectrum and a dlta^2_j+{\cal O}(\Delta^3_j). \]
\bergivatin{align*}
(D_fe^{vq})_j=v\,e w^{vq_j},
\end{alitgn*}
%
\[ (D_bg)_j=\frac{g_j-g_{j-1}}{\chi_{b,j}},
\quad
\chi_{b,j}=\frac{1-ep^{-v\Dends on thlta_{j-1}}}{v}
=\De lta_{j-1}-\functionsrac{v}{2}\Delta^2_{j-1}
+{\cal O}(\Delthata^3_{j-1}), \]
%
wehich want to consider. In also complies with
%
\begis n{articllign*}
(D_be^{vq})_j=v\, we^{vq_j}.
\e ndiscuss{align*}Properties
%
$ eigen\function is thrac{\chi_{b,j+1}}{\chi_{f,j}}
=e r^{-v\Deal exponentiallta_j} function.
$ This will provide with amplies that
%
\[ mom(D_fg)_j
=e^{-v\Denltum operator to bea_j}\,(D_bg)_{j+1}
. us\]The partition
The separatfions between mesh pointsnite differences versions of $\int_a^xdy\,g'(y)=g(x)-g(a)$ are
%
\begin{align*}
&\sum_{j=-N}^n\Delta_j(D_fg)_j
%
&=
-\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}
+\sepaum_{j=-N+1}^n\left(
\frac{\Deltionsa_{j-1}}{\chi_{f,j-1}}
-\fracan{\Delta_j}{\chi_{f,j}}\right)g_j
o+\fr cannot bac{\Delta_n}{\chi_{f,n}}g_{n+1}
,
\e eqund{al ign*}
%
and ar
%
\be fgini{align*}
&\Delta_{-N}(D_fg)_{-N}
+\sum_{j=-N+1}^n\De. Tlta_{j-1}(D_bg)_j
%
&=
-\left(\frac{1}{\che dis_{f,-N}}
+\frac{1}{\chi_{b,-N+1}}\rete vaight)\Delta_{-N}g_{-N}
+\friable $q_j$ ic{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}
+\s the ium_{j=-N+1}^{ndependent-1}\left(
va\friable withac{\Delta_{j-1}}{\chi_{b,j}}
-\frespect to we will cac{\Delta_j}{\chi_{b,{j+1}}}\right)g_j
+\fralculate th{\Delta_{n-1}}{\chi_{b,n}}g_n
%,
\e nd{align*}
%
wherivative of a function.The exact finite differences derivative
twhere the denominators are defined as
%
\begindepe{alignd*}
&
\sum_{j=-N}^{N-1}\Denlta_j(D_fg)_j
v+\Delta_{N-1}(D_bg)_N
\\
%
&=
-\friable withc{\Delta_{-N}}{\chi_{f,-N}}g_{-N}
re+\spectum_{j=-N+1}^{N-1}\left(
\frac{\Delto which we willa_{j-1}}{\chi_{f,j-1}}
c-\fralculatec{\Delta_j}{\chi_{f,j}}\right)g_j
the de-\frivative of aac{\Delta_{N-1}}{\chi_{b,N}}g_{N-1}+\left(\frac{1}
fun{\ction.hi_{f,N-1}}
The +\functions in thrac{1}{\chi_{b,N}}\right)\Delta_{N-1}g_N
,
\e nd{align*}
%
\beginomin{align*}
&
\Delta_{-N}(D_fg)_{-N}
+\sum_{j=-N+1}^N\Deltors oa_{j-1}(D_bg)_j\\
&=-\lef these(\frac{1}{\chi_{f,-N}}
exp+\fressions,ac{1}{\chi_{b,-N+1}}\right)
\Delthe a_{-N}g_{-N}
+\funraction{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}
+\s \( \eta_{j} \) and \( \chi_{j} \), umak_{j=-N+1}^{N-1}\lesft(
su\fre that theac{\Delta_{j-1}}{\chi_{b,j}}
-\freal exponentiac{\Delta_j}{\chi_{b,j+1}}\right)g_j
+\frac{\Del function \( e^{vq} \) ta_{N-1}}{\chi_{be,N}}g_N
,
\exnd{actly,lign*}
%
\begin{al eign*}
\sum_{j=-N}^n\Denlta_jv\,e^{valq_j}
&=\sum_{j=-N}^n\De \( v \)lta_j(D_f\,
%
whi&=-\frac{\Delta_{-N}}{\ch is the i_{f,-N}}e^{vq_{-N}}
+\saume_{-N+1}^n\left(
p\froperty as theac{\Delta_{j-1}}{\chi_{f,j-1}}
-\fracontinuous variable de{\Delta_j}{\chi_{f,j}}\right)e^{vq_j}
+\frivativc{\Delta_n}{\chi_{f,n}}e^{vq_{n+1}},
\e hnd{as.
%
and \( |\chi_{j}-\Delta_j| \)
%
\begin{arlign*}
&\De less thanta_{-N}v\,e^{vq_{-N}}
+\some smaum_{j=-N+1}^n\Delta_{j-1}v\,e^{vq_j}
%=\Dell \( \epsilon \)ta_{-N}(D_f\,e^{vq})_{-N}
% o+\sur method and m_{j=-N+1}^n\Delta_{j-1}(D_b\,e^{vq})_j\\
%
&=-\left(\frac{1}{\chei_{f,-N}}
usu+\fral finitc{1}{\chi_{b,-N+1}}\right)
\Delta_{-difN}e^{vq_{-N}}
+\ferences derivative give ac{\Delta_{-N}}{\chi_{f,-N}}e^{vq_{-N+1}}
+\siumilar_{j=-N+1}^{n-1}\left(
\fresultsac{\Delta_{j-1}}{\chi_{b,j}}
-\for any vectorac{\Delta_j}{\chi_{b,j+1}}\right)e^{vq_j}
+\fr dac{\Definelta_{n-1}}{\chi_{b,n}}e^{vq_n},
\end o{align t*}
%
whe mresh. Not $-N<n<N$.
%
\beginomin{align*}
(D_f\,1)_j=0,
\quad\tors \( \eta_{j} \) ext{and \( \chi_{j} \) }\quad
(D_b\,1)_j=0,
\econd{align*}Properties of the finite-differences derivative
%
\begin{align*}
(D_f\,q)_j
=\frac{q_{j+1}-q_j}{\chi_{f,j}}
=\frac{\De exltact fini_j}{\chi_{f,j}},
\quad\text{and}\quad
(D_b\,q)_j
=\frac{q_j-dq_{j-1}}{\chi_{b,j}}
=\ffrac{\Derlta_{j-1}}{\chi_{b,j}}.
\encd{align*}
%
These derivative are
%
The summation of the derivative. Th\be fginit{align*}
(D_fg(h(q)))_j
=({\cal D}_fg)_j\,(D_fh)_j,
\e ndiff{align*}
%
where
%
\begin{align*}
({\cesal vD}_fg)_j
=\frac{g(h(q_{j+1}))-g(h(q_j))}{h(q_{j+1})-h(q_j)}.
\ersnd{alions of \( \displaystyle \int_a^xdy\,g'(y)=g(x)-g(a) \) gn*}
%
arnd
%
\be
(D_bg(h(q)))_j
=({\cal D}_bg)_j\,(D_bh)_j,
\end\( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_j
=-\frac{\Delta_{-N}}{\eta_{-N+1}}g_{-N}
+\sum_{j=-N+1}^{n-1}\left(
\frac{\Delta_{j-1}}{\eta_{j}}
-\frac{\Delta_j}{\eta_{{j+1}}}\right)g_j
+\frac{\Delta_{n-1}}{\eta_{n}}g_n, \){align*}
%
where
%
\begin{align*}
({\cal \( \)\( n \)<\( N \)D}_bg)_j
=\frac{g(h(q_j))-g(h(q_{j-1}))}{\chi_{b,j}}. Th
\e summnd{ation tlign*}
%
\be gin{align*}
(D_f\,gh)_j
=g_{j+1}(D_fh)_j+(D_fg)_jh_j
.
\eqund{alitign*}
%
\bes ginvolv{align*}
(D_f\,gh)_j
=(D_fg)_jh_{j+1}+g_j(D_fh)_j
.
\e nd{asymmlign*}
%
\betry tgin{align*}
(D_b\,gh)_j
=g_j(D_bh)_j+(D_bg)_jh_{j-1}
.
\erms thnd{at vlign*}
%
\begin{alignis*}
(D_b\,gh)_j
=(D_bg)_jh w_j+g_{j-1}(D_bh)_j
.
\end{align*}
%
\lefte di(D_fferences versions o\,\frac{1}{h}\right)_j
=-\f \( \displaystyle \int_a^xdx\,v\,e^{vx}=e^{vx}-e^{va} \) rare\( \displaystyle \sum_{j=-N}^n\Delta_jv\,e^{vq_j}
=\sum_{j=-N}^n\Delta_j(D_f\,e^{vq})_j
=-\frac{\Delta_{-N}}{\chi_{-N}}e^{vq_{-N}}
+\sum_{-N+1}^n\left(
\frac{\Delta_{j-1}}{\chi_{j-1}}
-\frac{\Delta_j}{\chi_{j}}\right)e^{vq_j}
+\frac{\Delta_n}{\chi_{n}}e^{vq_{n+1}}, \)c{(D_fh)_j}{h_jh_{j+1}},
%
\quad\text{and\( \displaystyle \sum_{j=-N+1}^n\Delta_{j-1}v\,e^{vq_j}
=\sum_{j=-N+1}^n\Delta_{j-1}(D_b\,e^{vq})_j
=-\frac{\Delta_{-N}}{\eta_{-N+1}}e^{vq_{-N}}
+\sum_{j=-N+1}^{n-1}\left(
\frac{\Delta_{j-1}}{\eta_{j}}
-\frac{\Delta_j}{\eta_{j+1}}\right)e^{vq_j}
+\frac{\Delta_{n-1}}{\eta_{n}}e^{vq_n}, \)w}\quad
%
\left(D_b\,\frac{1}{h}\righet)_j
=-\frac{(D_bh)_j}{h_jh_{j-1}}.
\e \( -N \)<\( n \)<\( N \).nd{align*}
The derivatives of \( q \).
These derivatives will approach to o of a ratio of vectors.\/}
%
\begine {alin gn*}
\left(D_f\,\frac{g}{he limit}\right)_j
&=\frac{(D_f\,g)_j}{h_j}
o-\frac{(D_fh)_j}{h_jh_{j+1}}g_{j+1}
%=\f smrall \( \Delta_j \)c{h_{j+1}(D_f\,g)_j-(D_fh)_jg_{j+1}}{h_jh_{j+1}}
. In p
%
\quad\text{orticu}\quad
%
\left(D_f\,\frarc{g}{h}\right)_j
=\frac{(D_f\, botg)_j}{h_{j+1}}
\( (D_f\,q)_j \) -\frac{(D_fh)_j}{h_jh_{j+1}}g_j\\
.
\end \( (D_b\,q)_j \) {align*}
%
arnd
%
\be equgin{al to onign*}
\le wft(D_b\,\frac{g}{hen \( v=0 \).
=\frac{(D_b\,g)_j}{he_{j-1}}
-\forward schrac{(D_bh)_j}{h_jh_{j-1}}g_j
,
%
\quad\temxt{or}\quad
%
\le fthis(D_b\,\frac{g}{h}\right)_j
=\frac{(D_b\,g)_j}{h_j}
-\fruac{(D_bh)_j}{h_jh_{j-1}}g_{j-1}
.
\end{align*}
%
\begin{align*}
w&\sum_{j=-N}^{N-1}\Delta_jg_{j+1}(D_fh)_j
+\Derlta_{N-1}g_N(D_bh)_N
+\sum_{j=-N}^{N-1}\Delta_j(D_fg)_jh_j
\( \displaystyle ({\cal D}_fg(h))_j
=\frac{g(h(q_{j+1}))-g(h(q_j))}{h(q_{j+1})-h(q_j)}. \)+\Delta_{N-1}(D_bg)_Nh_{N-1}
For =\the backward finitext{b.i.t},
\e ndiff{align*}
%
where
%
\beginc{align*}
&\text{b.i.t.}
=\sum_{j=-N}^{N-1}\Delta_j(D_f\,gh)_j
w+\De hlta_{N-1}(D_b\,gh)_N\\
&=-\fravc{\De
\( \displaystyle (D_bg(h(q)))_j=({\cal D}_bg(h))_j\,(D_bh)_j, \)
wh+\sum_{-N+1}^{N-1}\lereft(
\( \displaystyle ({\cal D}_bg(h))_j
=\frac{g(h(q_j))-g(h(q_{j-1}))}{h(q_{j})-h(q_{j-1})}. \)
-\four equalities
\( \displaystyle (D_f\,gh)_j=(D_fg)_jh_{j+1}+g_j(D_fh)_j, \)
\( \displaystyle (D_b\,gh)_j=g_j(D_bh)_j+(D_bg)_jh_{j-1}, \)
\( \displaystyle (D_b\,gh)_j=(D_bg)_jh_j+g_{j-1}(D_bh)_j. \)
prac{\Delta_j}{\chi_{f,j}}\rovided \( h_j,h_{j\pm 1} \neq 0 \).
-\frac{\De lta_{N-1}}{\chi_{b,N}}h_{N-1}g_{N-1}\\
&\quard+\le four versions oft(\frac{1}{\chi_{b,{N-1}}}
+\f trac{1}{\chis property
\ere \( \displaystyle {\cal I}_1
=\sum_{j=-N}^{N-1}\Delta_j(D_f\,gh)_j
=-\frac{\Delta_{-N}}{\chi_{-N}}g_{-N}h_{-N}
+\sum_{-N+1}^{N-1}\left(
\frac{\Delta_{j-1}}{\chi_{j-1}}
-\frac{\Delta_j}{\chi_{j}}\right)h_jg_j
+\frac{\Delta_{N-1}}{\chi_{{N-1}}}h_Ng_N. \)nd{align*}
%
The terms in the summation vanish when \( \Delta_j\to0 \),$\Delta_j\to0$ i.e.or when the separation between the mesh points is the same. Bu
%
\begin{align*}
&\Delt wa_{-N}g_{-N+1}(D_fh)_{-N}
+\De clta_{-N}(D_fg)_{-N}h_{-N}
+\sum_{j=-N+1}^n c\Delta_{j-1}g_j(D_bho)_j
+\sum_{j=-N+1}^n\De thlta_{j-1}(D_bg)_jh_{j-1}
\\
\\
%
&=
-\le vecftors \( \mathbf{h} \) (\frand \( \mathbf{g} \) c{1}{\chis su_{f,-N}}
+\frac{1}{\ch i_{b,-N+1}}\right)\Delta w_{-N}g_{-N}h_{-N}
+\fray c{\Delthaa_{-N}}{\chi_{f,-N}}g_{-N+1}h_{-N+1}
+\sum_{j=-N+1}^{N-1}\left(
\frac{\Deltha_{j-1}}{\chi_{b,j}}
-\frac{\De last sta_j}{\chi_{b,{j+1}}}\right)g_jh_j\\
&\qum vad+\frac{\Deltanish_{N-1}}{\chi_{b,N}}g_Nh_N
,
\es. And{alsoign*}
{\em The commutator between $q$ \( \displaystyle \cal{I}_2
=-\frac{\Delta_{-N}}{\chi_{b,-N+1}}g_{-N}h_{-N}
+\sum_{j=-N+1}^{N-1}\left(
\frac{\Delta_{j-1}}{\chi_{b,j}}
-\frac{\Delta_j}{\chi_{b,{j+1}}}\right)g_jh_j
+\frac{\Delta_{N-1}}{\chi_{b,N}}g_Nh_N. \)
\( \displaystyle (D_b\,qh)_j-q_j(D_b\,h)_j=\frac{\Delta_j}{\eta_{j}}q_{j-1}. \)%
\begin{align*}
=\fracr{\Detlta_j}{\chi_{f,j}}q_j,
\e nd{align*}
%
\bergivativn{align*}
(D_b\,qh)_j-q_j(D_b\,h)_j
=\frac{\De is ltha_j}{\chi_{b,j}}q_{j-1}
\e nd{aligenn*}
%
\( \displaystyle (e^{s\,D_{b,f}}e^{vq})_j=e^{v(q_j+s)}. \)
(e^{s\,D_{b,f}}e^{vq})_j
=e^{v(q_j+s)}
\end{align*}
{\em The bouneed ofded Fourier transform\/} the bounded Fourier transform of a given function \( g(v) \)$g(v)$ which is defined as
%
o\begin{align *}
\thilde meg_j=(Fg)_j
=\frac{1}{\sh.
\int_{-{\cal V}/2}^{{\calso V}/2}e^{ivq_j}g(v)dv,
\end{align*}
%
\begin{align*}
\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.
%
\begistant case in whi{align*}
p({\chal \( q_{j+1}-q_j=2\pi/(N-1) \) tF}he)_j
&=\sum_{j=-N}^{N-1}\sqrt{\frac{\De transforms canlta_j}{\cal V}}\,
be iden^{-ipq_{j+1}}(-iD_fh)_j
+\sqrti{\fied withrac{\Delta_{N-1}}{\cal V}}e^{-ipq_{N-1}}(-iD_bh)_N
the us\\
&\quald Fourier transform on \( [-{\cal V},{\cal V}] \)+i\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\,
and with the Four(D_fe^{-ipq}h)_j
+ier \series, respectively.
\e eqund{ality \( \displaystyle v({\cal F}h)_j=\sum_{j=-N}^{N-1}\sqrt{\frac{\Delta_j}{\cal V}}\,
e^{-ivq_{j+1}}(-iD_fh)_j
+i\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\,
(D_fe^{-ivq}h)_j. \)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*}
%
provided inthat the asymmeterference and boundary term
s \( \displaystyle \sum_{j=-N}^{N-1}\sqrt{\frac{\Delta_j}{\cal V}}\,
(D_fe^{-ivq}h)_j, \)
vanishes.
%
\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
\( \displaystyle e_{q_n}(v)=\frac{e^{-ivq_n}}{\sqrt{\cal V}}. \)
ete^*_{q_m}(v)
sen_{q_n}(v)
=\text{se, i.e. nc}
\( \displaystyle \int_{-{\cal V}/2}^{{\cal V}/2}dv\,e^*_{q_m}(v) e_{q_n}(v)=\text{sinc} \left[\frac{\cal V}{2}(q_m-q_n)\right] \)
%
\begin{a lign*}
e_{q_n}(m)
=\freac{1}{\sult whichqrt{2a}}\int_{-{\cal V}/2}^{{\cal V}/2}
becomes ^{ivq_m}\frac{e^{-ivq_n}}{\sqrt{\cal V}}dv
=\sqrthe K{\froneckerac{{\cal V}}{2a}}\,
d\teltaxt{sinc}\left[\frac{\cal \( \delta_{jn} \) wV}{2}(q_m-q_n)\right],
\en \( {\cal V}\to\infty \).d{align*}Eigenvectors of the derivative operator
%
\be coordginat{align*}
e _{q_n}(m)
=\frepac{1}{\sqresentat{\Delta_m}}\,
\text{sion isnc}\left[\frac{\cal V}{2}(q_m-q_n)\right],
\( \displaystyle e_{v}(n)=\frac{e^{ivq_n}}{\sqrt{\Delta_n(2N+1)}}, \)
%
\begin{align*}
\s
um_{m=-N}^N\Delta_me^*_{q_j}(31m) \( \displaystyle \sum_{n=-N}^N\Delta_ne^*_{v'}(n)e_{v}(n)
=\frac{1}{2N+1}\sum_{n=-N}^Ne^{i(v-v')q_n}
\mathop{\longrightarrow}_{N\to\infty}
\delta(v-v'). \)
\( \displaystyle e_{v}(v')
=\sum_{j=-N}^N\frac{e^{i(v'-v)q_j}} {\sqrt{{\cal V}(2N+1)}} , \)
\text{sion approximac}\left(\frac{\cal V}{2}(q_m-q_j)\right)
\text{s a delta funcinc}\left(\frac{\cal V}{2}(q_m-q_n)\right)\\
%
&\matihop{\lon with small noise in it.grightarrow}_{{\cal V}\to\infty}
\delta_{jn}
,
\end{align*}Concluding remarks
%
\be ugin{align*}
\sum_{n=-N}^N\Deltal ope_ne^*_{v'}(n)e_{v}(n)
=\frator of continu{1}{2N+1}\sum_{n=-N}^Ne^{i(v-v')q_n}
\mathop{\lous vangriable theory. Wghtarrow}_{N\to\infty}
\de will ta(v-v')
.
\expnd{alorign*}
%
\begin{align*}
e_{v}(v')
=\sum_{j=-N}^N\s qrt{\frac{\Delta_j}{\calike its V}}
e^{-inverse, itvq_j}\frac{e^{iv'q_j}}{\sqrt{2N+1}}
=\s use m_{j=-N}^N\sqrt{\frac{\Delta_j}{{\cal V}(2N+1)}}
e^{i(v'-v)q_j},
\en obtd{aininglign*}
{\em sThelf-adjoint ext normalized vector is}
%
\beginsion{align*}
e_{v}(v')
=\s, um_{j=-N}^N\forrac{e^{i(v'-v)q_j}}
in{\staqrt{{\cal V}(2N+1)}}
,
\ence.d{align*}
