\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small {\em
Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 129, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/129\hfil Spectral bisection algorithm]
{Spectral bisection algorithm for solving Schr\"odinger equation
using upper and lower solutions}
\author[Q. D. Katatbeh\hfil EJDE-2007/129\hfilneg]
{Qutaibeh Deeb Katatbeh}
\address{ Qutaibeh Deeb Katatbeh \newline
Department of Mathematics and Statistics,
Jordan University of Science and Technology,
Irbid, Jordan 22110}
\email{qutaibeh@yahoo.com}
\thanks{Submitted July 18, 2007. Published October 4, 2007.}
\subjclass[2000]{34L16, 81Q10}
\keywords{Schr\"odinger equation;
lower solution; upper solution; spectral bounds;
\hfill\break\indent envelope method}
\begin{abstract}
This paper establishes a new criteria for obtaining a sequence of
upper and lower bounds for the ground state eigenvalue of
Schr\"odinger equation
$ -\Delta\psi(r)+V(r)\psi(r)=E\psi(r)$ in $N$ spatial dimensions.
Based on this proposed criteria, we prove a new comparison theorem
in quantum mechanics for the ground state eigenfunctions of
Schr\"odinger equation. We determine also lower and upper
solutions for the exact wave function of the ground state
eigenfunctions using the computed upper and lower bounds for the
eigenvalues obtained by variational methods. In other words, by
using this criteria, we prove that the substitution of the
lower(upper) bound of the eigenvalue in Schr\"odinger equation
leads to an upper(lower) solution. Finally, two proposed iteration
approaches lead to an exact convergent sequence of solutions. The
first one uses Raielgh-Ritz theorem. Meanwhile, the second
approach uses a new numerical spectral bisection technique. We
apply our results for a wide class of potentials in quantum
mechanics such as sum of power-law potentials in quantum
mechanics.
\end{abstract}
\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
In quantum mechanics, many comparison theorems have been proved for
the spectrum of schr\"odinger equations of the form
\cite{qhc,hdc,rhallc,hnc},
\begin{equation}
-\Delta
\psi(r)+V(r)\psi(r)=E\psi(r),\quad \text{where }
r=\|\mathbf{r}\|,\; {\mathbf{r}}\in \mathbb{R}^N. \label{e1.1}
\end{equation}
The standard comparison theorem of quantum mechanics states that
the ordering $V_10$. In all previous
works in the literature, the researchers were interested in using
these comparison theorems to improve the upper and lower bounds
for the eigenvalues of Schr\"odinger equations without any
information about the corresponding wave function. For example,
using min-max principles \cite{reed} and the envelope method
\cite{hallen,halle}, we can find analytical upper and lower bounds
for the eigenvalue without any information about the corresponding
wave function. Many authors
\cite{seqa,nseqc,nseqd,nseqe,nseqf,nseqh,nseqi,nseqj,nseqk} have
developed iterative methods to solve differential equations of
first order, second order and higher orders. Throughout this
paper, we use the spectral bounds obtained by spectral
approximation methods for monotone increasing potentials having
discrete spectrum in quantum mechanics, to find upper and lower
solution for schr\"odinger equation for the ground state
eigenfunction. We say that $\phi(r)$ is an upper (lower) solution
to an eigenvalue problem $H\psi(r)=E\psi(r)$, where $H$ is a
differential operator if and only if $\phi(r)\ge(\le) \psi(r)$
for all $r$.
We propose here two iterative approaches which yield to an
exact convergent sequence of solutions. The first one uses a
modified and generalized Raielgh-Ritz theorem. Meanwhile, the
second approach uses a new numerical bisection technique. Our
results for Schr\"odinger equations allow us to prove the modified
Rayleigh-Ritz theorem which connects the upper bound for the
eigenvalue of the ground state with the corresponding solutions
for their eigenvalues. This paper is organized as follows: In
section 2, we recall some of the methods available in the
literature to find upper and lower solutions for the eigenvalues,
such as variational methods and the envelope method. In section
3, we prove the new comparison theorem for the upper and lower
solution of schr\"odinger equation. In section 4, we apply our
method to anharmonic oscillator potential in quantum mechanics to
verify the proposed criteria. More specifically, we modify the old
version of Rayleigh-Ritz and generalize it in a way that our
iterative algorithm together with the well-known variational
methods such as the envelope method yields to a more efficient
approximation method. In section 5, we introduce the spectral
bisection algorithm and explain how it can be applied. Finally, in
section 6, we apply our results to some examples of sum of
power-law potentials.
\section{Upper and lower bounds for the eigenvalues of
Sch\"odinger equations}
Upper bounds are easy to find over finite dimensional spaces.
In \cite{qhv}, we have developed new variational methods using
different kinds of bases to find bounds for the eigenvalues for a
sum of power-law potentials. Furthermore, we use comparison
theorems based on solvable models, to find upper and lower bounds
for the eigenvalues.
Moreover, in \cite{qcif,qhf,qcu,qhell,qhan}, we have developed
the envelope method and used it widely in finding upper and lower
bound for wide range of unsolvable potentials in quantum
mechanics. The min-max principal \cite{reed} plays an important
rule in obtaining the formula for the upper and lower bounds for
the eigenvalues in the envelope method. The minimization is
performed over two stages: first we fix the mean of the kinetic
energy $(\psi,-\Delta \psi)=s$, then we minimize over $s>0$. The
mean of the potential energy under the constrain $(\psi,-\Delta
\psi)=s$ is called the ' kinetic potential' $\overline{V}(s)$
associated with the potential $V(r)$. Accordingly, the eigenvalues
can be written in the form,
\begin{equation}
E=\min_{s>0}\{s+ \overline{V}(s)\}, \label{e2.1}
\end{equation}
where,
\begin{equation}
\overline{V}(s) = \inf\big\{(\psi, V\psi) :
\psi \in \mathcal{D}(H),\; (\psi,\psi) = 1,\;
(\psi, -\Delta\psi) = s \big\}. \label{e2.2}
\end{equation}
The kinetic potential can be derived using the coupling of the
potential in the original problem and can be found using Legendre
transformation \cite{gelfand},
\begin{equation}
H\psi=-\Delta\psi+V(r)\psi=E\psi. \label{e2.3}
\end{equation}
For simplicity, let $E=F(v)$; then
$ s=F(v)-vF'(v)$ and $\overline{V}(s)=F'(v) $. It is worth to
mention that Kinetic potentials are applicable to find upper or
lower bounds for the eigenvalues in case of concave or convex
transformation of a solvable model. We can apply this method to
find upper and lower bounds for many examples, such as Hamiltonian
of the form
\begin{equation}
H=-\Delta +\sum_q a(q)r^q,\quad q>0.\label{e2.4}
\end{equation}
In fact, the energy bounds can be expressed in natural way using
the P-repre\-sentation \cite{qhf,qhc} by minimizing over $r$, so we
have obtained new formulation for the energy in terms of the
potential $V(r)$,
\begin{equation}
E= \min_{r>0} \{K^{(V)}(r)+vV(r)\} \label{e2.5}
\end{equation}
where
\begin{equation}
K^{(V)}(r)=(\bar{V}^{-1}o V)(r). \label{e2.6}
\end{equation}
This function is known concretely for certain potentials. For
example, if we consider the pure power $V(r)=\mathop{\rm sgn}(q)r^q $ in $N$
dimensional space, we find that
\begin{equation}
K^{(q)}(r)=(P(q)/r)^2, \label{e2.7}
\end{equation}
\cite{qhf,qhc}, where
\begin{equation}
P(q)=|E{(q)}|^{(2+q)/2q} \big[{2\over {2+q}} \big]^{1/q}
\big[{|q|\over {2+q}}\big]^{1/2},\quad q\ne 0.\label{e2.8}
\end{equation}
Therefore, \eqref{e2.1} can be written in the form
\begin{equation}
E=\min_{r>0}\big\{\big({P(q)\over r}\big)^2+ V(r)\big\}.\label{e2.9}
\end{equation}
We established in our previous work \cite{qcif,qhf,qhc}, how to
choose the suitable values for the $P(q)$ to obtain upper and
lower bounds for the eigenvalues for the power-law potentials in
quantum mechanics.
In the next section, we prove our main new comparison theorem for
the upper and lower solution corresponding to the ground state of
Schr\"odinger equation in quantum mechanics.
\section{Upper and lower bounds for the solution of Sch\"odinger equation}
We need to prove some results before we derive the new method for
finding envelope for the solutions of Schr\"odinger equation in
Theorem \ref{thm2}.
\begin{theorem} \label{thm1}
Consider the eigenvalue problem
\[
H\psi=-\Delta\psi(r)+V(r)\psi(r)=E\psi(r),
\]
subject to $\psi(0)=1$ and $\psi'(0)=0$.
If $H\phi=\lambda\phi$, where $E\ne \lambda$, then
\[
\int_0^\infty \phi(r)\psi(r)r^{n-1}dr=0\,.
\]
\end{theorem}
\begin{proof}
First let $\phi$ and $\psi$ be solutions of
\begin{gather}
-\Delta\psi(r)+V(r)\psi(r)=E\psi(r), \label{e3.1}\\
-\Delta\phi(r)+V(r)\phi(r)=\lambda\phi(r). \label{e3.2}
\end{gather}
Multiply \eqref{e3.1} by $\phi$ and \eqref{e3.2} by $\psi$. After subtracting,
we obtain
\begin{equation}
\Delta\phi(r)\psi(r)-\Delta\psi(r)\phi(r)=(E-\lambda)\phi(r)\psi(r).
\label{e3.3}
\end{equation}
Integrating equation \eqref{e3.3} form $0$ to $\infty$, we get that the
left hand side is zero and the result follows directly.
\end{proof}
In \cite{qhc}, we proved the following lemma which plays an
important rule in our proof.
\begin{lemma}[\cite{qhc}] \label{lem1}
Suppose that $\psi=\psi(r)$, $r=\|\mathbf{r}\|$,
$\mathbf{r}\in \mathbb{R}^{N}$, satisfies Schr\"odinger's equation:
\begin{equation}
H\psi(r)=(-\Delta+V(r))\psi(r)=E\psi(r),\label{e3.4}
\end{equation}
where $V(r)$ is a central potential which is monotone
increasing, $r > 0$. Suppose that $E$ is a discrete eigenvalue at
the bottom of the spectrum of the operator $H = -\Delta + V$
defined on some suitable domain $\mathcal{D}(H)$ in $L^{2}(\mathbb{R}^N)$.
Suppose that $\psi(r)$ has no nodes, so that, without loss of
generality, we can assume that $\psi(r) > 0$, $r > 0$. Then
$\psi'(r) \leq 0$, $r > 0$.
\end{lemma}
Assume now that upper and lower bounds for the eigenvalues of
Schr\"odinger equation have been found as discussed earlier in
section 2, then we can state our main theorem for the first
eigenfunction as follows:
\begin{theorem} \label{thm2}
Consider a Schr\"odinger equation in $N$
spatial dimensions,
\[
H\psi=-\Delta\psi(r)+V(r)\psi(r)=E\psi(r),
\]
subject to $\psi(0)=1$ and $\psi'(0)=0$. If $E^L\psi(a)$. Suppose first that the solution of
$H\phi=E^U\phi$ is less than $\psi$ for $00$. Now, integrating equation \eqref{e3.7} form $0$
to $r_0$(the intersection point), we obtain:
\begin{equation}
\psi(r_0)\phi'(r_0)-\phi(r_0)\psi'(r_0)=-\epsilon\int_0^{r_0}\psi(t)\phi(t)dt.
\label{e3.8}
\end{equation}
In this step, we have to deal with two cases:
\noindent\textbf{First case:}
If $\phi'(r_0)>0$, this contradicts the
fact that the first eigenfunction of the Schr\"odinger equation is
always decreasing i.e $\psi'(r_0)<0$. Since we will have
\begin{equation}
0<{\psi(r_0)\phi'(r_0)+\epsilon\int_0^{r_0}\psi(t)\phi(t)dt\over \phi(r_0)}
=\psi'(r_0)<0. \label{e3.9}
\end{equation}
\noindent\textbf{Second case:}
If $\phi'(r_0)<0$, we have to note first that $\phi(r)$ intersects $\psi(r)$
at $r_0$, only if $\phi'(r_0)$ is greater than $\psi'(r_0)$. Now,
using equation \eqref{e3.8} and the fact that $\psi(r_0)=\phi(r_0)$, we
can take a common factor $\psi(r_0)$ in the left hand side to get:
\begin{equation}
0<(\phi'(r_0)-\psi'(r_0))
=-{\epsilon\int_0^{r_0}\psi(t)\phi(t)dt\over \psi(r_0)}<0 \label{e3.10}
\end{equation}
which is a contradiction.
The other case, if $\phi(r)>\psi(r)$ for $0\int_0^{\infty}\psi^2(r)dr>0$
(see Theorem \ref{thm1} (orthogonality)). Consequently, we have only to consider
$\phi(r)>\psi(r)$ for $00$ for $0$Tol do step 3-6
\item[Step 3:] Let $E_m={(E^L+E^U)\over 2}$
\item[Step 4:] Solve the differential equation
$H\phi^{(m)}=E_m\phi^{(m)}$
\item[Step 5:] If $\phi(x_{b})<0$, then $E^U=E_m$ else
$E^L=E_m$
\item[Step 6:] $i=i+1$
\item[Step 7:] If $i>$Nmax, then Print [``Method Fail to get
accurate approximation for the eigenvalue within Nmax iterations'',
$E_{app}=E_m$], else Print[$E_{app}=E_m$]
\item[Step 8:] End.
\end{itemize}
We can develop many forms of algorithms based on our new
comparison theorem, where we can transform our eigenvalue problem
to a simple algebraic problem.
The Spectral Bisection Algorithm always converges to the exact
eigenvalue. The following theorem allows us to obtain accurate
number of iterations needed to compute the eigenvalue within a
given error bound.
\begin{theorem} \label{thm5}
Suppose that $H\phi=E\phi$, where $E\in [E^L,E^U]$ is the ground
state eigenvalue for Schr\"odinger
equation in $N$ spatial dimensions. The Spectral Bisection
Algorithm generates a sequence $\{E_n\}$ approximating the exact
eigenvalue with
\begin{equation}
|E_n -E|\le {{E^U-E^L}\over 2^n},\ n\ge 1. \label{e5.1}
\end{equation}
\end{theorem}
\begin{proof} For each $n\ge 1$, we have
\begin{equation}
E^U_n-E^L_n={{E^U-E^L}\over 2^{n-1}} \label{e5.2}
\end{equation}
and $E\in (E_n^L,E_n^U)$. Because $E_n={{E^L_n+E^U_n}\over 2}$ for
all $n\ge 1$. Therefore,
\begin{equation}
|E_n-E|\le {1\over 2}(E^U_n-E^L_n)={E^U-E^L\over 2^n}. \label{e5.3}
\end{equation}
It is clear that the generated sequence converges to the exact
eigenvalue $E$ with rate of convergence $O(1/2^n)$.
\end{proof}
We can determine the number of iterations needed to approximate
the ground stat eigenvalue within given accuracy for wide class of
problems in quantum mechanics. For the anharmonic oscillator, if
${\rm Tol}=10^{-6}$ using Theorem \ref{thm5} with $E^L=1$ and $E^U=1.4$ we can
iterate to approximate the eigenvalue of
$-\psi''(x)+x^4\psi(x)=E\psi(x)$, where the exact eigenvalue (using
the shooting method) is given by $Ex=1.06303600$. We find that
$E_{18}({\rm app})=1.0603622$ with absolute error less than
$1.66953\times 10^{-6}$. If we choose $x_{\rm large}=15$, we achieve
this result after $18$ steps using our algorithm. Similarly for
all potentials in $N$ spatial dimensions, with discrete spectrum,
Mathematica software or any other Mathematical softwares can be
used to write the above algorithm, generate and analyze these
spectral properties for Schr\"odinger equation in quantum
mechanics.
\section{Applications}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{figure1}
\end{center}
\caption{Upper and lower solutions for
$-\psi''(x)+(x^2+x^4)\psi(x)=E\psi(s)$ using the upper and lower
bounds for the eigenvalues calculated using envelope method. The
exact eigenvalue is $EX=1.39235$ calculated using shooting method,
$EL=1.18226$ and $EU=1.65098$ obtained using envelope method}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{figure2}
\end{center}
\caption{Upper solutions $\{\phi_n,\; n\in N\}$ for the wave
function of $-\psi''(x)+x^4\psi(x)=E_0\psi(x)$ for $E_{0}$
calculated using Theorem \ref{thm2}, using $El^{(n)},\, n\in N$ calculated
using generalized Rayleigh-Ritz sequence of upper bounds in finite
dimensional subspace. $\psi_0$ denotes the exact wave function
corresponds to the bottom of the spectrum of the anharmonic
oscillator}
\end{figure}
\begin{table}[ht]
\caption{Approximation for the first eigenvalue
$E_{1}$ for $-\Delta +x+x^2$. Linear potential using the second
numerical Algorithm (Spectral Bisection Algorithm) as an
application for the new comparison theorem. The exact eigenvalue
using shooting method is $Ex=1.52789748$. In the right column,
appears Sign$(\phi^{(m)}(x_{b}))$.
}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
$i$ & $E^L$& $E^m={(E^U+E^L)\over 2}$& $E^U $& Sign \\ \hline
1 & 1.000000000000000 & 1.350000000000000 & 1.700000000000000 & +1\\ \hline
2 & 1.350000000000000 & 1.525000000000000 & 1.700000000000000 & +1\\ \hline
3 & 1.525000000000000 & 1.612500000000000 & 1.700000000000000 & +1\\ \hline
4 & 1.525000000000000 & 1.568749999999999 & 1.612500000000000 & $-1$\\ \hline
5 & 1.525000000000000 & 1.546880000000000 & 1.568749999999999 & $-1$\\ \hline
6 & 1.525000000000000 & 1.535937500000000 & 1.546875000000000 & $-1$\\ \hline
10& 1.527734374999999 & 1.528417968750000 & 1.529101562499999 & $-1$\\ \hline
15& 1.527862548828124 & 1.527883911132812 & 1.527905273437499 & +1\\ \hline
20& 1.527897262573242 & 1.527897930145263 & 1.527898597717285 & $-1$\\ \hline
25& 1.527897471189498 & 1.527897492051124 & 1.527897512912750 & +1\\ \hline
30& 1.527897498570382 & 1.527897499222308 & 1.527897499874234 & +1\\ \hline
\end{tabular}
\end{center}
\end{table}
Schr\"odingers equations with power-law potentials have enjoyed
wide attention in the literature of quantum mechanics
\cite{qcif,plaa,qhv,qhf,qhc,hallen,halle,plb,reed,pla,qhan,pld,plf,ple,plc}. We
can apply this method to find a lower and upper solutions for a
wide class of non-solvable problems in quantum mechanics in $N$
spatial dimensions as well as to approximate the eigenvalue using
SBA. The form for such Hamiltonian is given by,
\begin{equation}
-\Delta\psi(r)+\sum_q a(q) r^q \psi(r)=E(q)\psi(r),\quad q>0, \label{e6.1}
\end{equation}
where $r=\|\mathbf{r}\|,\ {\mathbf{r}}\in \mathbb{R}^N$.
We can apply our method to find upper and lower solutions
using the envelope method for anharmonic oscillator model:
$-\psi''(x)+(x^2+x^4)\psi(x)=\lambda \psi(x)$, subject to
$\psi(0)=1$ and $\psi'(0)=0$, as we see in Figure 1. As another
application for our results, we can generate a sequence of lower
bounds that converges to the exact solution for a Hamiltonian of
the form $-\psi''(x)+x^4\psi(x)=\lambda \psi(x)$, with the use of
variational methods to obtain a sequence of upper bounds as we see
in Figure 2.
Now, for a wide class of potentials studied in the literature,
we can use the obtained upper and lower bounds for the eigenvalues
to obtain lower and upper bounds for the corresponding solutions.
Moreover, the Spectral Bisection Algorithm can be used efficiently
to approximate the ground state eigenvalues for the corresponding
eigenvalues, as it is clear in Table 1. Transforming our spectral
problem to a problem similar to an algebraic problem, is the first
step in spectral analysis to find the first eigenvalue as well
eigenfunctions using simple algebraic algorithm in quantum
mechanics.
\subsection*{Extensions and further remarks}
Upper and lower sequence of solutions that converge to the exact
solution for Schr\"odinger equation can be constructed easily
using our new comparison theorem. We can use the upper and lower
bounds using modified Rayleigh-Ritz theorem, envelope method to
find lower and upper solutions for the first eigenfunction. This
method is efficient and reliable in solving eigenvalue problems in
quantum mechanics. Numerical applications and iterative methods
are recognized to be useful in computing the eigenvalues and
verifying the analytical results. Many applications and
interesting examples can be applied and analyze the solutions of
Schr\"odinger equations in $N$ spatial dimensions.
In a forthcoming work, we will generalize this iterative method to
approximate the eigenvalues and eigenfunctions for higher states.
Moreover, we will combine the proved comparison theorems with the
sum approximation{ \cite{qhc}} and the generalized Temple's bounds
to develop a new algebraic approach for the eigenvalue problem.
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\end{thebibliography}
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