\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 $V_1<V_2$ of the potentials implies the ordering
$E_1<E_2$ of the eigenvalues. Hall in  \cite{rhallc}, proved a new
comparison theorem by allowing the potentials to intersect and to
keep the ordering of the eigenvalues. In \cite{qhc}, we have
generalized the comparison theorem for higher dimensions and
proved it for higher angular momentum $\ell >0$. 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 $E^L<E_1<E^U$, then
\begin{enumerate}
\item The solution of $H\phi=E^U\phi$, subject to the same
boundary conditions is a lower solution for $\psi(r)$; i.e.,
$\phi(r)\le \psi(r)$.

\item The solution of $H\phi=E^L\phi$, subject to the same
boundary conditions is an upper solution for $\psi(r)$; i.e.,
$\phi(r)\ge \psi(r)$.
\end{enumerate}
\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{e3.5}\\
 -\Delta\phi(r)+V(r)\phi(r)=E^U\phi(r).\label{e3.6}
\end{gather}
We prove the first part of the theorem; the second part can be
proved similarly. Suppose that there exists $a\in \mathbb{R}$ such that
$\phi(a)>\psi(a)$. Suppose first that the solution of
$H\phi=E^U\phi$ is less than $\psi$ for $0<r<r_0<a$, where $r_0$
is the intersection point. Then, by expressing $\Delta \psi(r)$ by
$\psi''(r)+{(N-1)\over r}\psi(r)$ in $N$ spatial dimensions,
multiplying \eqref{e3.5} by $\phi$ and \eqref{e3.6} by $\psi$,
we obtain after subtracting
\begin{equation}
\phi''(r)\psi(r)-\psi''(r)\phi(r)=(E-E^U)\phi(r)\psi(r)
=-\epsilon\psi(r)\phi(r) \label{e3.7}
\end{equation}
for some $\epsilon >0$. 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 $0<r<r_0$: since it
can not be greater than $\psi(r)$ for all $r$, then this
contradicts the fact that
$0=\int_0^{\infty}\psi(r)\phi(r)dr>\int_0^{\infty}\psi^2(r)dr>0$
(see Theorem \ref{thm1} (orthogonality)). Consequently, we have only to consider
$\phi(r)>\psi(r)$ for $0<r<r_0$. Using the Mean Value Theorem,
there exists $0<c<r_0$ such that $\phi'(c)=\psi'(c)$. Now
similarly, integrating equation \eqref{e3.7} form $0$ to $c$, we obtain,
\begin{equation}
\psi(c)\phi'(c)-\phi(c)\psi'(c)=-\epsilon\int_0^{c}\psi(t)\phi(t)dt.
 \label{e3.11}
\end{equation}
Since $\alpha=\phi'(c)=\psi'(c)<0$, we get
\begin{equation}
\alpha(\psi(c)-\phi(c))=-\epsilon\int_0^{c}\psi(t)\phi(t)dt
\label{e3.12}
\end{equation}
and $(\psi(c)-\phi(c))<0$. This implies that
$\int_0^{c}\psi(t)\phi(t)dt<0$ which contradicts the fact that
$\psi \phi>0$ for $0<r<c$. This completes the proof of our main
theorem.
\end{proof}

 Next, we will use this theorem to develop two algorithms to
approximate the solution for the ground state eigenfunction and
its corresponding eigenvalue. We should emphasize the fact that
the monotonicity behavior of the wave function of the ground state
allows us to develop these two algorithms. It can not be applied
to higher eigenfunctions in this paper to approximate higher
eigenvalues problems.


\section{An Iterative method to construct a sequence converges
to the exact eigenfunction and the exact eigenvalue}

 The Rayleigh-Ritz theorem is the most well-known theorem to
find upper bounds for the first eigenvalue. We restrict the
operator to a finite dimensional subspace, then we approximate the
eigenvalues of the Hamiltonian by the eigenvalues of the
constrained operator.

 Now, after proving Theorem \ref{thm2}, we can state a generalized
modified form for Rayleigh-Ritz theorem, where the proof can be
directly deduced from the standard Rayliegh-Ritz theorem and
Theorem \ref{thm2}.



\begin{theorem}[Modified Rayliegh-Ritz Theorem] \label{thm3}
 For an arbitrary function $\psi$ in $\mathcal{D}(H)$, the
expectation value (mean value) of $H$ in the state $\psi$ is such
that
\begin{equation}
E={(\psi,H\psi)\over(\psi,\psi)}\ge E_{1},\label{e4.1}
\end{equation}
where the equality holds if and only if $\psi$ is the eigenstate
of $H$ with the eigenvalue $E_{1}$. Moreover, the corresponding
solution $\phi$ for $H\phi=E\phi$, is a lower bound for the exact
eigenfunction, i.e $\phi\le \psi_{\rm exact}$.
\end{theorem}

 In the more general case, if the trial function is
chosen as a linear combination of a finite number of linearly
independent functions $\phi_i$:
\begin{equation}
\psi=\sum_{i=1}^{n}c_i\phi_i,\label{e4.2}
\end{equation}
the restriction of the eigenvalue problem of $H$ to the
$n$-dimensional subspace $\mathcal{D}_n$ yields to a good approximate
solutions for the eigenvalue problem. If
$\mathcal{D}_n=\mathop{\rm span}\{\phi_1,\phi_2,\dots,\phi_n\}
\subset \mathcal{D}(H)$, then in
a sense, we reduce the problem to a matrix problem
$HC={\mathcal{E}}C$, where $H$ is the $n\times n$ matrix
$H_{ij}=(\phi_i,H\phi_j)$ with eigenvalues
$\{ \mathcal{E}_1, \mathcal{E}_2,\dots\}$ such that
 $\mathcal{E}_1\le \mathcal{E}_2\dots\le\mathcal{E}_n$. So, we obtain
upper bounds using the following theorem.


\begin{theorem}[Generalized Ritz Theorem \cite{reed}]
\label{thm4} $                           $
\begin{enumerate}
\item  $E_{i}\le {\mathcal{E}}_{i}^{(n)},~i=1,\dots ,n $ provided
the $E_{i}$ exist. \item $\lim_{n\to
\infty}{\mathcal{E}}_{i}^{(n)}=E_{i}$, provided $\mathop{\rm
span}\{\phi_n:  n\in N \}$ is dense in $\mathcal{D}(H)$.
\end{enumerate}
\end{theorem}

 A consequent practical result from the above theorems is the
convergence of the monotone sequence of solutions to the exact
wave function corresponding to the ground state eigenvalue.


\section{Computing eigenvalues using upper and lower solutions}

Based on the previous analysis and proposed theorems, a new
converging numerical criteria is proposed as follows:

\subsection*{Numerical Convergence criteria}
 The sequence of upper
bounds for the first eigenvalue obtained using Ragileh-Ritz
method, generate a monotone sequence of lower solutions to the
exact eigenvalue problem and converges to the exact eigenfunction
for Schr\"odinger equation.

 This criteria provides us with a simple numerical technique to
construct a sequence of monotone solutions for eigenvalue problems
which converges to the exact solution for a wide class of
potentials of interest in quantum mechanics. Having upper and
lower bounds for the eigenvalues provides us with lower and upper
bounds for the solutions for the eigenfunction. These results
connect the spectral results obtained in quantum mechanics, such
as, the envelope method, and many other variational methods.
Theorems \ref{thm2}, \ref{thm3}, and \ref{thm4}
 considered the first step to generate monotone
sequences for the corresponding solution for Schr\"odinger
equations.


 This method combined with the envelope method is powerful and
improves spectral approximation bounds for the eigenvalues
see \cite{qhf,hallen,halle,qhan}. Now, our bounds have a
meaning in terms of upper and lower solution. The second algorithm
is explained as follows:


\subsection*{Spectral Bisection Algorithm (SBA) to approximate the bottom
of the spectrum of Schr\"odinger equation} \quad

\noindent\textbf{Input} $E^L$ and $E^U$: Lower and Upper bounds for the
eigenvalue calculated using the envelope method or the generalized
comparison theorems, or any other variational approach.
\\
 $x_b$: Large number to solve the problem in the given domain.
\\
 Nmax: Maximum number of iterations.
\\
 Tol: Tolerance
\begin{itemize}
\item[Step 1:] $i=1$

\item[Step 2:] While $i\le $Nmax  or $|E^U-E^L|>$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{document}