\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 64, pp. 1--15.\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 2008 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2008/64\hfil Matrix elements]
{Matrix elements for sum of power-law potentials in
quantum mechanic using generalized hypergeometric functions}

\author[Q. D. Katatbeh, M. E. Abu-Amra,\hfil EJDE-2008/64\hfilneg]
{Qutaibeh D. Katatbeh, Ma'zoozeh E. Abu-Amra}

\address{Qutaibeh Deeb Katatbeh \newline
Department of Mathematics and Statistics \\
Faculty of Science and Arts \\
Jordan University of Science and Technology \\
Irbid 22110, Jordan} 
\email{qutaibeh@yahoo.com}

\address{Ma'zoozeh E. Abu-Amra \newline
Department of Mathematics and Statistics \\
Faculty of Science and Arts \\
Jordan University of Science and Technology \\
Irbid 22110, Jordan}


\thanks{Submitted February 19, 2008 Published April 28, 2008.}
\subjclass[2000]{34L15, 34L16, 81Q10, 35P15}
\keywords{Schr\"odinger equation; variational technique;
eigenvalues; \hfill\break\indent upper bounds; analytical
computations}

\begin{abstract}
 In this paper we derive close form for the matrix elements for
 $\hat H=-\Delta +V$, where $V$ is a pure power-law potential.
 We use trial functions of the form
 $$
 \psi _{n}(r)=  \sqrt{{\frac{2\beta ^{\gamma/2}(\gamma )_{n}}
 {n!\Gamma(\gamma )}}} r^{\gamma - 1/2}
 e^{-\frac{\sqrt{\beta }}{2}r^q} \ _{p}F_{1}
 ( -n,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta } r^q),
 $$
 for $\beta, q,\gamma >0$ to obtain the matrix elements for $\hat H$.
 These formulas are then optimized with respect to variational
 parameters $\beta ,q$ and $\gamma $ to obtain accurate upper
 bounds for the given nonsolvable eigenvalue problem in quantum mechanics.
 Moreover, we write the matrix elements in terms of the generalized
 hypergeomtric functions. These results are generalization of those
 found earlier in \cite{a2}, \cite{h1}--\cite{h9} for power-law potentials.
 Applications and comparisons with earlier work are presented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}\label{sec:sig-amrt:sig-amrt}

 In $ 1998$ Hall et al \cite{h8} found a closed form
expressions for the singular-potential integrals
$\langle m\mid r^{-\alpha }\mid n\rangle $ that obtained with
respect to the Gol'dman and Krivchenkov
eigenfunctions for the singular Hamiltonian with
$\hbar^2=2m=1$,
\begin{equation} \label{e1}
 \hat H=-\frac{d^2}{dr^2} +\beta r^{2}+\frac{A}{r^2},\quad
(\beta >0, A\geq 0).
\end{equation}
We present a variational analysis of the generalized spiked
harmonic oscillator Hamiltonian \cite{a2,a3,b1},
\cite{g1}--\cite{z2} of the form
\begin{equation} \label{e2}
 \hat H=-\frac{d^2}{dr^2} +\beta r^{2}+\frac{A}{r^2}
+\frac{\lambda }{r^\alpha },\quad 0\leq r<\infty ,
\end{equation}
where $ \alpha $ and $\lambda $ are positive real numbers. By
writing $\hat H=H^{(0)}+\lambda V$ with $H^{(0)}$
standing for the generalized  spiked
harmonic-oscillator Hamiltonian, and $V(r)=r^{-\alpha } $, Hall et
al \cite{h1}--\cite{h9} used the basis set
\[
\psi _{n}(r) =  T_{n} r^{\gamma - 1/2} e^{\frac{-\sqrt{\beta
}}{2}r^2}\   _{1}F_{1}( -n;\gamma;\sqrt {\beta } r^2), \quad
T_{n}=\sqrt{\frac{2\beta ^{\frac{\gamma }{2}} (\gamma
)_{n}}{n!\Gamma(\gamma )}},
\]
$ n=0,1,2,3,\ldots$,
constructed from the normalized solutions of $H^{(0)}\psi_n =E_n\psi_n $
to evaluate the matrix elements of $\hat H$. They found that
\[
H_{mn}=\langle m\mid \hat H\mid n\rangle
=2\sqrt{\beta }(2n+\gamma )\delta _{mn}+\lambda \langle m\mid r^{-\alpha }
\mid n\rangle \quad    m,n=0,1,2,\ldots ,N-1
\]
where
\begin{align*}
\langle m\mid r^{-\alpha }\mid n\rangle
&=(-1)^{m+n}\beta ^{\frac{\alpha }{4}}
 \sqrt{\frac{(\gamma )_n(\gamma )_m}{n!m!}}
 \frac{\Gamma (\gamma -\frac{\alpha }{2})
 (\frac{\alpha }{2})_n}{(\gamma )_n\Gamma (\gamma )} \\
&\quad \times  _3F_2(-m,\gamma -\frac{\alpha }{2},1
 -\frac{\alpha }{2};\gamma ,1 -\frac{\alpha }{2}-n;1).
\end{align*}
 Now, by considering the problem in a finite dimensional subspace and
choosing a suitable trial functions as a linear combination of the form
\begin{equation} \label{e3}
\psi _{n}(r)=  T_{n}  r^{\gamma - 1/2}
 e^{\frac{-\sqrt{\beta }}{2}r^q} \  _{p}F_{1}( -n,a_{2},\ldots
,a_{p};\gamma;\sqrt {\beta } r^q), \quad
 T_{n}=\sqrt{\frac{2\beta ^{\frac{\gamma }{2}}
 (\gamma )_{n}}{n!\Gamma(\gamma )}}
\end{equation}
This basis is more general than the basis used in \cite{a2} and
\cite{h7}, because the number of variational parameters
for the exponential function and the generalized hypergeometric
function play an important rule for improving the
numerical results for our eigenvalue problem.
 Herein, $_{p}F_{1} $ stands for the hypergeometric function defined by
 \begin{equation} \label{e4}
_pF_q(\alpha _1,\alpha _2,\ldots ,\alpha _p;\beta _1,\beta _2,\ldots
 ,\beta _q; x)= \sum_{k=0}^{\infty }{\frac{\Pi _{i=1}^{p}
 (\alpha _i)_k}{\Pi _{j=1}^{q}(\beta _j)_k}\frac{x^k}{k!}};
\end{equation}
where $p$ and $q$ are non-negative integers and
$\beta _j (j=1,2,\ldots ,q)$ cannot be a non-positive
integer \cite{a1}. The expression $(a)_n$ is the Pochhammer symbol
defined by the relations
\begin{align*}
(a)_n    &=a(a+1)\ldots (a+n-1)
         =\frac{\Gamma (a+n)}{\Gamma (a)} \\
 (a)_0   &=   1 \\
 (a)_{-n}&=\frac{(-1)^n}{(1-a)_n},
\end{align*}
where n is an integer (positive, negative or zero).

\section{Variational Technique}

 In this section we review the well known variational
technique for eigenvalue problem in quantum mechanics
\cite{d1}. We consider a non solvable eigenvalue problem of the
form $\hat H=-\triangle +V(r)$ in $N$-dimensional
space over a finite subspace in the domain of its
angular momentum subspace. Our variational technique is based on
forming trial wave function from a linear combination
of some of orthonormal basis function $\psi
_n(r),   n=$ $1,2,\ldots ,D$ such that,
\begin{equation} \label{e5}
\Psi (r)=\sum_{n=1}^{D}{c_n \psi _n(r)}
\end{equation}
where the $c_n$ are variational parameters. The variational energy
$E_\Psi  $ we obtain with this trial function is given by,
\begin{equation} \label{e6}
E_\Psi =\frac{\int{\Psi  ^* \hat H \Psi  d\tau }}{\int{\Psi ^* \Psi  d\tau }}
       =\frac{\sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m   \int\psi _n^*
  \hat H \psi _m d\tau }}}{\sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m
  \int\psi _n^*\psi _m d\tau }}}
\end{equation}
Now, defining the matrix element of $\hat H$ as
\begin{equation} \label{e7}
H_{nm}= \int{\psi _n^* \hat H \psi_m  d\tau }
\end{equation}
and using the orthonormal property of the $\psi $ functions, we obtain that
\begin{equation} \label{e8}
E_\Psi =\frac{\sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m H_{nm}}}}{\sum_{n=1}^D
{\mid c_n\mid ^2}}
\end{equation}
or
\[
E_\Psi \sum_{n=1}^D{\mid c_n\mid ^2}= \sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m
H_{nm}}}
\]
We seek the minimum value of $E_\Psi $ as a function of all the $c_n$.
By differentiating with respect to $c_i$, we obtain
\[
\frac{\partial E}{\partial c_i} \sum_{n=1}^D{\mid c_n\mid ^2}
+Ec_i^*=\sum_{n=1}^D{c_n^* H_{ni}}
\]
and setting ${\frac{\partial E}{\partial c_i}=0},$ we have
\begin{equation} \label{e9}
\sum_{n=1}^D{c_n^*H_{ni}-Ec_i^*}=0.
\end{equation}
We can derive one equation similar to \eqref{e9} for each possible
value of $i, i=$ $1,\ldots ,D,$ and we may take \eqref{e9}
to represent a set of linear equations with each equation characterized
 by a different value of $i$. According to the theory of linear
algebraic equations, there is a nontrivial solution, if and only if
the determinant of the coefficients vanishes or if and only if
\[
\left|  \begin{matrix}
H_{11}-E & H_{12}   & H_{13}   & \cdots & H_{1D}\\
H_{21}   & H_{22}-E & H_{23}   & \cdots & H_{2D}\\
\vdots   & \vdots   & \vdots   &        &  \vdots\\
H_{D1}   &  H_{D2}  & H_{D3}   & \cdots &   H_{DD}-E
\end{matrix}\right| =0
\]
This determinant is called a \emph{secular   determinant}.
If we use a basis set that is not orthonormal, we define the matrix
element $S_{nm}$ by
\[
S_{nm}=\int {\psi _n \psi _m d\tau },
\]
in this case the equation corresponding to \eqref{e9} and the secular
 determinant are
\begin{equation} \label{e10}
\sum_{n=1}^D{c_n H_{in}-E S_{in}c_n}= 0
\end{equation}
and the secular determinant can be written as
\[
\left|  \begin{matrix}
H_{11}-ES_{11} & H_{12}-ES_{12}   &  \cdots  & H_{1D}-ES_{1D}\\
H_{21}-ES_{21} & H_{22}-ES_{22}   &  \cdots  & H_{2D}-ES_{2D}\\
\vdots         & \vdots           &          &         \vdots\\
H_{D1}-ES_{D1} & H_{D2}-ES_{D2}   & \cdots  &  H_{DD}-ES_{DD}
\end{matrix}\right| =0.
\]

\section{Derivation of the Matrix Elements}

We divide the problem of computing the matrix elements
for Schr\"odinger equation into simple parts. In the first lemma we
compute the matrix elements for singular potential in three-spatial
dimensions, it will be used to compute the matrix elements for
the kinetic energy.

\begin{lemma} \label{lem1}
For the generalized spiked harmonic oscillator Hamiltonian \eqref{e2},
the expectation  values of the operator $V(x)=r^{-\alpha }$ with respect
to a trial basis \eqref{e3} are given by
 \begin{equation} \label{e11}
\begin{aligned}
\langle \psi _{m}\mid r^{-\alpha }\mid \psi _{n}\rangle
&=\frac{1}{q}T_{m}T_{n}\beta ^{\frac{\alpha -2\gamma }{2q}}
 \Gamma (\frac{2\gamma -\alpha }{q})
 \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}
 (\frac{2\gamma -\alpha }{q})_{k}}{(\gamma )_{k}k!} \\
&\quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},
 \frac{2\gamma -\alpha }{q}+k;\gamma; 1)
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof} Using \eqref{e3},   it  follows that
\begin{equation} \label{e12}
\begin{aligned}
\langle \psi _{m}\mid r^{-\alpha }\mid \psi _{n} \rangle
&= \int_{0}^\infty {r^{2\gamma -\alpha -1}e^{-\sqrt{\beta }r^q}}  _{p}F_{1}( -m,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta } r^q) \\
&\quad \times  _{p}F_{1}( -n,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta } r^q)
 dr
\end{aligned}
\end{equation}
 Using the series representation \eqref{e4} of the hypergeometric function
 $_{p}F_{1}$, yields
\begin{equation} \label{e13}
\begin{aligned}
r_{mn}^{-\alpha }
&=T_{m}T_{n}\sum_{k=0}^m
\sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}
\cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}k!l!} \\
&\quad \times \beta ^{\frac{l+k}{2}}\int_{0}^\infty {r^{2\gamma -\alpha
+qk+ql-1}e^{-\sqrt{\beta }r^q}} dr
\end{aligned}
\end{equation}
By restoring to the integral representation of gamma function,
we obtain under the condition ${\frac{2\gamma -\alpha }{q}+}$ $k+l >0$,
that
\begin{equation} \label{e14}
\begin{aligned}
&r_{mn}^{-\alpha }\\
&=\frac{1}{q}T_{m}T_{n}\sum_{k=0}^m \sum_{l=0}^n \frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}
  (\gamma )_{l}} \frac {\beta ^{\frac{l+k}{2}}}{k!l!}
  \Gamma (\frac{2\gamma -\alpha }{q}+k+l) \\
&=\frac{1}{q}T_{m}T_{n} \beta ^{\frac{\alpha -2\gamma }{2q}}
  \sum_{k=0}^m \Big[ \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}
 \cdots(a_{p})_{l}}{(\gamma )_{l}l!}\Gamma(\frac{2\gamma -\alpha }{q}+k+l)
\Big]  \\
& \quad \times  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
\end{aligned}
\end{equation}
On the other hand, using the definition of the Pochhammer symbols and
the series representation of the hypergeometric functions \eqref{e4},
the finite sum inside the bracket collapses to
\begin{equation} \label{e15}
\begin{aligned}
&\sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{l}l!}
 \Gamma(\frac{2\gamma -\alpha }{q}+k+l) \\
&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}
  (\frac{2\gamma -\alpha }{q} +k)_{l}}{(\gamma )_{l}l!}
 \Gamma(\frac{2\gamma -\alpha }{q}+k) \\
&= _{p+1}F_{1}(-n,a_{2},  \ldots ,a_{p},\frac{2\gamma -\alpha  }{q}
 +k;\gamma; 1)  \Gamma(\frac{2\gamma -\alpha }{q}+k)
\end{aligned}
\end{equation}
 Consequently, we arrive at
% \label{eq:overhead-gen}
\begin{align*}
 r_{mn}^{-\alpha }
&=\frac{1}{q}T_{m}T_{n}\beta ^{\frac{\alpha -2\gamma }{2q}}
 \sum_{k=0}^m \Gamma (\frac{2\gamma -\alpha }{q}+k)
 \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!} \\
&\quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma -\alpha }{q}
 +k;\gamma; 1) \\
&= \frac{1}{q}T_{m}T_{n}\beta ^{\frac{\alpha -2\gamma }{2q}}
 \Gamma (\frac{2\gamma -\alpha }{q})  \sum_{k=0}^m
 \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(\frac{2\gamma
 -\alpha }{q})_{k}}{(\gamma )_{k}k!} \\
&\quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma
 -\alpha }{q}+k;\gamma; 1).
\end{align*}
This completes the proof.
\end{proof}

 For the case of $\gamma >1$ and $\alpha =2$,  we have
\begin{equation} \label{e16}
\begin{aligned}
r^{-2}_{mn}
&=\frac{1}{q}T_{m}T_{n}\beta ^{\frac{1 -\gamma }{q}} \sum_{k=0}^m
 \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
 \Gamma (\frac{2(\gamma -1) }{q}+k) \\
&\quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2(\gamma
-1) }{q}+k;\gamma; 1).
\end{aligned}
\end{equation}

\subsection{Matrix Elements for $V(r)= r^\alpha $}

We now use the suggested basis \eqref{e3} to  compute the matrix
elements for the power-law potential operators $r^\alpha$ $,\
\alpha >0$. This kind of computation is important for may problems
in the literature, such as Kratzer potential \cite{f1}. Whose
calculation is achieved by means of following lemma.

\begin{lemma} \label{lem2}
For the generalized spiked harmonic oscillator Hamiltonian \eqref{e2},
the expectation values of the operator $V(x)=r^{\alpha}$ with respect
to a trial basis \eqref{e3} are given by
\begin{equation} \label{e17}
\begin{aligned}
\langle \psi _{m}\mid r^{\alpha }\mid \psi _{n}\rangle
&=\frac{1}{q}T_{m}T_{n}\beta ^{-\frac{\alpha +2\gamma }{2q}}
  \Gamma (\frac{2\gamma +\alpha }{q})  \sum_{k=0}^m
  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(\frac{2\gamma
  +\alpha }{q})_{k}}{(\gamma )_{k}(k!} \\
&\quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma
 +\alpha }{q}+k;\gamma; 1)
\end{aligned}
\end{equation}
\end{lemma}


\begin{proof} Using \eqref{e3}, it immediately follows that
\begin{equation} \label{e18}
\begin{aligned}
\langle \psi _{m}\mid r^{\alpha }\mid \psi _{n} \rangle
&= \int_{0}^\infty {r^{2\gamma +\alpha -1}e^{-\sqrt{\beta }r^q}}  _{p}F_{1}
 ( -m,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta } r^q) \\
&\quad \times  _{p}F_{1}( -n,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta }
r^q) dr
\end{aligned}
\end{equation}
 Using the series representation \eqref{e4} of the hypergeometric function
$ _{p}F_{1}$, yields
\begin{equation} \label{e19}
\begin{aligned}
r_{mn}^\alpha
&=T_{m}T_{n}\sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
 \cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}
 (\gamma )_{l}k!l!} \\ &\quad  \times \beta ^{\frac{l+k}{2}}
 \int_{0}^\infty {r^{2\gamma +\alpha +qk+ql-1}e^{-\sqrt{\beta }r^q}} dr.
\end{aligned}
\end{equation}
Using the integral representation of gamma function, we obtain under
the condition
 ${\frac{2\gamma +\alpha }{q}}+$ $k+l >0$ that
 \begin{equation} \label{e20}
\begin{aligned}
r_{mn}^{\alpha }
&=T_{m}T_{n}\frac{\beta ^{-\frac{\alpha +2\gamma }{2q}}}{q}
  \sum_{k=0}^m \sum_{l=0}^n \frac{(-m)_{k}(a_{2})_{k}\cdots
  (a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}
  (\gamma )_{l}} \frac {1}{k!l!} \\
&\quad\times \Gamma (\frac{2\gamma +\alpha }{q}+k+l) \\
&= T_{m}T_{n}\frac{\beta ^{-\frac{\alpha +2\gamma }{2q}}}{q}\sum_{k=0}^m
 \Big[\sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{l}l!}
  \Gamma(\frac{2\gamma +\alpha }{q}+k+l)\Big] \\
 &\quad\times \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
\end{aligned}
\end{equation}
On the other hand, by using the definition of the Pochhammer symbols and
the series representation of the hypergeometric functions \eqref{e4},
the finite sum inside the bracket collapses to
 \begin{equation} \label{e21}
\begin{aligned}
&\sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{l}l!}
  \Gamma(\frac{2\gamma +\alpha }{q}+k+l) \\
&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}(\frac{2\gamma
  +\alpha }{q} +k)_{l}}{(\gamma )_{l}l!}\Gamma(\frac{2\gamma
  +\alpha +2}{q}+k) \\
&= _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma +\alpha }{q} +k;
   \gamma; 1)\Gamma(\frac{2\gamma +\alpha }{q}+k).
\end{aligned}
\end{equation}
Finally, we arrive at
\begin{align*}
 r_{mn}^{\alpha }
&=T_{m}T_{n}\frac{\beta ^{-\frac{\alpha +2\gamma }{2q}}}{q}\sum_{k=0}^m
  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
  \Gamma (\frac{2\gamma +\alpha }{q}+k) \\
& \quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma
  +\alpha }{q}+k;\gamma; 1) \\
&=T_{m}T_{n}\frac{\beta ^{-\frac{\alpha +2\gamma }{2q}}}{q}
  \Gamma (\frac{2\gamma +\alpha }{q})  \sum_{k=0}^m
   \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(\frac{2\gamma
   +\alpha }{q})_{k}}{(\gamma )_{k}k!} \\
& \quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma
  +\alpha }{q}+k;\gamma; 1). %\label{eq:overhead-gen}
\end{align*}
The proof is complete.
\end{proof}

On the other hand, for the case of $ \gamma >0$ and $ \alpha =2>0$,
we have that
\begin{equation} \label{e22}
\begin{aligned}
\langle \psi _{m}\mid r^2\mid  \psi _{n} \rangle
&= T_{m}T_{n}\frac{1}{q}
\beta ^{-\frac{1 +\gamma }{q}} \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}
\cdots(a_{p})_{k}}{(\gamma )_{k}k!}\Gamma (\frac{2(\gamma +1) }{q}+k) \\
&\quad \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2(\gamma +1) }{q}+k;
\gamma; 1).
\end{aligned}
\end{equation}


\begin{lemma} \label{lem3}
For  $\gamma >0$, and $m,n=0,1,2,\dots $,
\begin{equation} \label{e23}
\begin{aligned}
 S_{mn}&=\langle \psi _{m}\mid \psi _{n}\rangle   \\
&=T_{m}T_{n}\frac{\beta ^{-\frac{\gamma }{q}}}{q}\Gamma (\frac{2\gamma  }{q})  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(\frac{2\gamma  }{q})_{k}}{(\gamma )_{k}k!} \\
&\quad  \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma  }{q}
+k;\gamma; 1)
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
 Using \eqref{e3}, it  follows that
\begin{equation} \label{e24}
\begin{aligned}
S_{mn}
&=\int_{0}^\infty {r^{2\gamma -1}e^{-\sqrt{\beta }r^q}} _{p}F_{1}
( -m,a_{2},\ldots ,a_{p};\gamma; \sqrt {\beta } r^q) \\
&\quad \times  _{p}F_{1}( -n,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta }
r^q) dr
\end{aligned}
\end{equation}
 By means of the series representation \eqref{e4} of the hypergeometric
function $_{p}F_{1}$, that
\begin{equation} \label{e25}
\begin{aligned}
S_{mn}&= T_{m}T_{n}\sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}
 (\gamma )_{l}}\frac{\beta ^{\frac{l+k}{2}}}{k!l!} \\
&\quad  \times   \int_{0}^\infty {r^{2\gamma +qk+ql-1}e^{-\sqrt{\beta }r^q}}
dr.
\end{aligned}
\end{equation}
Further, after restoring  the integral representation of the gamma
function and a change of variables, we obtain for
$ {\frac{2\gamma }{q}+k+l >0}$, that
\begin{equation} \label{e26}
\begin{aligned}
S_{mn}&=T_{m}T_{n}\frac{\beta ^{-\frac{\gamma }{q}}}{q}\sum_{k=0}^m
 \sum_{l=0}^n \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}
 (a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}}
 \frac {\Gamma (\frac{2\gamma }{q}+k+l)}{k!l!}  \\
&=T_{m}T_{n}\frac{\beta ^{-\frac{\gamma }{q}}}{q}\sum_{k=0}^m
\Big[ \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}
 {(\gamma )_{l}l!}\Gamma(\frac{2\gamma  }{q}+k+l)\Big]
\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
\end{aligned}
\end{equation}
On the other hand, by using the definition of the pochhammer symbols
and the series representation of the hypergeometric functions \eqref{e4},
the finite sum inside the bracket collapses to
\begin{equation} \label{e27}
\begin{aligned}
&\sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{l}l!}
 \Gamma(\frac{2\gamma  }{q}+k+l)  \\
&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}
 (\frac{2\gamma }{q} +k)_{l}}{(\gamma )_{l}l!} \Gamma(\frac{2\gamma }{q}+k) \\
&= _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma }{q} +k;\gamma; 1)
 \Gamma(\frac{2\gamma }{q}+k)
\end{aligned}
\end{equation}
 Consequently,
\begin{align*}
 S_{mn}
&=T_{m}T_{n}\frac{\beta ^{-\frac{\gamma }{q}}}{q}\sum_{k=0}^m
\Gamma (\frac{2\gamma  }{q}+k)\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}
 {(\gamma )_{k}k!} \\
&\quad\times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},
 \frac{2\gamma }{q}+k;\gamma; 1) \\
 &= T_{m}T_{n}\frac{\beta ^{-\frac{\gamma }{q}}}{q}
 \Gamma (\frac{2\gamma  }{q})\sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}
 \cdots(a_{p})_{k}(\frac{2\gamma  }{q})_{k}}{(\gamma )_{k}k!} \\
&\quad\times _{p+1}  F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma }{q}+k;\gamma; 1)
% \label{eq:overhead-gen}
\end{align*}
This completes the proof.
\end{proof}


\begin{lemma} \label{lem4}
For $\gamma >1$, and $m,n=1,2,\dots $,
the matrix elements of the generalized spiked harmonic
oscillator Hamiltonians \eqref{e2}  can be written as
\begin{align}
&H_{mn} \nonumber\\
&= T_{m}T_{n}[\beta ^{\frac{1-\gamma  }{q}}\frac{(A-\gamma ^2+2\gamma
 -\frac{3}{4})}{q}
  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
  \Gamma (\frac{2\gamma  -2}{q}+k)   \nonumber\\
&\quad  \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma  -2}{q}
 +k;\gamma; 1)  \nonumber\\
&\quad  +B^{-\frac{1+\gamma }{q}}  \frac{1 }{q}  \sum_{k=0}^m
 \frac{(-m)_{k}(a_{2})_{k}\cdots (a_{p})_{k}}{(\gamma )_{k}k!}
 \Gamma (\frac{2\gamma  +2}{q}+k)  \nonumber\\
&\quad  \times _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma  +2}{q}+k;
 \gamma; 1)   \nonumber\\
&\quad  -\beta ^{\frac{1-\gamma }{q}} \frac{q}{4}   \sum_{k=0}^m
 \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}\Gamma
 (\frac{2\gamma  -2}{q}+2+k)  \nonumber\\
&\quad  \times _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma  -2}{q}+2+k;
 \gamma; 1) \nonumber\\
&\quad  +\beta ^{\frac{1-\gamma }{q}}(\gamma -1+\frac{q}{2})
 \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
 \Gamma (\frac{2\gamma  -2}{q}+1+k) \nonumber\\ &\quad  \times  _{p+1}F_{1}(-n,a_{2},
 \ldots ,a_{p},\frac{2\gamma  -2}{q}+1+k;\gamma; 1) \nonumber\\
&\quad  +\beta ^{\frac{1-\gamma }{q}} (q+2(\gamma -1)) \frac{na_{2}
 \cdots a_{p}}{\gamma }  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}
 \cdots(a_{p})_{k}}{(\gamma )_{k}k!}  \Gamma (\frac{2\gamma  -2}{q}+1+k) \nonumber\\
&\quad \times  _{p+1}F_{1}(1-n,1+a_{2},\ldots ,1+a_{p},\frac{2\gamma  -2}{q}
 +1+k;1+\gamma; 1) \nonumber\\
&\quad  -\beta ^{\frac{1-\gamma }{q}} q \frac{na_{2}\cdots a_{p}}{\gamma }
  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}
 \Gamma (\frac{2\gamma  -2}{q}+2+k) \nonumber\\
&\quad  \times _{p+1}F_{1}(1-n,1+a_{2},\ldots ,1+a_{p},\frac{2\gamma  -2}{q}
 +2+k;1+\gamma; 1) \nonumber\\
&\quad -\beta ^{\frac{1-\gamma }{q}} q  \frac{n(-1+n)a_{2}(1+a_{2})
 \cdots a_{p}(1+a_{p})}{\gamma (1+\gamma )}  \sum_{k=0}^m
 \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}  \nonumber\\
&\quad  \times \Gamma (\frac{2\gamma  -2}{q}+2+k)  _{p+1}F_{1}(2-n,2+a_{2},
 \ldots ,2+a_{p},\frac{2\gamma  -2}{q}+2+k;2+\gamma; 1) \nonumber\\
&\quad  +\lambda \beta ^{\frac{\alpha -2\gamma }{2q}}\frac{1}{q}
 \Gamma (\frac{2\gamma -\alpha }{q})  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}
 \cdots(a_{p})_{k}(\frac{2\gamma -\alpha }{q})_{k}}{(\gamma )_{k}k!} \nonumber\\
&\quad  \times  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2\gamma -\alpha }{q}
 +k;\gamma; 1) \label{e28}
\end{align}
\end{lemma}

\begin{proof}
 By writing the Hamiltonian $H$ in compactified form as
\begin{align*}
H_{mn}=\prec \psi _{m}\mid -\frac{d^2}{dr^2}\mid \psi _{n}\succ
+\beta  r^2_{mn}+ A r^{-2 }_{mn} +\lambda r^{-\alpha }_{mn}
\end{align*}
where $ r^{2 }_{mn} $ is given by \eqref{e22} $ r^{-2 }_{mn} $ is given by
\eqref{e16}, and $r^{-\alpha  }_{mn}$ is given by \eqref{e11}.
By using  \eqref{e3}  it is follows that
\begin{equation} \label{e29}
\begin{aligned}
H_{mn}
&=-\int_{0}^\infty {r^{\gamma -\frac{1}{2}}e^{-\frac{\sqrt{B}}{2}r^q}} _{p}
 F_{1}( -m,a_{2},\ldots ,a_{p};\gamma; \sqrt {B} r^q)   \\
&\quad \times\frac{d^{2}}{d r^2}(r^{\gamma -\frac{1}{2}}
 e^{-\frac{\sqrt{B}}{2}r^q}\  _{p}F_{1}( -n,a_{2},\ldots ,a_{p};\gamma;
 \sqrt {B} r^q))  dr \\
&\quad  + \beta  r^2_{mn}+ A r^{-2 }_{mn} +\lambda r^{-\alpha }_{mn}
\end{aligned}
\end{equation}
 We denote the first term on the right-hand side of \eqref{e29} by
$ I_{mn} $and have by means of the series representation \eqref{e4}
of the hypergeometric function $_{p}F_{1}$, that
\begin{align*} %\label{e30}
I_{mn}
&=(-\gamma^2 +2\gamma -\frac{3}{4})\sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}
  (a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}
  {(\gamma )_{k}(\gamma )_{l}}\frac{B^{\frac{l+k}{2}}}{k!l!} \\
&\quad \times  \int_{0}^\infty {r^{2\gamma -3 +qk+ql}e^{-\sqrt{B}r^q}} dr \\
&\quad -B\frac{q^2}{4}\sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}
 (\gamma )_{l}}\frac{B^{\frac{l+k}{2}}}{k!l!} \\
&\quad  \times \int_{0}^\infty {r^{2\gamma -3+2q+qk+ql}e^{-\sqrt{B}r^q}} dr\\
&\quad +\sqrt{B}q(\gamma -1+\frac{q}{2})\sum_{k=0}^m \sum_{l=0}^n
  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}
  \cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}}\frac{B^{\frac{l+k}{2}}}{k!l!} \\
&\quad \times \int_{0}^\infty {r^{2\gamma -3+q+qk+ql}e^{-\sqrt{B}r^q}} dr
     +\sqrt{B}q(2(\gamma -1)+q) \frac{na_{2}\cdots a_{p}}{\gamma }
  \sum_{k=0}^m \\
&\quad\times \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(1-n )_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}}
  {(\gamma )_{k}(1+\gamma )_{l}}  \\
&\quad  \times  \frac{B^{\frac{l+k}{2}}}{k!l!} \int_{0}^\infty
 {r^{2\gamma -3+q+qk+ql}e^{-\sqrt{B}r^q}} dr \\
&\quad -B q^2 \frac{na_{2}\cdots a_{p}}{\gamma }\sum_{k=0}^m
  \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(1-n )_{l}
  (1+a_{2})_{l}\cdots(1+a_{p})_{l}}{(\gamma )_{k}(1+\gamma )_{l}}\\
&\quad \times \frac{B^{\frac{l+k}{2}}}{k!l!}
   \int_{0}^\infty  {r^{2\gamma -3+2q+qk+ql}e^{-\sqrt{B}r^q}} dr \\
&\quad  -B q^2
  \frac{n(-1+n)a_{2}(1+a_{2})\cdots a_{p}(1+a_{p})}{\gamma (1+\gamma )} \\
&\quad  \times \sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(2-n )_{l}(2+a_{2})_{l}\cdots(2+a_{p})_{l}}
  {(\gamma )_{k}(2+\gamma )_{l}}\\
&\quad\times  \frac{B^{\frac{l+k}{2}}}{k!l!}
 \int_{0}^\infty {r^{2\gamma -3+2q+qk+ql}e^{-\sqrt{B}r^q}} dr
\end{align*}
 Further, after restoring the integral representation of the gamma
 function and a change of variables, we obtain for
$ \frac{2(\gamma -1)}{q}+k+l>0, \frac{2(\gamma -1)}{q}+1+k+l>0 $
and $ \frac{2(\gamma -1)}{q}+$ $2+k+l>0$ that
\begin{align} % \label{e31}
&I_{mn}\nonumber\\
&=B^{\frac{1-\gamma }{q}}\frac{(-\gamma^2 +2\gamma -\frac{3}{4})}{q}
 \sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}
  {(\gamma )_{k}(\gamma )_{l}}\nonumber\\
&\quad\times  \frac{\Gamma(\frac{2(\gamma -1)}{q}+k+l)}{k!l!} \nonumber\\
&\quad  -B^{\frac{l-\gamma }{q}}\frac{q}{4}\sum_{k=0}^m \sum_{l=0}^n
  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}
  \cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}}
  \frac{\Gamma (\frac{2(\gamma -1)}{q}+2+k+l)}{k!l!} \nonumber\\
&\quad  +B^{\frac{l-\gamma }{q}}(\gamma -1+\frac{q}{2})\sum_{k=0}^m
 \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}(a_{2})_{l}
  \cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}}\nonumber\\
&\quad\times  \frac{\Gamma(\frac{2(\gamma -1)}{q}+1+k+l)}{k!l!}
 + B^{\frac{1-\gamma }{q}}(2(\gamma -1)+q)
  \frac{na_{2}\cdots a_{p}}{\gamma }\sum_{k=0}^m \nonumber\\
&\quad\times  \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(1-n )_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}}{(\gamma )_{k}
  (1+\gamma )_{l}}
    \frac{\Gamma (\frac{2(\gamma -1)}{q}+1+k+l)}{k!l!}   \nonumber\\
 &\quad  -B^{\frac{1-\gamma }{q}} q \frac{na_{2}\cdots a_{p}}{\gamma }
  \sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
  \cdots(a_{p})_{k}(1-n )_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}}{(\gamma )_{k}
  (1+\gamma )_{l}}  \nonumber\\
&\quad  \times\frac{\Gamma (\frac{2(\gamma -1)}{q}+2+k+l)}{k!l!}
  -B^{\frac{1-\gamma }{q}} q  \frac{n(-1+n)a_{2}(1+a_{2})\cdots
  a_{p}(1+a_{p})}{\gamma (1+\gamma )}   \nonumber\\
&\quad  \times \sum_{k=0}^m \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}
 \cdots(a_{p})_{k}(2-n )_{l}(2+a_{2})_{l}\cdots(2+a_{p})_{l}}{(\gamma )_{k}
  (2+\gamma )_{l}}\nonumber\\
&\quad\times\frac{\Gamma (\frac{2(\gamma -1)}{q}+2+k+l)}{k!l!} \nonumber\\
&=-B^{\frac{1-\gamma }{q}}\frac{(-\gamma^2 +2\gamma
  -\frac{3}{4})}{q}\sum_{k=0}^m \Big[\sum_{l=0}^n\frac{(-n )_{l}(a_{2})_{l}
  \cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}}
  \frac{\Gamma(\frac{2(\gamma -1)}{q}+k+l)}{l!}\Big]\nonumber\\
&\quad\times \frac{(-m)_{k}(a_{2})_{k}  \cdots(a_{p})_{k}}{k!} \nonumber\\
&\quad -B^{\frac{l-\gamma }{q}}\frac{q}{4}\sum_{k=0}^m\Big[
  \sum_{l=0}^n\frac{(-n )_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}
  (\gamma )_{l}}\frac{\Gamma (\frac{2(\gamma -1)}{q}+2+k+l)}{l!}\Big]\nonumber\\
&\quad\times  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{k!}
  +B^{\frac{l-\gamma }{q}}(\gamma -1+\frac{q}{2})\sum_{k=0}^m\nonumber\\
&\quad\times  \Big[ \sum_{l=0}^n\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}(-n )_{l}
  (a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{k}(\gamma )_{l}}
   \frac{\Gamma(\frac{2(\gamma -1)}{q}+1+k+l)}{l!}\Big] \nonumber\\
&\quad \times \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{k!} \nonumber\\
&\quad  + B^{\frac{1-\gamma }{q}}(2(\gamma -1)+q) \frac{na_{2}
  \cdots a_{p}}{\gamma }\sum_{k=0}^m\Big[ \sum_{l=0}^n\frac{(1-n )_{l}
  (1+a_{2})_{l}\cdots(1+a_{p})_{l}}{(\gamma )_{k}(1+\gamma )_{l}}  \nonumber\\
&\quad  \times \frac{\Gamma (\frac{2(\gamma -1)}{q}+1+k+l)}{k!}\Big]
  \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k} }{k!}  \nonumber\\
&\quad  -B^{\frac{1-\gamma }{q}} q \frac{na_{2}\cdots a_{p}}{\gamma }
  \sum_{k=0}^m\Big[ \sum_{l=0}^n\frac{(1-n )_{l}(1+a_{2})_{l}\cdots
  (1+a_{p})_{l}}{(\gamma )_{k}(1+\gamma )_{l}}\frac{\Gamma
  (\frac{2(\gamma -1)}{q}+2+k+l)}{l!}\Big]   \nonumber\\
&\quad  \times\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{k!}
  -B^{\frac{1-\gamma }{q}} q  \frac{n(-1+n)a_{2}(1+a_{2})
  \cdots a_{p}(1+a_{p})}{\gamma (1+\gamma )} \nonumber\\
&\quad\times \sum_{k=0}^m
  \Big[\sum_{l=0}^n\frac{(2-n )_{l}(2+a_{2})_{l}\cdots(2+a_{p})_{l}}
  {(\gamma )_{k}(2+\gamma )_{l}}
 \frac{\Gamma (\frac{2(\gamma -1)}{q}+2+k+l)}{l!}\Big]\nonumber\\
&\quad\times
\frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{k!} \label{e31}
\end{align}
We denote the finite sum inside the brackets in the expression above
as  $ I_{1mn}$, $I_{2mn}$, \ldots, $I_{6mn}$. On the
other hand, by using of the definition of the pochhammer symbols
and the series representation of the hypergeometric
functions \eqref{e4}, the finite sum inside the bracket collapses to
 \begin{equation} \label{e32}
\begin{aligned}
 I_{1mn}&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}
 \cdots(a_{p})_{l}}{(\gamma )_{l}l!} \Gamma(\frac{2(\gamma  -1)}{q}+k+l) \\
&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}
  (\frac{2(\gamma -1)}{q} +k)_{l}}{(\gamma )_{l}l!}
  \Gamma(\frac{2(\gamma  -1)}{q}+k)   \\
&=  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2(\gamma  -1)}{q} +k;\gamma; 1)
 \Gamma(\frac{2(\gamma  -1)}{q}+k)
\end{aligned}
 \end{equation}
\begin{equation}\label{e33}
\begin{aligned}
I_{2mn}&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{l}l!}  \Gamma(\frac{2(\gamma  -1)}{q}+1+k+l) \\
       &= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}(\frac{2(\gamma -1)}{q}+1 +k)_{l}}{(\gamma )_{l}l!} \Gamma(\frac{2(\gamma  -1)}{q}+1+k) \\
       &=  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2(\gamma  -1)}{q}+1 +k;
\gamma; 1) \Gamma(\frac{2(\gamma  -1)}{q}+1+k)
\end{aligned}
\end{equation}
\begin{equation} \label{e34}
\begin{aligned}
I_{3mn}&= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}}{(\gamma )_{l}l!} \Gamma(\frac{2(\gamma  -1)}{q}+2+k+l)  \\
       &= \sum_{l=0}^n\frac{(-n)_{l}(a_{2})_{l}\cdots(a_{p})_{l}(\frac{2(\gamma -1)}{q}+2 +k)_{l}}{(\gamma )_{l}l!} \Gamma(\frac{2(\gamma  -1)}{q}+2+k) \\
       &=  _{p+1}F_{1}(-n,a_{2},\ldots ,a_{p},\frac{2(\gamma  -1)}{q}+2+k;
\gamma; 1)  \Gamma(\frac{2(\gamma  -1)}{q}+2+k)
\end{aligned}
\end{equation}
\begin{equation} \label{e35}
\begin{aligned}
I_{4mn}&= \sum_{l=0}^n\frac{(1-n )_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}}{(1+\gamma )_{l}l!} \Gamma (\frac{2(\gamma -1)}{q}+1+k+l) \\
       &= \sum_{l=0}^n\frac{(1-n)_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}(\frac{2(\gamma -1)}{q}+1 +k)_{l}}{(1+\gamma )_{l}l!} \Gamma(\frac{2(\gamma  -1)}{q}+1+k) \\
       &=    _{p+1}F_{1}(1-n,1+a_{2},\ldots ,1+a_{p},
\frac{2(\gamma  -1)}{q}+1+k;1+\gamma; 1)\\
&\quad\times \Gamma(\frac{2(\gamma  -1)}{q}+1+k)
\end{aligned}
\end{equation}
\begin{equation} \label{e36}
\begin{aligned}
I_{5mn}&= \sum_{l=0}^n\frac{(1-n )_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}}{(1+\gamma )_{l}l!} \Gamma (\frac{2(\gamma -1)}{q}+2+k+l) \\
       &= \sum_{l=0}^n\frac{(1-n)_{l}(1+a_{2})_{l}\cdots(1+a_{p})_{l}(\frac{2(\gamma -1)}{q}+2 +k)_{l}}{(1+\gamma )_{l}l!} \Gamma(\frac{2(\gamma  -1)}{q}+2+k) \\
       &=  _{p+1}F_{1}(1-n,1+a_{2},\ldots ,1+a_{p},
  \frac{2(\gamma  -1)}{q}+2+k;1+\gamma; 1) \\
&\quad\times \Gamma(\frac{2(\gamma  -1)}{q}+2+k)
\end{aligned}
\end{equation}
\begin{equation} \label{e37}
\begin{aligned}
I_{6mn}&= \sum_{l=0}^n\frac{(2-n )_{l}(2+a_{2})_{l}\cdots(2+a_{p})_{l}}{(2+\gamma )_{l}}  \frac{\Gamma (\frac{2(\gamma -1)}{q}+2+k+l)}{l!} \\       &= \sum_{l=0}^n\frac{(2-n)_{l}(2+a_{2})_{l}\cdots(2+a_{p})_{l}(\frac{2(\gamma -1)}{q}+2 +k)_{l}}{(2+\gamma )_{l}l!}\Gamma(\frac{2(\gamma  -1)}{q}+2+k) \\
       &=    _{p+1}F_{1}(2-n,2+a_{2},\ldots ,2+a_{p},
  \frac{2\gamma  -2}{q}+2+k;2+\gamma; 1)\\
&\quad\times \Gamma(\frac{2(\gamma  -1)}{q}+2+k)
\end{aligned}
 \end{equation}

Substituting \eqref{e32},  \eqref{e33},  \eqref{e34},  \eqref{e35},
\eqref{e36} and \eqref{e37} into \eqref{e31} we obtain
\begin{align*}
I_{mn}
&= -B^{\frac{1-\gamma }{q}}\frac{(\gamma ^2-2\gamma +\frac{3}{4})}{q}
  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!} I_{1mn}  \\
&\quad  -B^{\frac{1-\gamma }{q}} \frac{q}{4}   \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!} I_{2mn} \\
&\quad  +B^{\frac{1-\gamma }{q}}(\gamma -1+\frac{q}{2})  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!} I_{3mn} \\
&\quad  +B^{\frac{1-\gamma }{q}} (q+2(\gamma -1)) \frac{na_{2}\cdots a_{p}}{\gamma }  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}   I_{4mn} \\
&\quad -B^{\frac{1-\gamma }{q}} q \frac{na_{2}\cdots a_{p}}{\gamma }  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}   I_{5mn}  \\
&\quad  -B^{\frac{1-\gamma }{q}} q  \frac{n(-1+n)a_{2}(1+a_{2})\cdots a_{p}(1+a_{p})}{\gamma (1+\gamma )}  \sum_{k=0}^m \frac{(-m)_{k}(a_{2})_{k}\cdots(a_{p})_{k}}{(\gamma )_{k}k!}   I_{6mn}
\end{align*}
This completes the proof.
\end{proof}

In the following tables, we used the derived formulas
to compute upper bounds for many problems of interest
in the literature. We compare our results with other
authors who analyze the spectrum for this kind of
problems. One of the advantage and the purpose of such
formulas is to provide the researcher with simple
method to compute the spectrum for sum of power-law potentials in
the literature other than the saving in the time in these
computations.

Table \ref{tabl1} shows the eigenvalues of $H = -\frac{d^2}{dr^2} + V(r)$,
where $V(r)=r^2 +\frac{\lambda }{r^{2.5}}$ anharmonic oscillator
potential for  different values of $ \lambda $ using
\[
 \psi _n(r)=\sqrt{\frac{2\beta ^{\frac{\gamma }{2}}(\gamma )_n}
{n! \Gamma (\gamma )}} r^{\gamma -\frac{1}{2}}
e^{-\frac{\sqrt{\beta }r^q}{2}}\  _3F_1(-n,0.2,0.07;\gamma
;\sqrt{\beta }r^q)
\]
obtained using the computed matrix elements \eqref{e28} and minimizing
with respect to $q , \gamma ,  \rm{and} \beta $. The exponent $(n)$
refer to the dimensions of the matrix used for the variational computations.

{\footnotesize
\begin{table}[ht]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline
   $\lambda$ & $q$ & $\gamma $ & $\beta $   & $ E^{(3)}$ & $ E^{(4)}$&
$ E^{(5)}$ & $E$\\ \hline
    100   & 1.824 & 11.117 & 2.4115  & 17.541 916  & 17.541 912  & 17.541 912 &17.541 890\\ \hline
    10    & 1.811 & 4.797  & 2.154   &  7.735 317  &  7.735 256  & 7.735 254 & 7.735 111\\ \hline
    1     & 1.801 & 2.553  & 1.917   & 4.318 485   &  4.318 181  & 4.318 180 & 4.317 312\\ \hline
    0.5   & 1.807 & 2.229  & 1.8169  & 3.850 244   & 3.850 028   & 3.849 862 & 3.848 553\\ \hline
    0.01  & 1.95  & 1.55   & 1.132   & 3.037 422   & 3.037 421   & 3.037 415 & 3.036 665\\ \hline
    0.001 &1.993  & 1.506  & 1.017   & 3.004 046 2 & 3.004 046 2 & 3.004 046 1& 3.004 014\\ \hline
  \end{tabular}
   \end{center}
\caption{} \label{tabl1}
\end{table}}


Table \ref{tabl2}
Shows a comparison between the results of$  E^{HS} $and the results of
the present work$ E^U $ using
\[
 \psi _n(r)=\sqrt{\frac{2\beta ^{\frac{\gamma }{2}}(\gamma )_n}{n! \Gamma
(\gamma )}} r^{\gamma -\frac{1}{2}} e^{-\frac{\sqrt{\beta
}r^q}{2}}\  _3F_1(-n,0.2,0.07;\gamma ;\sqrt{\beta }r^q)
\]
 and
$ H = -\frac{d^2}{dr^2} +V(r) $ where $V(r)= r^2 + \frac{\lambda
}{r^{2.5}}$ anharmonic oscillator potential for different values
of $ \lambda $, obtained using the computed matrix elements
\eqref{e28} and minimizing with respect to $q , \gamma ,\
\rm{and} \beta  $.
 The exponent $(n)$ refer to the dimensions of the matrix used for
the variational computations.

{\footnotesize
\begin{table}[ht]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|}\hline
   $\lambda$ & $q$ & $\gamma $ & $\beta $ & $E^{HS}$ & $E_m^{HS}$ & $E^U$ & $E$\\ \hline
100   & 1.824 & 11.117 & 2.4115 & 17.541 890$^{(30)}$  & 17.542 040$^{(5)}$   &17.541 912$^{(5)}$  &17.541 890\\ \hline
10    & 1.811 & 4.797  & 2.154  & 7.735 637$^{(30)}$   & 7.735 596$^{(5)}$    & 7.735 254$^{(5)}$  &7.735 111\\ \hline
1     & 1.801 & 2.553  & 1.917  & 4.323 263$^{(30)}$   & 4.318 963$^{(5)}$    &  4.318 141$^{(7)}$ &4.317 312\\ \hline
0.5   & 1.807 & 2.229  & 1.8169 & 3.869 547$^{(30)}$   & 3.850 823$^{(5)}$    & 3.849 759$^{(7)}$  & 3.848 553\\ \hline
0.01  & 1.95  & 1.55   & 1.132  & 3.039 244$^{(30)}$   & 3.037 474$^{(5)}$    &  3.037 399$^{(8)}$ & 3.036 665\\ \hline
0.001 & 1.993 & 1.506  & 1.017  & 3.004 074$^{(30)}$   & 3.004 047$^{(5)}$     & 3.004 046$^{(8)}$  & 3.004 014\\ \hline
\end{tabular}
   \end{center}
\caption{$E^{HS}$ from  \cite{h7}. $E^{HS}_m$ from \cite{h7} after
minimizing with respect to one parameter.}\label{tabl2}
\end{table}}

Table \ref{tabl3} shows a
 comparison between the results of $  E^{B} $and the results of
the present work $ E^U $ using
\[
 \psi _n(r)=\sqrt{\frac{2\beta ^{\frac{\gamma }{2}}(\gamma )_n}{n! \Gamma
(\gamma )}} r^{\gamma -\frac{1}{2}} e^{-\frac{\sqrt{\beta
}r^q}{2}}\  _3F_1 (-n,0.2,0.07;\gamma ;\sqrt{\beta }r^q)
\]
 and
$ H = -\frac{d^2}{dr^2} +V(r) $where $V(r)= r^4-\lambda  r^2 $
anharmonic oscillator potential for different values of $ \lambda
$, obtained using the computed matrix elements \eqref{e28} and
minimizing with respect to $q , \gamma ,  \rm{and}\  \beta  $. The
exponent $(n)$ refer to the dimensions of the matrix used for the
variational
 computations.

{\footnotesize
\begin{table}[ht]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|}\hline
   $\lambda$ & $q$ & $\gamma $ & $\beta $   & $E^{B}$     & $E^U$ & $E$\\ \hline
2.0   & 2.928 & 1.472 & 0.425 & 1.726 29$^{(10)}$  & 1.713 36$^{(9)}$   & 1.713 03 \\ \hline
1.0   & 2.023 & 1.469 & 0.927 & 2.838 91$^{(10)}$  & 2.835 34$^{(8)}$   & 2.834 54\\ \hline
0.9   & 2.616 & 1.469 & 0.989 & 2.941 23$^{(10)}$  & 2.937 73$^{(9)}$   & 2.937 30\\ \hline
0.8   & 2.595 & 1.470 & 1.052 & 3.042 10$^{(10)}$  & 3.039 06$^{(8)}$   & 3.038 56\\ \hline
0.7   & 2.574 & 1.470 & 1.117 & 3.141 55$^{(10)}$  & 3.138 86$^{(8)}$   & 3.138 37\\ \hline
0.6   & 2.536 & 1.740 & 1.184 & 3.239 62$^{(10)}$  & 3.237 24$^{(9)}$   & 3.236 76\\ \hline
0.5   & 2.536 & 1.470 & 1.253 & 3.336 36$^{(10)}$  & 3.334 12$^{(9)}$   & 3.333 78\\ \hline
0.4   & 2.518 & 1.471 & 1.323 & 3.431 79$^{(10)}$  & 3.429 94$^{(9)}$   & 3.429 47\\ \hline
0.3   & 2.500 & 1.471 & 1.394 & 3.525 96$^{(10)}$  & 3.524 02$^{(9)}$   & 3.523 87 \\ \hline
0.2   & 2.484 & 1.472 & 1.467 & 3.618 90$^{(10)}$  & 3.617 47$^{(8)}$   & 3.617 01\\ \hline
0.1   & 2.468 & 1.472 & 1.542 & 3.710 64$^{(10)}$  & 3.709 39$^{(8)}$   & 3.708 93\\ \hline
\end{tabular}
   \end{center}
 \caption{$E^{B}$ from \cite{b1}.}\label{tabl3}
\end{table}}


\subsection*{Conclusion}

In this paper we compute closed formula for the matrix
elements for Schr\"odinger equation using generalized
hypergeometric function. We used general basis using special kind
of functions obtained from the exact solutions of the
Gol'dman and Krivchenkov eigenvalue problem \cite{z2}.
In fact, for specific values of the
constants $p $ and $q $ in \eqref{e3} we retrieve to the old results
found in the literature \cite{h1,h5,h6,h7,h8}. One of the
advantages of our results is to obtain analytical formulas that
provide us with accurate bounds for the non-solvable singular
eigenvalues problems as well as saves time of the
computations using the derived formulas.


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 \end{document}