\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 $\hbar^2=2m=1$, $$\label{e1} \hat H=-\frac{d^2}{dr^2} +\beta r^{2}+\frac{A}{r^2},\quad (\beta >0, A\geq 0).$$ We present a variational analysis of the generalized spiked harmonic oscillator Hamiltonian \cite{a2,a3,b1}, \cite{g1}--\cite{z2} of the form $$\label{e2} \hat H=-\frac{d^2}{dr^2} +\beta r^{2}+\frac{A}{r^2} +\frac{\lambda }{r^\alpha },\quad 0\leq r<\infty ,$$ 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 $$\label{e3} \psi _{n}(r)= T_{n} r^{\gamma - 1/2} e^{\frac{-\sqrt{\beta }}{2}r^q} \ _{p}F_{1}( -n,a_{2},\ldots ,a_{p};\gamma;\sqrt {\beta } r^q), \quad T_{n}=\sqrt{\frac{2\beta ^{\frac{\gamma }{2}} (\gamma )_{n}}{n!\Gamma(\gamma )}}$$ 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 $$\label{e4} _pF_q(\alpha _1,\alpha _2,\ldots ,\alpha _p;\beta _1,\beta _2,\ldots ,\beta _q; x)= \sum_{k=0}^{\infty }{\frac{\Pi _{i=1}^{p} (\alpha _i)_k}{\Pi _{j=1}^{q}(\beta _j)_k}\frac{x^k}{k!}};$$ 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, $$\label{e5} \Psi (r)=\sum_{n=1}^{D}{c_n \psi _n(r)}$$ where the $c_n$ are variational parameters. The variational energy $E_\Psi$ we obtain with this trial function is given by, $$\label{e6} E_\Psi =\frac{\int{\Psi ^* \hat H \Psi d\tau }}{\int{\Psi ^* \Psi d\tau }} =\frac{\sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m \int\psi _n^* \hat H \psi _m d\tau }}}{\sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m \int\psi _n^*\psi _m d\tau }}}$$ Now, defining the matrix element of $\hat H$ as $$\label{e7} H_{nm}= \int{\psi _n^* \hat H \psi_m d\tau }$$ and using the orthonormal property of the $\psi$ functions, we obtain that $$\label{e8} E_\Psi =\frac{\sum_{n=1}^D{\sum_{m=1}^D{c_n^*c_m H_{nm}}}}{\sum_{n=1}^D {\mid c_n\mid ^2}}$$ 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 $$\label{e9} \sum_{n=1}^D{c_n^*H_{ni}-Ec_i^*}=0.$$ 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 $$\label{e10} \sum_{n=1}^D{c_n H_{in}-E S_{in}c_n}= 0$$ 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 \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{lemma} \begin{proof} Using \eqref{e3}, it follows that \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} Using the series representation \eqref{e4} of the hypergeometric function $_{p}F_{1}$, yields \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} By restoring to the integral representation of gamma function, we obtain under the condition ${\frac{2\gamma -\alpha }{q}+}$ $k+l >0$, that \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} 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 \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} 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 \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} \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 \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{lemma} \begin{proof} Using \eqref{e3}, it immediately follows that \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} Using the series representation \eqref{e4} of the hypergeometric function $_{p}F_{1}$, yields \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} Using the integral representation of gamma function, we obtain under the condition ${\frac{2\gamma +\alpha }{q}}+$ $k+l >0$ that \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} 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 \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} 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 \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} \begin{lemma} \label{lem3} For $\gamma >0$, and $m,n=0,1,2,\dots$, \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{lemma} \begin{proof} Using \eqref{e3}, it follows that \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} By means of the series representation \eqref{e4} of the hypergeometric function $_{p}F_{1}$, that \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} 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 \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} 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 \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} 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 \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} 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 \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} \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} \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} \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} \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} \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} 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. \begin{thebibliography}{00} \bibitem{a1} M. Abramowitz, I. A. Stegun. \emph{Handbook of Mathmatical Functions with Formulas, Graphs, and Mathematical Tables}. New York: Dover; 1972: 555-566. \bibitem{a2} V. C. Aguilera-Navarro, R. Guardiola. \emph{Nonsingular spiked harmonic oscillator}. J. Math. Phys. 1991; 31: 2135. \bibitem{a3} V. C. Aguilera-Navarro, R. Guardiola. \emph{Variational and perturbative schemes for a spiked harmonic oscillator}. J. Math. Phys. 1990; 31: 99. \bibitem{b1} G. R. P. Broges, A. de Souza Dutra, Elso Drigo, and J. R. Ruggiero. \emph{Variational method for excited states from supersymmetric techniques}. Can. J. phys. 2003; 81: 1283. \bibitem{d1} R. H. Dicke, J. P. Wittke. \emph{Introduction to Quantum Mechanics}. 6th ed. Amsterdam: Addison-wesley; 1973: 226-237. \bibitem{f1} S. Fl\"ugge. \emph{Practical Quantum Mechanics}. New York: Springer; 1971. \bibitem{g1} R. Guardiola; J. Ros. \emph{Singular potentials with quasi-exact Dirac bound states}. J. Phys. A: Math. Gen. 1992; 25: 335. \bibitem{h1} R. L. Hall, N. Saad, Q. D. Katatbeh. \emph{Study of anharmonic singular potentials}. J. Math. Phsy. 022104 (2005); 46. \bibitem{h2} Richard L. Hall, Qutaibeh D. Katatbeh, Nasser Saad. \emph{A basis for variational calculations in d dimensions}. J. Phys. A 2004; 37: 11629-11644. \bibitem{h3} R. L. Hall, N. Saad, B. Attila von Keviczky. \emph{Friedrich extensions of Schr\"odinger operators with singular potentials}. J. Math. Anal. Appl. 2004; 292: 274-293. \bibitem{h4} R. L. Hall, N. Saad, A.von Keviczky. \emph{Energy bounds for a class of singular potentials some related series}. J. Phys. A: Math. Gen. 2003; 36: 487-498. \bibitem{h5} R. L. Hall, N. Saad, A.von Keviczky. \emph{Spiked harmonic oscillators}. J. Math. Phys. 2002; 43: 94-112. \bibitem{h6} R. L. Hall, N. Saad, A. von Keviczky. \emph{Generalized spiked harmonic oscillator}. J. Phys. A: Math. Gen. 2001; 34: 1169-1179. \bibitem{h7} R. L. Hall, N. Saad. \emph{Variational analysis for a generalized spiked harmonic oscillator}. J. Phys. A: Math. Gen. 2000; 33: 569-578. \bibitem{h8} R. L. Hall, N. Saad, A. von Keviczky. \emph{Matrix elements for a generalized spiked harmonic oscillator}. J. Math. Phys. 1998; 39: 6345-6351. \bibitem{h9} E. M. Harrel. \emph{Singular perturbation potentials}. Ann. Phys. 1977; 105: 339-406. \bibitem{v1} Y. P. Varshni, N. Nag, R. Roychoudhury. \emph{Study of anharmonic singular potentials}. Can. J. Phys. 1995; 73: 519. \bibitem{z1} M. Znojil. \emph{Potential $V(r)=ar^2+br^{-4}+cr^{-6}$ and a new method of solving the Schr\"odinger equation}. J. Phy. Lett. A 1991; 158: 436. \bibitem{z2} M. Znojil. \emph{Singular anharmonicities and the analytic continued fractions. II. The force $V(r)=ar^2+br^{-4}+cr^{-6}$}. J. Math. Phys. 1990; 31: 108. \end{thebibliography} \end{document}