Matrix elements for sum of power-law potentials in quantum mechanic using generalized hypergeometric functions

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 $$ psi _n(r)= sqrt{{frac{2eta ^{gamma/2}(gamma )_n} {n!Gamma(gamma )}}} r^{gamma - 1/2} e^{-frac{sqrt{eta }}{2}r^q} _pF_1 ( -n,a_2,ldots ,a_p;gamma;sqrt {eta } r^q), $$ for $eta, q,gamma >0$ to obtain the matrix elements for $hat H$. These formulas are then optimized with respect to variational parameters $eta ,q$ 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 [2], [8-16] for power-law potentials. Applications and comparisons with earlier work are presented.