Perurbation expansions for the spiked harmonic oscillator and related series involving the gamma function

TL;DR

This paper develops perturbation expansions for the ground-state energy of spiked harmonic oscillators and derives closed-form expressions for related series involving gamma functions, advancing analytical methods in quantum mechanics.

Contribution

It introduces a method to obtain closed-form expressions for series involving gamma functions related to spiked harmonic oscillators, extending previous perturbation techniques.

Findings

Derived perturbation series for the ground-state energy.

Obtained closed-form expressions for gamma function series.

Applied the method to multi-dimensional and generalized potentials.

Abstract

We study weak-coupling perturbation expansions for the ground-state energy of the Hamiltonian with the generalized spiked harmonic oscillator potential V(x) = Bx^2 + A/x^2 + lambda/x^alpha, and also for the bottoms of the angular momentum subspaces labelled by ell = 0,1,2 ..., in N-dimensions corresponding to the spiked harmonic oscillator potential: V(x) = x^2 + lambda/x^alpha, where alpha is a real positive parameter. A method of Znojil is then applied to obtain closed form expressions for the sums of some infinite series whose terms involve ratios and products of gamma functions.