Generalized spiked harmonic oscillator

TL;DR

This paper develops a variational and perturbative approach for a family of generalized spiked harmonic oscillator Hamiltonians, providing explicit matrix elements and convergence properties of the wave function series.

Contribution

It introduces a novel method combining variational and perturbative techniques with closed-form matrix elements for these oscillators.

Findings

Explicit matrix elements derived in closed form.

Wave function series shown to converge for alpha <= 2.

Perturbation series provides accurate approximations.

Abstract

A variational and perturbative treatment is provided for a family of generalized spiked harmonic oscillator Hamiltonians H = -(d/dx)^2 + B x^2 + A/x^2 + lambda/x^alpha, where B > 0, A >= 0, and alpha and lambda denote two real positive parameters. The method makes use of the function space spanned by the solutions |n> of Schroedinger's equation for the potential V(x)= B x^2 + A/x^2. Compact closed-form expressions are obtained for the matrix elements <m|H|n>, and a first-order perturbation series is derived for the wave function. The results are given in terms of generalized hypergeometric functions. It is proved that the series for the wave function is absolutely convergent for alpha <= 2.