Perturbation expansions for a class of singular potentials

TL;DR

This paper extends Harrell's modified perturbation theory to derive non-power series expansions for a class of singular Hamiltonians, specifically generalized spiked harmonic oscillators, valid for small coupling constants and including excited states.

Contribution

It introduces extended perturbation expansions for singular potentials, generalizing previous results to include a broader class of Hamiltonians and excited states.

Findings

Derived non-power perturbation series for generalized spiked harmonic oscillators.

Extended perturbation formulas to excited states.

Validated expansions for small coupling constants.

Abstract

Harrell's modified perturbation theory [Ann. Phys. 105, 379-406 (1977)] is applied and extended to obtain non-power perturbation expansions for a class of singular Hamiltonians H = -D^2 + x^2 + A/x^2 + lambda/x^alpha, (A\geq 0, alpha > 2), known as generalized spiked harmonic oscillators. The perturbation expansions developed here are valid for small values of the coupling lambda > 0, and they extend the results which Harrell obtained for the spiked harmonic oscillator A = 0. Formulas for the the excited-states are also developed.