Small-energy series for one-dimensional quantum-mechanical models with non-symmetric potentials

TL;DR

This paper extends a small-energy expansion method for one-dimensional quantum models from symmetric to non-symmetric potentials, enabling accurate ground-state energy calculations through perturbation series convergence.

Contribution

It introduces a generalized approach for non-symmetric potentials by matching logarithmic derivatives, expanding the applicability of the small-energy series method.

Findings

Method converges in all tested models

Accurate ground-state energies obtained

Applicable to non-symmetric potentials

Abstract

We generalize a recently proposed small-energy expansion for one-dimensional quantum-mechanical models. The original approach was devised to treat symmetric potentials and here we show how to extend it to non-symmetric ones. Present approach is based on matching the logarithmic derivatives for the left and right solutions to the Schr\"odinger equation at the origin (or any other point chosen conveniently) . As in the original method, each logarithmic derivative can be expanded in a small-energy series by straightforward perturbation theory. We test the new approach on four simple models, one of which is not exactly solvable. The perturbation expansion converges in all the illustrative examples so that one obtains the ground-state energy with an accuracy determined by the number of available perturbation corrections.