The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation

TL;DR

This paper investigates how a Gaussian perturbation affects the two lowest eigenvalues of a one-dimensional quantum harmonic oscillator, using Fredholm determinants and eigenfunction scalar products for precise calculations.

Contribution

It introduces a method to accurately compute the perturbed eigenvalues of a harmonic oscillator with Gaussian potential using trace class operators and scalar product techniques.

Findings

Accurate computation of the two lowest eigenvalues as functions of the coupling constant.

Application of Fredholm determinant approach to perturbed quantum systems.

Utilization of Wang's results for eigenfunction scalar products to improve calculations.

Abstract

In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wang's results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling constant $\lambda$.