A New Two-Parameter Family of Potentials with a Tunable Ground State

TL;DR

This paper introduces a two-parameter family of quantum potentials that modify the ground state energy of the harmonic oscillator while preserving its spectrum, with applications to Bose-Einstein condensation.

Contribution

It extends previous work by deriving all real partner potentials of the harmonic oscillator with a tunable ground state energy.

Findings

Complete set of real partner potentials depending on two parameters

Ground state energy can be tuned independently

Potential applications to Bose-Einstein condensation analysis

Abstract

In a previous paper we solved a countably infinite family of one-dimensional Schr\"odinger equations by showing that they were supersymmetric partner potentials of the standard quantum harmonic oscillator. In this work we extend these results to find the complete set of real partner potentials of the harmonic oscillator, showing that these depend upon two continuous parameters. Their spectra are identical to that of the harmonic oscillator, except that the ground state energy becomes a tunable parameter. We finally use these potentials to analyse the physical problem of Bose-Einstein condensation in an atomic gas trapped in a dimple potential.