Harmonic Oscillator with a Step and its Isospectral Properties

TL;DR

This paper studies a modified harmonic oscillator with a step potential, revealing special solutions related to Hermite polynomials and constructing infinitely many isospectral potentials through Darboux transformations.

Contribution

It introduces a novel analysis of the harmonic oscillator with a step, deriving explicit solutions for specific step values and generating a family of isospectral potentials.

Findings

Solutions expressed by Hermite polynomials for specific step values

Construction of infinitely many isospectral Hamiltonians

Extension of harmonic oscillator spectral properties

Abstract

We investigate the one-dimensional Schr\"{o}dinger equation for a harmonic oscillator with a finite jump $a$ at the origin. The solution is constructed by employing the ordinary matching-of-wavefunctions technique. For the special choices of $a$, $a=4\ell$ ($\ell=1,2,\ldots$), the wavefunctions can be expressed by the Hermite polynomials. Moreover, we explore isospectral deformations of the potential via the Darboux transformation. In this context, infinitely many isospectral Hamiltonians to the ordinary harmonic oscillator are obtained.