Two simple models derived from a quantum-mechanical particle on an elliptical path

TL;DR

This paper explores two quantum models based on a particle moving along an elliptical path, analyzing their spectral properties and symmetries, with one model being non-Hermitian yet equivalent to a Hermitian form.

Contribution

It introduces and compares two simple quantum models derived from elliptical particle paths, highlighting their spectral degeneracies and symmetry group descriptions.

Findings

Non-Hermitian model has the same degeneracy as circular path case.

Hermitian model shows energy level splitting at perturbation order.

Both models are characterized by symmetry point groups.

Abstract

We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but equivalent to an Hermitian operator. It appears to exhibit the same two-fold degeneracy as the particle on a circular path. More precisely, the spectrum is $E_{n}=n^{2}E_{1},\ n=0,\pm 1,\pm 2,\ldots $, $E_{1}>0$. The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of symmetry point groups with one-dimensional irreducible representations.