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arXiv:2512.09905

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

Francisco M. Fernández

quant-phSubmitted Dec 10, 2025• Announced Dec 11, 2025• 0 citations• Revised Sep 9, 2026 (v3)
v1 · Dec 10, 2025v2 · Mar 15, 2026v3 · Sep 9, 2026
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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 exhibits 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 but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schrödinger equation for the first example is exactly solvable.

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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 exhibits the same two-fold degeneracy as the particle on a circular path. More precisely, the spectrum is En=n2E1, n=0,±1,±2,…E_{n}=n^{2}E_{1},\ n=0,\pm 1,\pm 2,\ldots En​=n2E1​, n=0,±1,±2,…, E1>0E_{1}>0E1​>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 but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schrödinger equation for the first example is exactly solvable.
We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but isomorphic to an Hermitian operator. It appears to exhibit the same two-fold degeneracy as the particle on a circular path. More precisely, En=n2E1,n=1,2,…{E_{n} = {n^{2}E_{1}}},{n = {1,2,\ldots}}En​=n2E1​,n=1,2,… (in addition to an exact eigenvalue E0=0E_{0} = 0E0​=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 the same point-group symmetry.
We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but isomorphic to an Hermitian operator. It appears to exhibit the same two-fold degeneracy as the particle on a circular path. More precisely, En=n2E1,n=1,2,…{E_{n} = {n^{2}E_{1}}},{n = {1,2,\ldots}}En​=n2E1​,n=1,2,… (in addition to an exact eigenvalue E0=0E_{0} = 0E0​=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.
In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h , obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT -invariant Hamiltonian, with potential V = i gx 3 . In this case the corresponding h , which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P .
In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V = igx 3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P. 1
This chapter presents a detailed examination of a number of relatively simple problems pertaining to the scattering and bound states of two structureless particles interacting through a central field. It describes the spectrum of the angular momentum operators by a method that does not refer at all to the structure of the system. The spherical harmonics are the eigenfunctions appropriate to a single particle that can be described by the three classical degrees of freedom, or for a pair of such particles in their center-of-mass frame. The formulation of collision theory that is based on the one-body Schrodinger equation with a Hermitian interaction. The chapter introduces and discusses the observables pertaining to the two-body problem and explores the eigenvalues and eigenfunctions of these observables. Since the degeneracies of the energy spectrum are a consequence of the symmetries of the Hamiltonian, they can be determined experimentally by destroying this symmetry.