Isoperiodic classical systems and their quantum counterparts

TL;DR

This paper explores classical isoperiodic systems and their quantum counterparts, revealing how classical equivalences often break down upon quantization and identifying conditions for spectral preservation or divergence.

Contribution

It extends Abel's classical characterization of isoperiodic systems to singular and partial potentials, and analyzes the quantum spectral anomalies arising from classical isoperiodic equivalences.

Findings

Classical isoperiodic potentials are connected by translations, reflections, or Joukowski transformations.

Quantum spectra of isoperiodic classical systems often differ at order O(h^2) due to semiclassical corrections.

Some systems maintain spectral identity upon quantization, while others exhibit quantum anomalies.

Abstract

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel's characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(h^2) because semiclassical corrections of energy levels of order O(h) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.