Unitary transformations of a family of two-dimensional anharmonic oscillators

TL;DR

This paper investigates two-dimensional anharmonic oscillators using unitary transformations, revealing separability, issues with unbounded models, and calculating resonances to improve understanding of their spectral properties.

Contribution

It demonstrates how unitary transformations can simplify the analysis of anharmonic oscillators and addresses challenges in perturbation theory for unbounded models.

Findings

Two models are shown to be separable.

An unbounded model requires complex harmonic frequency for perturbation.

Resonance calculations align with previous eigenvalue estimates.

Abstract

In this paper we analyze a recent application of perturbation theory by the moment method to a family of two-dimensional anharmonic oscillators. By means of straightforward unitary transformations we show that two of the models studied by the authors are separable. Other is unbounded from below and therefore cannot be successfully treated by perturbation theory unless a complex harmonic frequency is introduced in the renormalization process. We calculate the lowest resonance by means of complex-coordinate rotation and compare its real part with the eigenvalue estimated by the authors. A pair of the remaining oscillators are equivalent as they can be transformed into one another by unitary transformations.