Quantization of classical integrable systems. Part IV: systems of resonant oscillators

TL;DR

This paper constructs and analyzes classes of quantum integrable systems based on resonant oscillators with nonlinear couplings, exploring symmetries and special cases like the 1:1:2 resonance.

Contribution

It introduces new quantum integrable models for resonant oscillators, including an exceptional case with unique integrability properties not solely based on symmetry.

Findings

Constructed integrable quantum systems for fully resonant oscillators.

Identified special integrable cases with additional symmetries.

Discovered an exceptional 1:1:2 resonance case with modified symmetrization.

Abstract

By applying methods already discussed in a previous series of papers by the same authors, we construct here classes of integrable quantum systems which correspond to n fully resonant oscillators with nonlinear couplings. The same methods are also applied to a series of nontrivial integral sets of functions, which can be constructed when additional symmetries are present due to the equality of some of the frequencies. Besides, for n=3 and resonance 1:1:2, an exceptional integrable system is obtained, in which integrability is not explicitly connected with this type of symmetry. In this exceptional case, quantum integrability can be realized by means of a modification of the symmetrization procedure.