On the stable eigenvalues of perturbed anharmonic oscillators in dimension two

TL;DR

This paper analyzes the asymptotic spectrum of perturbed 2D anharmonic oscillators, showing that many eigenvalues can be approximated using Bohr-Sommerfeld quantization and asymptotic expansions, extending normal form methods.

Contribution

It extends the normal form approach to analyze the spectrum of perturbed anharmonic oscillators in two dimensions, providing asymptotic eigenvalue descriptions.

Findings

Eigenvalues approximated by Bohr-Sommerfeld rule

Asymptotic expansion of eigenvalues at infinity

Extension of normal form techniques to 2D oscillators

Abstract

We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and also admit an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in [BLM20b] (see also [PS10]) for the particular case of $T^d$.