Band invariants for perturbations of the harmonic oscillator

TL;DR

This paper investigates the spectral properties of perturbed harmonic oscillators, computes key invariants, and demonstrates that certain potentials can be uniquely determined from spectral data, especially in two dimensions.

Contribution

It introduces the concept of band invariants for semiclassical harmonic oscillators and establishes inverse spectral results for determining potentials from spectral information.

Findings

Eigenvalue clusters form as  approaches zero

First two band invariants are computed

Spectral data uniquely determines certain potentials in 2D

Abstract

We study the direct and inverse spectral problems for semiclassical operators of the form $S = S_0 +\h^2V$, where $S_0 = \frac 12 \Bigl(-\h^2\Delta_{\bbR^n} + |x|^2\Bigr)$ is the harmonic oscillator and $V:\bbR^n\to\bbR$ is a tempered smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as $\h$ tends to zero, and compute the first two associated "band invariants". We derive several inverse spectral results for $V$, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation).