Band invariants for perturbations of the harmonic oscillator

2011/09/02 by Victor Guillemin, Guillemin, Victor, Alejandro Uribe +3
Mathematics · Physics and Astronomy · #35P20 #81Q20 #FOS: Mathematics #Numerical methods in inverse problems #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1109.0567

openalex publication_date 2011/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the direct and inverse spectral problems for semiclassical operators of the form S = S₀ +\h²V, where S₀ = \frac 12 (-\h²Δ_\bbRⁿ + |x|²) is the harmonic oscillator and V:\bbRⁿ→\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).

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