Band invariants for perturbations of the harmonic oscillator
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Guillemin_Band invariants.pdf
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Author(s)
Uribe, A.
Wang, Z.
Guillemin, Victor W.
Date Issued
June 2012
Journal
Journal of Functional Analysis
Publisher
Elsevier
Citation
Guillemin, V., A. Uribe, and Z. Wang. “Band Invariants for Perturbations of the Harmonic Oscillator.” Journal of Functional Analysis 263, no. 5 (September 2012): 1435–1467.
Version
Author's final manuscript
Abstract
We study the direct and inverse spectral problems for semiclassical operators of the form S = S[subscript 0] + ℏ[superscript 2]V, where S[subscript 0] = 1/2(−ℏ[superscript 2]Δ[subscript Rn] + |x|[superscript 2]) is the harmonic oscillator and V:R[superscript n] → R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ℏ 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).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-NoDerivatives
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DOI of Published Version
https://doi.org/10.1016/j.jfa.2012.05.022
