Band invariants for perturbations of the harmonic oscillator
Author(s)Uribe, A.; Wang, Z.; Guillemin, Victor W.
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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).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Functional Analysis
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.
Author's final manuscript