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A Fulling-Kuchment theorem for the 1D harmonic oscillator

We prove that there exists a pair of "non-isospectral" 1D semiclassical Schrödinger operators whose spectra agree modulo h^\infty. In particular, all their semiclassical trace invariants are the same. Our proof is based on an idea of Fulling-Kuchment and Hadamard's variational formula applied to suitable perturbations of the harmonic oscillator. Keywords: Inverse spectral problems, semiclassical Schrödinger operators, trace invariants, Hadamard's variational formula, harmonic oscillator, Penrose mushroom, Sturm-Liouville theory.

preprint2011arXivOpen access
Victor GuilleminHamid Hezari
math.APmath-phmath.MPmath.SPmath.CA
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Victor GuilleminHamid Hezari

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