A Fulling-Kuchment theorem for the 1D harmonic oscillator

TL;DR

This paper constructs a pair of 1D semiclassical Schr"odinger operators with different potentials that share identical spectra modulo all orders of h, revealing limitations in spectral uniqueness for inverse problems.

Contribution

It establishes a Fulling-Kuchment type theorem for the harmonic oscillator, showing non-isospectral operators can have identical spectral invariants in the semiclassical regime.

Findings

Existence of non-isospectral operators with identical spectra modulo h^ -infinity

Spectral invariants do not distinguish certain different potentials

Application of Hadamard's variational formula to harmonic oscillator perturbations

Abstract

We prove that there exists a pair of "non-isospectral" 1D semiclassical Schr\"odinger 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 Schr\"odinger operators, trace invariants, Hadamard's variational formula, harmonic oscillator, Penrose mushroom, Sturm-Liouville theory.