A Fulling-Kuchment theorem for the 1D harmonic oscillator

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.