  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.
