A Pair of Non-Isometric Potentials With the Same Semiclassical Invariants

We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schr\"odinger operator whose spectra agree up to $O(h^\infty)$, yet their corresponding eigenvalues differ no less than exponentially. This result was conjectured by Guillemen and Hezari in [GH12], where they prove a very similar result, yet cannot remove the possibility of a subsequence $h_k\to 0$ where the ground state eigenvalues may agree.