Abstract
We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schrödinger operator whose spectra agree up to O(h∞), yet their corresponding eigenvalues differ no less than exponentially. This result was conjectured in the work of Guillemin and Hezari [Inverse Probl. 28, 045009 (2012)], where they prove a very similar result, yet cannot remove the possibility of a subsequence hk → 0 where the ground state eigenvalues may agree.
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