UC Irvine

In this work we study inverse spectral problems for bounded domains, smooth closed mani- folds, and semiclassical Schro ̈dinger operators, with particular concern towards the latter. A central tool in the analysis of inverse spectral problems are trace invariants, however these are not without limitations. We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schro ̈dinger operator whose spectra agree up to O(h∞), and hence have the same semiclassical trace invariants, yet all corresponding eigenvalues differ no less than exponentially. This result was conjectured in the work of Guillemin and Hezari [GH12], where they prove a very similar result for the ground state eigenvalues, however cannot remove the possibility of a subsequence hk → 0 where the ground state eigenvalues may agree.