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.

The purpose of this dissertation is to develop the spectral theory for bounded planar domains. A particularly effective method for understanding the Laplace spectrum is via the wave equation. Chapters 3-6 develop a general theory for explicitly solving the wave equation in terms of dynamical data associated to the billiard map on the underlying domain. In particular, we develop a microlocal Hadamard-Riesz type parametrix for the wave propagator near reflecting rays in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in [44] and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works of [44] and [60], is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this dissertation (in addition to the author’s previous works [69] and [68]) contain the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard-Riesz type parametrix for the wave propagator appears to be new in the literature. This is also based on the author’s prior work [69] in the special case of elliptical domains and [68] for general convex billiard tables. It allows us to circumvent the symbol calculus in [12] and [25] when computing trace formulas, which are instead derived from our explicit parametrix and a rescaling argument via Hadamard’s variational formula for the wave trace. These techniques also appear to be new in the literature. In Chapter 7, we investigate C1 isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. If the deformation is in fact smooth, reparametrizing allows us to show that the first variation actually vanishes to infinite order. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard’s variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we use the explicit parametrix for the wave propagator developed in the interior, microlocally near geodesic loops (see Chapter 5).