Year of Publication

2017

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Peter Hislop

Abstract

Bruning considered sets of isospectral Schrodinger operators with smooth real potentials on a compact manifold of dimension three. He showed the set of potentials associated to an isospectral set is compact in the topology of smooth functions by relating the spectrum to the trace of the heat semi-group. Similarly, we can consider the resonances of Schrodinger operators with real valued potentials on Euclidean space of whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an ``isoresonant" set of potentials. This isoresonant set of potentials is also compact in the topology of smooth functions for dimensions one and three. The basis of the result stems from the relation of a regularized wave trace to the resonances via the Poisson formula (also known as the Melrose trace formula). The second link is the small-t asymptotic expansion of the regularized wave trace whose coefficients are integrals of the potential function and its derivatives. For an isoresonant set these coefficients are equal due to the Poisson formula. The equivalence of coefficients allows us to uniformly bound the potential functions and their derivatives with respect to the isoresonant set. Finally, taking a sequence of functions in the isoresonant set we use the uniform bounds to construct a convergent subsequence using the Arzela-Ascoli theorem.

Digital Object Identifier (DOI)

https://doi.org/10.13023/ETD.2017.161

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