PhD Defense Martin Dam Larsen

Title: Spectral estimates for Wiener-Hopf operators with applications to area laws

Abstract:
This thesis studies spectral asymptotics for Wiener–Hopf operators with discontin uous symbols. We establish a two-term asymptotic formula for traces of the form Tr[f (1αΩ(x)P(−i∇)1αΩ(x))] as α → ∞ under essentially optimal regularity conditions, namely for P a function of bounded variation and Ω a set of finite perimeter, and show that these assumptions are sharp if P is an indicator function. We also study time-frequency limiting operators, which arise as the special case of Wiener–Hopf operators where the symbol is an indicator function, and obtain sharp uniform bounds on the plunge region when one of the underlying sets is a finite disjoint union of parallelepipeds. As an application, we extend the two-term asymptotic formula to very rough spectral functions f.

The thesis will be announced later.

Advisor: Professor Jan Philip Solovej, University of Copenhagen

Assessment committee:

Associate Professor Albert Werner (chair) University of Copenhagen,

Professor Alexander V. Sobolev, University College London, United Kingdom

Professor Peter Müller, Ludwig-Maximilians-Universität Muximilians-Universität München, Germany