PhD defence by Martin Dam Larsen

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

Assessment Committee:
Associate Professor Albert H. Werner, Department of Mathematical Sciences, University of Copenhagen (Chairperson)
Professor Alexander Sobolev, University College London
Professor Peter Müller, Ludwig Maximilians University Munich

Supervisor(s):
Professor Jan Philip Solovej
Professor Søren Fournais

Department:
Department of Mathematical Sciences

Place:
The defence is conducted as a hybrid defence.

To attend the defence in person:
HCØ, Room: Auditorium 5, Universitetsparken 5, 2100 København Ø

To attend the defence online:
Please follow the link to attend the defence online:
https://ucph-ku.zoom.us/j/7628840427?pwd=UTRMZkxDRURueTFyQmJnRlo2dnAxUT09&omn=66700349182 

MeetingID, if relevant: 762 884 0427
Password, if relevant: 549357

Email address to gain access to the thesis: mdl@math.ku.dk 
You will either receive a copy of the thesis or be informed where you can read a physical copy.

Short description of the thesis:

This thesis studies spectral asymptotics for Wiener–Hopf operators with discontinuous symbols. We establish a two-term asymptotic formula for traces of the form $\mathrm{T}\mathrm{r}\text{\hspace{0.17em}⁣}\left[f\text{\hspace{0.17em}⁣}(1_{\alpha \mathrm{\Omega }}(x)P(-i\mathrm{\nabla })1_{\alpha \mathrm{\Omega }}(x))\right]$ as $\alpha \to \mathrm{\infty }$ 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 $\mathrm{f.}$