Spectral geometry over the disk : Weyl's law and nodal sets

TL;DR

This thesis investigates spectral properties of the Dirichlet Laplacian on disks and sectors, focusing on nodal sets, eigenvalue ordering algorithms, and Weyl's law error estimates, with both theoretical proofs and numerical analysis.

Contribution

It provides a detailed proof of a van der Corput type estimate on the Weyl law remainder and develops an efficient eigenvalue ordering algorithm for disks.

Findings

Location of the second eigenfunction's nodal line in sectors

An efficient algorithm for eigenvalue ordering on disks

Numerical analysis of Weyl law error growth

Abstract

In this M.Sc. thesis (Universit\'e de Montr\'eal, 2007), we consider problems arising in the study of the spectrum of the Dirichlet Laplacian on a disk as well as on a circular sector. The first part of the thesis is concerned with the location of the nodal line of the second eigenfunction of a sector. In the second part of the thesis we develop an efficient algorithm for ordering the eigenvalues of a disk, and study numerically the growth of the error term in Weyl's law. We also give a detailed proof of a theorem due to Kuznetsov and Fedosov (1965), who obtained a van der Corput type estimate on the remainder. The result of Kuznetsov and Fedosov was rediscovered in 2011 by Y. Colin de Verdi\`ere using similar techniques, see http://arxiv.org/pdf/1104.2233v2.pdf .