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

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 .