  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 .
