Théorème de Pleijel pour l'oscillateur harmonique quantique
The aim of this thesis is to explore the geometric properties of eigenfunctions of the isotropic quantum harmonic oscillator. We focus on studying the nodal domains, which are the connected components of the complement of the nodal (i.e. zero) set of an eigenfunction. Assume that the eigenvalues are listed in an increasing order. According to a fundamental theorem due to Courant, an eigenfunction corresponding to the $n$-th eigenvalue has at most $n$ nodal domains. This result has been originally proved for the Dirichlet eigenvalue problem on a bounded Euclidean domain, but it also holds for the eigenfunctions of a quantum harmonic oscillator. Courant's theorem was refined by Pleijel in 1956, who proved a more precise result on the asymptotic behaviour of the number of nodal domains of the Dirichlet eigenfunctions on bounded domains as the eigenvalues tend to infinity. In the thesis we prove a similar result in the case of the isotropic quantum harmonic oscillator. To do so, we use  a combination of classical tools from spectral geometry (some of which were used  in Pleijel’s original argument) with a number of new ideas, which include applications of techniques from algebraic geometry and the study of unbounded nodal domains.
Pleijel Faber-Krahn Conditions de Dirichlet Quantum harmonic oscillator Nodal domain Eigenfunction Dirichlet boundary conditions Sobolev space