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We give a detailed proof for two discrete analogues of Courant's Nodal Domain Theorem. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

We give a detailed proof for two discrete analogues of Courant's Nodal Domain Theorem. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e., the connected components of the maximal induced subgraphs of G on which an eigenvector \psi does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of \psi in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.

Let G be a connected graph with n vertices and let x=(x1, ..., xn) be a real vector. A positive (negative) sign graph of the vector x is a maximal connected subgraph of G on vertices xi>0 (xi<0). For an eigenvalue of a generalized Laplacian of a tree: We characterize the maximal number of sign graphs of an eigenvector. We give an O(n2) time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an NP-complete problem. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

L'objectif de ce mémoire est de démontrer certaines propriétés géométriques des fonctions propres de l'oscillateur harmonique quantique. Nous étudierons les domaines nodaux, c'est-à-dire les composantes connexes du complément de l'ensemble nodal. Supposons que les valeurs propres ont été ordonnées en ordre croissant. Selon un théorème fondamental dû à Courant, une fonction propre associée à la $n$-ième valeur propre ne peut avoir plus de $n$ domaines nodaux. Ce résultat a été prouvé initialement pour le laplacien de Dirichlet sur un domaine borné mais il est aussi vrai pour l'oscillateur harmonique quantique isotrope. Le théorème a été amélioré par Pleijel en 1956 pour le laplacien de Dirichlet. En effet, on peut donner un résultat asymptotique plus fort pour le nombre de domaines nodaux lorsque les valeurs propres tendent vers l'infini. Dans ce mémoire, nous prouvons un résultat du même type pour l'oscillateur harmonique quantique isotrope. Pour ce faire, nous utiliserons une combinaison d'outils classiques de la géométrie spectrale (dont certains ont été utilisés dans la preuve originale de Pleijel) et de plusieurs nouvelles idées, notamment l'application de certaines techniques tirées de la géométrie algébrique et l'étude des domaines nodaux non-bornés. / 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.