M.Sc Thesis
M.Sc Student Bersudsky Michael Courant-Sharp Eigenvalues of Neumann 2-Rep-Tiles Department of Mathematics Professor Ram Band

Abstract

In this thesis we are concerned with Laplacian's eigenvalues and eigenfunctions. In particular we are interested in the relation between the nodal domains of eigenfunctions, which are the connected components on which the eigenfunction has a fixed sign, with respect to the spectral position of the corresponding eigenvalue, which is its position among the eigenvalues according to size. A well known result in this direction is Courant's theorem, which asserts that the number of nodal domains of an eigenfunction is bounded by the spectral position of the corresponding eigenvalue. A Courant-sharp eigenfunction is an eigenfunction with a maximal nodal count and in that case, the corresponding eigenvalue is said to be a Courant-sharp eigenvalue. We find the Courant-sharp eigenvalues of the Neumann Laplacian on some special domains known as 2-rep-tile domains. In \R^{2} the domains we consider are the isosceles right triangle and the rectangle with edge ratio \sqrt{2} (also known as the A4 paper). In \R^{d} the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\unfolding. This structure affects the nodal set of the eigenfunctions, which in turn allows to derive necessary conditions for Courant-sharpness. In addition, the eigenvalues of these domains are arranged as a lattice which allows for a comparison between the nodal count and the spectral position. The Courant-sharpness of most eigenvalues is ruled out using those methods.