Level Set Shape For Ground State Eigenfunctions On Convex Domains

Abstract

In this thesis, we study the ground state Dirichlet eigenfunction of two classes of Schr\"odinger operators on convex domains. The assumptions on these operators ensure that the level sets of the eigenfunction are convex sets, and we find length scales and an orientation of the domain which determine the shape of the level sets. An an intermediate step we establish bounds on the ground state eigenvalue.

We first consider a class of two dimensional Schr\"odinger operators with a convex potential on a convex domain. In this case, the length scales and orientation determining the shape of the level sets are defined in terms of the geometry of the domain, the properties of the potential, and an associated family of one dimensional Schr\"odinger operators. Once we have established the shape of the level sets, we begin to analyse the behaviour of the eigenfunction near to where it achieves its maximum.

In the second part of the thesis we study the ground state eigenfunction of the Dirichlet Laplacian for a class of three dimensional convex domains. For each of these domains we use an approximate separation of variables to find an associated two dimensional Schr\"odinger operator of the form above. This then allows us to obtain sufficiently precise bounds on the first eigenvalue and also to use the same length scales and orientation as for the two dimensional operator in order to determine the shape of the level sets of the three dimensional eigenfunction.