Spectral Analysis on Point Interactions

This thesis explores the spectral properties of Schrödinger-type operators on variousdomains such as R^2, rectangles, and metric graphs. In particular, we consider special types of operators called point scatterers that act as the Laplacian away from a discrete set of points. Such a model provides a simple tool to study how the presence of point-wise potentials perturbs the spectral properties of the Laplacian.In Chapter 1, we introduce the general procedure to properly define the point scattererson a general domain. The theory of self-adjoint extension and Krein’s formula play importantroles in the process.In Chapter 2, we formulate point scatterers in R^2 using the renormalization processmentioned in the previous chapter. We start with the one-point scatterer which is thesimplest case and then generalize the result to finitely many point scatterers and infinitelymany point scatterers. Then we consider a special case in which the scatterers are placedperiodically as a combination of infinitely many point scatterers and the Floquet-Blochtheory of solid-state physics for crystal structures. As an application inspired by carbonnano-structures such as graphene, we prove that honeycomb lattice point scatterers generateconic singularities on the dispersion relation.In Chapter 3, we consider a point scatterer on a rectangular domain to investigate howthe eigenfunctions on the rectangle are affected by the point-wise perturbation. We provethat a point scatterer eventually acts as a barrier confining the eigenfunction as the domaingets thinner.In Chapter 4, we introduce how the point scatterers can be incorporated with the notion ofquantum graphs. In addition, the resonances of quantum graphs are investigated. We providethe quantum graph version of a Fermi golden rule, which provides an explicit expression forthe infinitesimal change of states in terms of the scattering resonances