This thesis explores the spectral properties of Schrödinger-type operators on various
domains 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 scatterers
on a general domain. The theory of self-adjoint extension and Krein’s formula play important
roles in the process.
In Chapter 2, we formulate point scatterers in R^2 using the renormalization process
mentioned in the previous chapter. We start with the one-point scatterer which is the
simplest case and then generalize the result to finitely many point scatterers and infinitely
many point scatterers. Then we consider a special case in which the scatterers are placed
periodically as a combination of infinitely many point scatterers and the Floquet-Bloch
theory of solid-state physics for crystal structures. As an application inspired by carbon
nano-structures such as graphene, we prove that honeycomb lattice point scatterers generate
conic singularities on the dispersion relation.
In Chapter 3, we consider a point scatterer on a rectangular domain to investigate how
the eigenfunctions on the rectangle are affected by the point-wise perturbation. We prove
that a point scatterer eventually acts as a barrier confining the eigenfunction as the domain
gets thinner.
In Chapter 4, we introduce how the point scatterers can be incorporated with the notion of
quantum graphs. In addition, the resonances of quantum graphs are investigated. We provide
the quantum graph version of a Fermi golden rule, which provides an explicit expression for
the infinitesimal change of states in terms of the scattering resonances.
