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Spectral Analysis on Point Interactions

  • Author(s): Lee, Minjae
  • Advisor(s): Zworski, Maciej R
  • et al.

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.

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