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Spectral Analysis on Point Interactions.

Lee, Minjae.

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Spectral Analysis on Point Interactions.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Spectral Analysis on Point Interactions./
Author:	Lee, Minjae.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2016,
Description:	98 p.
Notes:	Source: Dissertation Abstracts International, Volume: 78-03(E), Section: B.
Contained By:	Dissertation Abstracts International78-03B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10150803
ISBN:	9781369055689

Spectral Analysis on Point Interactions.
Lee, Minjae.

Spectral Analysis on Point Interactions. - Ann Arbor : ProQuest Dissertations & Theses, 2016 - 98 p.

Source: Dissertation Abstracts International, Volume: 78-03(E), Section: B.

Thesis (Ph.D.)--University of California, Berkeley, 2016.

This thesis explores the spectral properties of Schrodinger-type operators on various domains such as R2, 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.

ISBN: 9781369055689Subjects--Topical Terms:

515831
Mathematics.

Spectral Analysis on Point Interactions.
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100 1 $a Lee, Minjae. $3 3284410
245 1 0 $a Spectral Analysis on Point Interactions.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2016
300 $a 98 p.
500 $a Source: Dissertation Abstracts International, Volume: 78-03(E), Section: B.
500 $a Adviser: Maciej R. Zworski.
502 $a Thesis (Ph.D.)--University of California, Berkeley, 2016.
520 $a This thesis explores the spectral properties of Schrodinger-type operators on various domains such as R2, 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.
520 $a 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.
520 $a 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.
520 $a 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.
520 $a 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.
590 $a School code: 0028.
650 4 $a Mathematics. $3 515831
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710 2 $a University of California, Berkeley. $b Mathematics. $3 1685396
773 0 $t Dissertation Abstracts International $g 78-03B(E).
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792 $a 2016
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10150803