Perturbation theory for embedded eigenvalues of asymptotically periodic operators
(2024) In Licentiate Thesis in Mathematical Sciences 2024(6).- Abstract
- This thesis explores the persistence of embedded eigenvalues of asymptotically periodic Schrödinger-type operators under perturbations on domains with only one unbounded direction. Embedded eigenvalues present unique challenges, as they lie within the continuous spectrum, making their stability under perturbations a nontrivial problem.
The aim of the first part of the thesis is to provide a general background on embedded eigenvalues and their perturbations as well as the central problem formulation for the thesis, outlining the motivation, key challenges and well-known results and examples, as well as some preliminaries on the theory needed to investigate problems of this kind.
The second part of the thesis includes the... (More) - This thesis explores the persistence of embedded eigenvalues of asymptotically periodic Schrödinger-type operators under perturbations on domains with only one unbounded direction. Embedded eigenvalues present unique challenges, as they lie within the continuous spectrum, making their stability under perturbations a nontrivial problem.
The aim of the first part of the thesis is to provide a general background on embedded eigenvalues and their perturbations as well as the central problem formulation for the thesis, outlining the motivation, key challenges and well-known results and examples, as well as some preliminaries on the theory needed to investigate problems of this kind.
The second part of the thesis includes the perturbation theory results of Paper I and Paper II and prove in the two different settings that, under the condition that the unperturbed operator’s embedded eigenvalue lies away from the thresholds of the continuous spectrum, the set of perturbations for which a simple embedded eigenvalue persists forms a smooth Banach manifold with a finite and even codimension in the set of admissible perturbations. Paper I considers the Schrödinger operator in one dimension with a continuous, matrix-valued, symmetric and asymptotically periodic electric potential, whereas Paper II considers a magnetic laplacian on an infinite cylindrical domain, where
the magnetic potential is C 2 and asymptotically periodic along the length of the cylinder. The main tools are Floquet theory, exponential dichotomies and Lyapunov-Schmidt reduction. Examples are provided for both of the investigated settings. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/f4c876b2-95ae-4582-aa6c-a5027157b030
- author
- Treschow, Wilhelm LU
- supervisor
- organization
- publishing date
- 2024
- type
- Thesis
- publication status
- published
- subject
- keywords
- embedded eigenvalues, perturbation theory, differential equations, periodicity
- in
- Licentiate Thesis in Mathematical Sciences
- volume
- 2024
- issue
- 6
- pages
- 107 pages
- publisher
- Centre for the Mathematical sciences, Lund University
- ISSN
- 1404-028X
- ISBN
- 978-91-8104-276-4
- 978-91-8104-277-1
- language
- English
- LU publication?
- yes
- id
- f4c876b2-95ae-4582-aa6c-a5027157b030
- date added to LUP
- 2024-11-26 12:22:06
- date last changed
- 2025-04-04 15:05:09
@misc{f4c876b2-95ae-4582-aa6c-a5027157b030,
abstract = {{This thesis explores the persistence of embedded eigenvalues of asymptotically periodic Schrödinger-type operators under perturbations on domains with only one unbounded direction. Embedded eigenvalues present unique challenges, as they lie within the continuous spectrum, making their stability under perturbations a nontrivial problem.<br/><br/>The aim of the first part of the thesis is to provide a general background on embedded eigenvalues and their perturbations as well as the central problem formulation for the thesis, outlining the motivation, key challenges and well-known results and examples, as well as some preliminaries on the theory needed to investigate problems of this kind.<br/><br/>The second part of the thesis includes the perturbation theory results of Paper I and Paper II and prove in the two different settings that, under the condition that the unperturbed operator’s embedded eigenvalue lies away from the thresholds of the continuous spectrum, the set of perturbations for which a simple embedded eigenvalue persists forms a smooth Banach manifold with a finite and even codimension in the set of admissible perturbations. Paper I considers the Schrödinger operator in one dimension with a continuous, matrix-valued, symmetric and asymptotically periodic electric potential, whereas Paper II considers a magnetic laplacian on an infinite cylindrical domain, where<br/>the magnetic potential is C 2 and asymptotically periodic along the length of the cylinder. The main tools are Floquet theory, exponential dichotomies and Lyapunov-Schmidt reduction. Examples are provided for both of the investigated settings.}},
author = {{Treschow, Wilhelm}},
isbn = {{978-91-8104-276-4}},
issn = {{1404-028X}},
keywords = {{embedded eigenvalues; perturbation theory; differential equations; periodicity}},
language = {{eng}},
note = {{Licentiate Thesis}},
number = {{6}},
publisher = {{Centre for the Mathematical sciences, Lund University}},
series = {{Licentiate Thesis in Mathematical Sciences}},
title = {{Perturbation theory for embedded eigenvalues of asymptotically periodic operators}},
url = {{https://lup.lub.lu.se/search/files/200730291/Wilhelm_Treschow_-_WEBB.pdf}},
volume = {{2024}},
year = {{2024}},
}