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Embedded eigenvalues for asymptotically periodic ODE Systems

Treschow, Wilhelm LU (2022) In Master’s Theses in Mathematical Sciences FMAM05 20221
Mathematics (Faculty of Engineering)
Abstract
In this thesis we investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L^2(\mathbb{R},\mathbb{R}^n), with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. The set of perturbations for which the embedded eigenvalue persists is shown to form a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov-Schmidt reduction.
In the end, as a way of showing that the... (More)
In this thesis we investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L^2(\mathbb{R},\mathbb{R}^n), with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. The set of perturbations for which the embedded eigenvalue persists is shown to form a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov-Schmidt reduction.
In the end, as a way of showing that the investigated setting exists, a concrete example is presented. The example itself relates to a problem from quantum mechanics and represents a system of electrons in an infinite one-dimensional crystal. (Less)
Popular Abstract
Differential equations arise in many different areas of science, such as mathematics, physics, mechanics, biology etc. One such example from quantum mechanics is the Schrödinger equation, describing the evolution and shape of the wave-function of an electron, or a system of electrons. In the time-independent case, we can find stationary states of the electrons representing bound states of the electrons subject to a potential, as well as the corresponding energy of the system. An interesting problem is to examine what happens to these bound states when we alter the potential slightly. This thesis studies this problem and aims to investigate under which perturbations of the potential these bound states still exist.
Please use this url to cite or link to this publication:
author
Treschow, Wilhelm LU
supervisor
organization
course
FMAM05 20221
year
type
H2 - Master's Degree (Two Years)
subject
keywords
embedded eigenvalue perturbation theory ODE theory
publication/series
Master’s Theses in Mathematical Sciences
report number
LUTFMA-3482-2022
ISSN
1404-6342
other publication id
2022:E34
language
English
id
9093620
date added to LUP
2022-06-30 15:06:53
date last changed
2022-06-30 15:06:53
@misc{9093620,
  abstract     = {{In this thesis we investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L^2(\mathbb{R},\mathbb{R}^n), with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. The set of perturbations for which the embedded eigenvalue persists is shown to form a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov-Schmidt reduction.
In the end, as a way of showing that the investigated setting exists, a concrete example is presented. The example itself relates to a problem from quantum mechanics and represents a system of electrons in an infinite one-dimensional crystal.}},
  author       = {{Treschow, Wilhelm}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master’s Theses in Mathematical Sciences}},
  title        = {{Embedded eigenvalues for asymptotically periodic ODE Systems}},
  year         = {{2022}},
}