Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Embedded eigenvalues for asymptotically periodic ODE systems

Maad Sasane, Sara LU and Treschow, Wilhelm LU (2024) In Arkiv for Matematik 62(1). p.103-126
Abstract

We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L2(R; Rn), 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. Our main result is that the set of perturbations for which the embedded eigenvalue persists forms 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. A second result is provided, where under an... (More)

We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L2(R; Rn), 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. Our main result is that the set of perturbations for which the embedded eigenvalue persists forms 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. A second result is provided, where under an extra assumption, it can be proved that the first result holds even when the space of perturbations is replaced by a much smaller space, as long as it contains a minimal subspace. 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)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Arkiv for Matematik
volume
62
issue
1
pages
24 pages
publisher
Springer
external identifiers
  • scopus:85196265696
ISSN
0004-2080
DOI
10.4310/ARKIV.2024.v62.n1.a6
language
English
LU publication?
yes
id
2acea749-589e-4e09-94ff-d48bd0f42bf4
date added to LUP
2024-09-09 11:34:26
date last changed
2025-05-23 10:36:29
@article{2acea749-589e-4e09-94ff-d48bd0f42bf4,
  abstract     = {{<p>We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L<sup>2</sup>(R; R<sup>n</sup>), 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. Our main result is that the set of perturbations for which the embedded eigenvalue persists forms 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. A second result is provided, where under an extra assumption, it can be proved that the first result holds even when the space of perturbations is replaced by a much smaller space, as long as it contains a minimal subspace. 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.</p>}},
  author       = {{Maad Sasane, Sara and Treschow, Wilhelm}},
  issn         = {{0004-2080}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{103--126}},
  publisher    = {{Springer}},
  series       = {{Arkiv for Matematik}},
  title        = {{Embedded eigenvalues for asymptotically periodic ODE systems}},
  url          = {{http://dx.doi.org/10.4310/ARKIV.2024.v62.n1.a6}},
  doi          = {{10.4310/ARKIV.2024.v62.n1.a6}},
  volume       = {{62}},
  year         = {{2024}},
}