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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.

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author
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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
2024-09-09 11:35:02
@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}},
}