Embedded eigenvalues for asymptotically periodic ODE systems
(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
- Maad Sasane, Sara LU and Treschow, Wilhelm LU
- organization
-
- Algebra, Analysis and Dynamical Systems (research group)
- LTH Profile Area: AI and Digitalization
- ELLIIT: the Linköping-Lund initiative on IT and mobile communication
- eSSENCE: The e-Science Collaboration
- Biomedical Modelling and Computation (research group)
- Partial differential equations (research group)
- publishing date
- 2024
- 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}}, }