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Perturbations of embedded eigenvalues for a magnetic Schrödinger operator on a cylinder

Laptev, Ari and Sasane, Sara Maad LU (2017) In Journal of Mathematical Physics 58(1).
Abstract

Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper, we study a perturbation problem for embedded eigenvalues for a magnetic Schrödinger operator, when the underlying domain is a cylinder. The magnetic potential is C2 with an algebraic decay rate as the unbounded variable of the cylinder tends to ±∞. In particular, no analyticity of the magnetic potential is assumed. We also assume that the embedded eigenvalue of the unperturbed problem is not the square of an integer, thus avoiding the thresholds of the continuous spectrum of the unperturbed operator. We show that the set of nearby potentials,... (More)

Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper, we study a perturbation problem for embedded eigenvalues for a magnetic Schrödinger operator, when the underlying domain is a cylinder. The magnetic potential is C2 with an algebraic decay rate as the unbounded variable of the cylinder tends to ±∞. In particular, no analyticity of the magnetic potential is assumed. We also assume that the embedded eigenvalue of the unperturbed problem is not the square of an integer, thus avoiding the thresholds of the continuous spectrum of the unperturbed operator. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension.

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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Mathematical Physics
volume
58
issue
1
article number
012105
publisher
American Institute of Physics (AIP)
external identifiers
  • scopus:85011297905
  • wos:000395279200018
ISSN
0022-2488
DOI
10.1063/1.4974360
language
English
LU publication?
yes
id
a6c7b57f-f27c-4a31-8ed9-19b552134678
date added to LUP
2017-02-16 06:59:17
date last changed
2025-01-07 07:08:55
@article{a6c7b57f-f27c-4a31-8ed9-19b552134678,
  abstract     = {{<p>Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper, we study a perturbation problem for embedded eigenvalues for a magnetic Schrödinger operator, when the underlying domain is a cylinder. The magnetic potential is C<sup>2</sup> with an algebraic decay rate as the unbounded variable of the cylinder tends to ±∞. In particular, no analyticity of the magnetic potential is assumed. We also assume that the embedded eigenvalue of the unperturbed problem is not the square of an integer, thus avoiding the thresholds of the continuous spectrum of the unperturbed operator. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension.</p>}},
  author       = {{Laptev, Ari and Sasane, Sara Maad}},
  issn         = {{0022-2488}},
  language     = {{eng}},
  month        = {{01}},
  number       = {{1}},
  publisher    = {{American Institute of Physics (AIP)}},
  series       = {{Journal of Mathematical Physics}},
  title        = {{Perturbations of embedded eigenvalues for a magnetic Schrödinger operator on a cylinder}},
  url          = {{http://dx.doi.org/10.1063/1.4974360}},
  doi          = {{10.1063/1.4974360}},
  volume       = {{58}},
  year         = {{2017}},
}