Weyl asymptotics for perturbed functional difference operators
We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = e−bP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 2N − 1. This class of potentials W includes polynomials of even degree wit...
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ftimperialcol:oai:spiral.imperial.ac.uk:10044/1/76568 2023-05-15T17:07:17+02:00 Weyl asymptotics for perturbed functional difference operators Laptev, A Schimmer, L Takhtajan, LA 2019-09-01 http://hdl.handle.net/10044/1/76568 https://doi.org/10.1063/1.5093401 English eng AIP Publishing Journal of Mathematical Physics 0022-2488 http://hdl.handle.net/10044/1/76568 doi:10.1063/1.5093401 © 2019 Author(s). 10 1 Science & Technology Physical Sciences Physics Mathematical EIGENVALUES 01 Mathematical Sciences 02 Physical Sciences Mathematical Physics Journal Article 2019 ftimperialcol https://doi.org/10.1063/1.5093401 2020-02-13T23:38:12Z We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = e−bP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 2N − 1. This class of potentials W includes polynomials of even degree with leading coefficient 1, which have recently been considered in Grassi and Mariño [SIGMA Symmetry Integrability Geom. Methods Appl. 15, 025 (2019)]. In this paper, we show that such operators have discrete spectrum and obtain Weyl-type asymptotics for the Riesz means and for the number of eigenvalues. This is an extension of the result previously obtained in Laptev et al. [Geom. Funct. Anal. 26, 288–305 (2016)] for W = V + ζV−1, where V = e2πbx, ζ > 0. Article in Journal/Newspaper laptev Imperial College London: Spiral Journal of Mathematical Physics 60 10 103505 |
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Open Polar |
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Imperial College London: Spiral |
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ftimperialcol |
language |
English |
topic |
Science & Technology Physical Sciences Physics Mathematical EIGENVALUES 01 Mathematical Sciences 02 Physical Sciences Mathematical Physics |
spellingShingle |
Science & Technology Physical Sciences Physics Mathematical EIGENVALUES 01 Mathematical Sciences 02 Physical Sciences Mathematical Physics Laptev, A Schimmer, L Takhtajan, LA Weyl asymptotics for perturbed functional difference operators |
topic_facet |
Science & Technology Physical Sciences Physics Mathematical EIGENVALUES 01 Mathematical Sciences 02 Physical Sciences Mathematical Physics |
description |
We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = e−bP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 2N − 1. This class of potentials W includes polynomials of even degree with leading coefficient 1, which have recently been considered in Grassi and Mariño [SIGMA Symmetry Integrability Geom. Methods Appl. 15, 025 (2019)]. In this paper, we show that such operators have discrete spectrum and obtain Weyl-type asymptotics for the Riesz means and for the number of eigenvalues. This is an extension of the result previously obtained in Laptev et al. [Geom. Funct. Anal. 26, 288–305 (2016)] for W = V + ζV−1, where V = e2πbx, ζ > 0. |
format |
Article in Journal/Newspaper |
author |
Laptev, A Schimmer, L Takhtajan, LA |
author_facet |
Laptev, A Schimmer, L Takhtajan, LA |
author_sort |
Laptev, A |
title |
Weyl asymptotics for perturbed functional difference operators |
title_short |
Weyl asymptotics for perturbed functional difference operators |
title_full |
Weyl asymptotics for perturbed functional difference operators |
title_fullStr |
Weyl asymptotics for perturbed functional difference operators |
title_full_unstemmed |
Weyl asymptotics for perturbed functional difference operators |
title_sort |
weyl asymptotics for perturbed functional difference operators |
publisher |
AIP Publishing |
publishDate |
2019 |
url |
http://hdl.handle.net/10044/1/76568 https://doi.org/10.1063/1.5093401 |
genre |
laptev |
genre_facet |
laptev |
op_source |
10 1 |
op_relation |
Journal of Mathematical Physics 0022-2488 http://hdl.handle.net/10044/1/76568 doi:10.1063/1.5093401 |
op_rights |
© 2019 Author(s). |
op_doi |
https://doi.org/10.1063/1.5093401 |
container_title |
Journal of Mathematical Physics |
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60 |
container_issue |
10 |
container_start_page |
103505 |
_version_ |
1766062648807391232 |