Parnovski, L;
Shterenberg, R;
(2019)
Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case.
Communications in Mathematical Physics
10.1007/s00220-019-03329-3.
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Abstract
We consider the family of operators H(ε):=-d2dx2+εV in R with almost-periodic potential V. We study the behaviour of the integrated density of states (IDS) N(H (ε) ; λ) when ε→ 0 and λ is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each λ the IDS has a complete asymptotic expansion in powers of ε; these powers are either integer, or in some special cases half-integer. These results are new even for periodic V. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set S of energies (which we call the super-resonance set) such that for any λ∉S there is a complete power asymptotic expansion of IDS, and when λ∈S, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set S is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of H (ε) has a complete asymptotic expansion in natural powers of ε when ε→ 0.
| Type: | Article |
|---|---|
| Title: | Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00220-019-03329-3 |
| Publisher version: | http://doi.org/10.1007/s00220-019-03329-3 |
| Language: | English |
| Additional information: | Copyright © The Author(s) 2019 Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10069681 |
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