Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential

Naboko, Serguei and Simonov, Sergey (2012) Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential. Mathematical Proceedings of the Cambridge Philosophical Society, 153 (01). pp. 33-58. ISSN 1469-8064. (doi:https://doi.org/10.1017/S030500411100079X) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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Official URL
http://dx.doi.org/10.1017/S030500411100079X

Abstract

We consider the Schrödinger operator α on the half-line with a periodic background potential and the Wigner–von Neumann potential of Coulomb type: csin(2ωx + δ)/(x + 1). It is known that the continuous spectrum of the operator α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator α has a subordinate solution, which can be either an eigenvalue or a “half-bound” state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.

Item Type: Article
Additional information: number of additional authors: 1;
Subjects: Q Science > QA Mathematics (inc Computing science)
Q Science > QA Mathematics (inc Computing science) > QA276 Mathematical statistics
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Stewart Brownrigg
Date Deposited: 07 Mar 2014 00:05 UTC
Last Modified: 10 Feb 2016 10:12 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/40459 (The current URI for this page, for reference purposes)
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