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Communications in Mathematical Physics

, Volume 358, Issue 2, pp 675–704 | Cite as

Formulas of Szegő Type for the Periodic Schrödinger Operator

  • Bernhard Pfirsch
  • Alexander V. Sobolev
Open Access
Article

Abstract

We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator \({H = -\frac{d^2}{dx^2}+V}\) in dimension one. Admitting fairly general functions h with \({h(0)=0}\), we study the trace of the operator \({h(\chi_{(-\alpha,\alpha)} \chi_{(-\infty,\mu)}(H)\chi_{(-\alpha,\alpha)})}\) and link its subleading behaviour as \({\alpha \to \infty}\) to the position of the spectral parameter μ relative to the spectrum of H.

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© The Author(s) 2018

Open AccessThis 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.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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