# Formulas of Szegő Type for the Periodic Schrödinger Operator

Formulas of Szegő Type for the Periodic Schrödinger Operator We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator $${H = -\frac{d^2}{dx^2}+V}$$ H = - d 2 d x 2 + V in dimension one. Admitting fairly general functions h with $${h(0)=0}$$ h ( 0 ) = 0 , we study the trace of the operator $${h(\chi_{(-\alpha,\alpha)} \chi_{(-\infty,\mu)}(H)\chi_{(-\alpha,\alpha)})}$$ h ( χ ( - α , α ) χ ( - ∞ , μ ) ( H ) χ ( - α , α ) ) and link its subleading behaviour as $${\alpha \to \infty}$$ α → ∞ to the position of the spectral parameter μ relative to the spectrum of H. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

# Formulas of Szegő Type for the Periodic Schrödinger Operator

, Volume 358 (2) – Mar 6, 2018
30 pages

/lp/springer_journal/formulas-of-szeg-type-for-the-periodic-schr-dinger-operator-MnSn1JiMJD
Publisher
Springer Berlin Heidelberg
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-018-3106-z
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Mar 6, 2018

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