Integral Equations and Operator Theory

, Volume 73, Issue 3, pp 351–364 | Cite as

Zeroes of the Spectral Density of Discrete Schrödinger Operator with Wigner-von Neumann Potential

Article

Abstract

We consider a discrete Schrödinger operator \({\mathcal{J}}\) whose potential is the sum of a Wigner-von Neumann term \({\frac{c\sin(2\omega n+\delta)}n}\) and a summable term. The essential spectrum of the operator \({\mathcal{J}}\) is equal to the interval [−2, 2]. Inside this interval, there are two critical points \({\pm2\cos\omega}\) where eigenvalues may be situated. We prove that, generically, the spectral density of \({\mathcal{J}}\) has zeroes of the power \({\frac{|c|}{2|\sin\omega|}}\) at these points.

Mathematics Subject Classification (2010)

47B36 34E10 

Keywords

Jacobi matrices Asymptotics of generalized eigenvectors orthogonal polynomials discrete Schrödinger operator Wigner-von Neumann potential pseudogaps 

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Chebyshev Laboratory, Department of Mathematics and MechanicsSaint-Petersburg State UniversitySaint-PetersburgRussia

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