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

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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.

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  1. Chebyshev Laboratory, Department of Mathematics and Mechanics, Saint-Petersburg State University, 14th Line, 29b, Saint-Petersburg, 199178, Russia

    Sergey Simonov

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  1. Sergey Simonov
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Correspondence to Sergey Simonov.

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Simonov, S. Zeroes of the Spectral Density of Discrete Schrödinger Operator with Wigner-von Neumann Potential. Integr. Equ. Oper. Theory 73, 351–364 (2012). https://doi.org/10.1007/s00020-012-1972-x

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  • DOI: https://doi.org/10.1007/s00020-012-1972-x

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