Zeroes of the spectral density of discrete Schroedinger operator with Wigner-von Neumann potential

TL;DR

This paper investigates the spectral density zeros of a discrete Schrödinger operator with a combined Wigner-von Neumann and summable potential, revealing generic power-type zeroes at critical spectrum points.

Contribution

It provides a rigorous analysis of the spectral density behavior at critical points for this class of operators, highlighting the generic nature of zeroes of power type.

Findings

Spectral density has zeroes of power type at critical points

The essential spectrum is the interval [-2,2]

Eigenvalues may be located at two critical points

Abstract

We consider a discrete Schroedinger operator whose potential is the sum of a Wigner-von Neumann term and a summable term. The essential spectrum of this operator equals to the interval [-2,2]. Inside this interval, there are two critical points where eigenvalues may be situated. We prove that, generically, the spectral density of the operator has zeroes of the power type at these points.