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

TL;DR

This paper analyzes the spectral density zeros of a Schrödinger operator with a periodic background and a slowly decaying Wigner-von Neumann potential, revealing exponential-type zeros at critical points.

Contribution

It establishes the existence and nature of spectral density zeros at critical points for Schrödinger operators with specific slowly decaying oscillatory potentials.

Findings

Spectral density has exponential zeros at critical points.

Critical points correspond to eigenvalues for specific boundary parameters.

The spectral structure mirrors the unperturbed periodic operator.

Abstract

We consider the Schr\"odinger operator $\mathcal L_{\alpha}$ on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner--von Neumann potential $\frac{c\sin(2\omega x+\delta)}{x^{\gamma}}$, where $\gamma\in(\frac12,1)$. The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point $\nu_{cr}$ is an eigenvalue of the operator $\mathcal L_{\alpha}$ for some value of the boundary parameter $\alpha=\alpha_{cr}$, specific to that particular point. We prove that for $\alpha\neq\alpha_{cr}$ the spectral density of the operator $\mathcal L_{\alpha}$ has a zero of the exponential type at $\nu_{cr}$.