Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential

Abstract

We consider the Schrodinger 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 \(\gamma \in (\frac{1}{2},1)\). 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 \(\alpha \ne \alpha _{cr}\) the spectral density of the operator \({\mathcal {L}}_{\alpha }\) has a zero of the exponential type at \(\nu _{cr}\).