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

2016/03/17 by Sergey V. Simonov, Simonov, Sergey, Sergey Simonov
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1603.05483

Abstract

We consider the Schrödinger operator \mathcal L_α 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 (csin(2ωx+δ))/(x^γ), where γ∈(\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 ν_cr is an eigenvalue of the operator \mathcal L_α for some value of the boundary parameter α=α_cr, specific to that particular point. We prove that for α≠α_cr the spectral density of the operator \mathcal L_α has a zero of the exponential type at ν_cr.

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