Mathematical Proceedings of the Cambridge Philosophical Society

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

a1 Department of Mathematical Physics, Institute of Physics, Saint-Petersburg State University, Ulianovskaia 1, St. Petergoff, Saint-Petersburg, 198904 Russia. e-mail: sergey.naboko@gmail.com

a2 Chebyshev Laboratory, Department of Mathematics and Mechanics, Saint-Petersburg State University, 14th Line, 29b, Saint-Petersburg, 199178 Russia. e-mail: sergey.a.simonov@gmail.com

Abstract

We consider the Schrödinger operator α on the half-line with a periodic background potential and the Wigner–von Neumann potential of Coulomb type: csin(2ωx + δ)/(x + 1). It is known that the continuous spectrum of the operator α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator α has a subordinate solution, which can be either an eigenvalue or a “half-bound” state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.