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

2016 ◽  
Vol 284 (1-2) ◽  
pp. 335-411 ◽  
Author(s):  
Sergey Simonov
Keyword(s):  
Spectral Density ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Von Neumann

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

2011 ◽  
Vol 153 (1) ◽  
pp. 33-58 ◽  
Author(s):  
SERGUEI NABOKO ◽  
SERGEY SIMONOV
Keyword(s):  
Spectral Density ◽  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Bound State ◽  
Periodic Operator ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Von Neumann ◽  
Band Gap Structure

AbstractWe 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.

scholarly journals Weyl—Titchmarsh-type formula for periodic Schrödinger operator with Wigner—von Neumann potential

2013 ◽  
Vol 143 (2) ◽  
pp. 401-425 ◽  
Author(s):  
Pavel Kurasov ◽  
Sergey Simonov
Keyword(s):  
Continuous Spectrum ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Von Neumann ◽  
Type Formula ◽  
Generalized Eigenvectors ◽  
Neumann Type ◽  
Periodic Schrödinger Operator

The Schrödinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner—von Neumann type is considered. The asymptotics of generalized eigenvectors for λ ϵ ℂ+ and on the absolutely continuous spectrum is established. The Weyl—Titchmarsh-type formula for this operator is proven.

scholarly journals On the two-dimensional Schrödinger operator with an attractive potential of the Bessel-Macdonald type

2020 ◽  
pp. 168385
Author(s):  
Wellisson B. De Lima ◽  
Oswaldo M. Del Cima ◽  
Daniel H.T. Franco ◽  
Bruno C. Neves
Keyword(s):  
Schrödinger Operator ◽  
Attractive Potential ◽  
Two Dimensional ◽  
Schrodinger Operator
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