Zeroes of the Spectral Density of Discrete Schrödinger Operator with Wigner-von Neumann Potential

2012 ◽  
Vol 73 (3) ◽  
pp. 351-364 ◽  
Author(s):  
Sergey Simonov
Keyword(s):  
Spectral Density ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Discrete Schrödinger 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.

ON THE SPECTRAL AND SCATTERING THEORY OF THE SCHRÖDINGER OPERATOR WITH SURFACE POTENTIAL

2000 ◽  
Vol 12 (04) ◽  
pp. 561-573 ◽  
Author(s):  
AYHAM CHAHROUR ◽  
JAOUAD SAHBANI
Keyword(s):  
Continuous Spectrum ◽  
Surface Potential ◽  
Schrödinger Operator ◽  
Scattering Theory ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Discrete Schrödinger Operator ◽  
Absolutely Continuous ◽  
Spectral And Scattering Theory

We consider a discrete Schrödinger operator H=-Δ+V acting in ℓ2 (ℤd+1), with potential V supported by the subspace ℤd×{0}. We prove that σ (-Δ)=[-2 (d+1), 2(d+1)] is contained in the absolutely continuous spectrum of H. For this we develop a scattering theory for H. We emphasize the fact that this result applies to arbitrary potentials, so it depends on the structure of the problem rather than on a particular choice of the potential.

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