Discrete Schrödinger Operator on a Tree, Angelesco Potentials, and Their Perturbations

2020 ◽  
Vol 311 (1) ◽  
pp. 1-9 ◽  
Author(s):  
A. I. Aptekarev ◽  
S. A. Denisov ◽  
M. L. Yattselev
Keyword(s):  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Discrete Schrödinger Operator

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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