scholarly journals Inverse scattering at a fixed energy for Discrete Schrödinger Operators on the square lattice

2015 ◽  
Vol 65 (3) ◽  
pp. 1153-1200 ◽  
Author(s):  
Hiroshi Isozaki ◽  
Hisashi Morioka
Keyword(s):  
Inverse Scattering ◽  
Schrödinger Operators ◽  
Square Lattice ◽  
Schrodinger Operators ◽  
Fixed Energy ◽  
Discrete Schrödinger Operators

scholarly journals On the Spectra of Separable 2D Almost Mathieu Operators

2021 ◽  
Author(s):  
Alberto Takase
Keyword(s):  
Positive Measure ◽  
Schrödinger Operators ◽  
Cantor Sets ◽  
Schrodinger Operators ◽  
Discrete Schrödinger Operators ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

AbstractWe consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator and utilizes the Newhouse Gap Lemma on sums of Cantor sets.

Inverse Scattering for Non-classical Impedance Schrödinger Operators

2012 ◽  
pp. 1-42
Author(s):  
Sergio Albeverio ◽  
Rostyslav O. Hryniv ◽  
Yaroslav V. Mykytyuk ◽  
Peter A. Perry
Keyword(s):  
Inverse Scattering ◽  
Schrödinger Operators ◽  
Schrodinger Operators
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