scholarly journals Inverse Problems, Trace Formulae for Discrete Schrödinger Operators

2011 ◽  
Vol 13 (4) ◽  
pp. 751-788 ◽  
Author(s):  
Hiroshi Isozaki ◽  
Evgeny Korotyaev
Keyword(s):  
Inverse Problems ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Discrete Schrödinger Operators ◽  
Trace Formulae

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.

On New Trace Formulae for Schrödinger Operators

KdV ’95 ◽  
1995 ◽  
pp. 315-333
Author(s):  
F. Gesztesy ◽  
H. Holden
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Trace Formulae

scholarly journals Wegner estimate for discrete Schrödinger operators with Gaussian random potentials

2019 ◽  
Vol 27 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Martin Tautenhahn
Keyword(s):  
Conditional Distribution ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Gaussian Random Process ◽  
Random Potentials ◽  
Wegner Estimate ◽  
Compactly Supported ◽  
Covariance Functions ◽  
Monotonicity Assumption ◽  
Discrete Schrödinger Operators

Abstract We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.

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