scholarly journals Continuous spectrum for a class of smooth mixing Schrödinger operators

2017 ◽  
Vol 39 (2) ◽  
pp. 357-369
Author(s):  
BASSAM FAYAD ◽  
YANHUI QU
Keyword(s):  
Dynamical System ◽  
Continuous Spectrum ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Sampling Function ◽  
Discrete Schrödinger Operators ◽  
Volume Preserving

We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schrödinger operators on the line defined with a potential generated by this system and a Hölder sampling function have almost surely a continuous spectrum.

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.

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