scholarly journals Examples of discrete Schrödinger operators with pure point spectrum

1983 ◽  
Vol 88 (4) ◽  
pp. 447-463 ◽  
Author(s):  
Jürgen Pöschel
Keyword(s):  
Point Spectrum ◽  
Schrödinger Operators ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Schrodinger Operators ◽  
Discrete Schrödinger Operators

scholarly journals Magnetic Schrödinger Operators with Discrete Spectra on Non-Compact Kähler Manifolds

2012 ◽  
Vol 20 (2) ◽  
pp. 11-20
Author(s):  
Nicolae Anghel
Keyword(s):  
Point Spectrum ◽  
Schrödinger Operators ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Schrodinger Operators ◽  
Type Problem ◽  
Dirac Type ◽  
Discrete Spectra ◽  
Parallel Spinor ◽  
Discreteness Criterion

Abstract We identify a class of magnetic Schrödinger operators on Käler manifolds which exhibit pure point spectrum. To this end we embed the Schröinger problem into a Dirac-type problem via a parallel spinor and use a Bochner-Weitzenböck argument to prove our spectral discreteness criterion

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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