scholarly journals Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators

2018 ◽  
Vol 9 (4) ◽  
pp. 1831-1847
Author(s):  
Keshav Raj Acharya
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Weyl Theory ◽  
Discrete Schrödinger Operators ◽  
Vector Valued

scholarly journals A Note on Vector Valued Discrete Schrödinger Operators

2016 ◽  
Vol 34 (1-2) ◽  
pp. 1-9 ◽  
Author(s):  
Keshav Raj Acharya
Keyword(s):  
Schrödinger Operators ◽  
One Dimension ◽  
Multidimensional Space ◽  
Schrodinger Operators ◽  
Higher Dimension ◽  
Resolvent Operators ◽  
Theoretic Analysis ◽  
Discrete Schrödinger Operators ◽  
Vector Valued

The main purpose of this paper is to extend some theory of Schr¨odinger operators from one dimension to higher dimension. In particular, we will give systematic operator theoretic analysis for the Schr¨odinger equations in multidimensional space. To this end, we will provide the detail proves of some basic results that are necessary for further studies in these areas. In addition, we will introduce Titchmarsh- Weyl m− function of these equations and express m− function in term of the resolvent 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.

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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