THE EXPONENTIAL LOCALIZATION AND STRUCTURE OF THE SPECTRUM FOR 1D QUASI-PERIODIC DISCRETE SCHRÖDINGER OPERATORS

1991 ◽  
Vol 03 (03) ◽  
pp. 241-284 ◽  
Author(s):  
V. A. CHULAEVSKY ◽  
YA. G. SINAI
Keyword(s):  
Spectral Measure ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Discrete Schrödinger Operators ◽  
Exponential Localization

We discuss main mechanisms of the exponential localization of the eigenfunctions for one-dimensional quasi-periodic Schrödinger operators with the potential of the form V(α + nω), where V(α) is a non-degenerate C2-function on the d-dimensional torus, and ω ∈ ℝd is a typical vector with rationally incommensurate components. The exponential localization is proved so far for d ≤ 2. We emphasize the different nature of the support of the spectral measure for d = 1 and for d > 1.

GAP LABELLING THEOREMS FOR ONE DIMENSIONAL DISCRETE SCHRÖDINGER OPERATORS

1992 ◽  
Vol 04 (01) ◽  
pp. 1-37 ◽  
Author(s):  
JEAN BELLISSARD ◽  
ANTON BOVIER ◽  
JEAN-MICHEL GHEZ
Keyword(s):  
Schrödinger Operators ◽  
Tight Binding ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Automatic Sequences ◽  
Discrete Schrödinger Operators ◽  
K Theory

We study one dimensional tight binding hamiltonians with potentials given by automatic sequences. By means of Shubin’s formula, we show how K-theory allows to prove gap labelling theorems for their spectrum. We apply them to some examples, for which we compare their predictions to previous results.

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 Reflection Probabilities of One-Dimensional Schrödinger Operators and Scattering Theory

2017 ◽  
Vol 18 (6) ◽  
pp. 2075-2085 ◽  
Author(s):  
Benjamin Landon ◽  
Annalisa Panati ◽  
Jane Panangaden ◽  
Justine Zwicker
Keyword(s):  
Scattering Theory ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
One Dimensional
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