GAP-LABELLING THEOREMS FOR SCHRÖDINGER OPERATORS ON THE SQUARE AND CUBIC LATTICE

1994 ◽  
Vol 06 (02) ◽  
pp. 319-342 ◽  
Author(s):  
ANDREAS VAN ELST
Keyword(s):  
Density Of States ◽  
Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Cubic Lattice ◽  
Higher Dimensional ◽  
K Theory

The spectra of Schrödinger operators on the square and cubic lattice are investigated by means of non-commutative topological K-theory. Using a general gap-labelling theorem, it is shown how to calculate the possible values of the integrated density of states on the gaps of the spectrum, provided some additional conditions hold. If the potential takes on only finitely many values, this reduces to the calculation of frequencies of patterns in the potential sequence. As an example, products of one-dimensional systems and potentials generated by higher-dimensional substitutions are considered.

scholarly journals Random Schrödinger operators with a background potential

2019 ◽  
Vol 27 (4) ◽  
pp. 253-259
Author(s):  
Hayk Asatryan ◽  
Werner Kirsch
Keyword(s):  
Inverse Scattering ◽  
Density Of States ◽  
Anderson Localization ◽  
Scattering Problem ◽  
Schrödinger Operators ◽  
Inverse Scattering Problem ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
One Dimensional

Abstract We consider one-dimensional random Schrödinger operators with a background potential, arising in the inverse scattering problem. We study the influence of the background potential on the essential spectrum of the random Schrödinger operator and obtain Anderson localization for a larger class of one-dimensional Schrödinger operators. Further, we prove the existence of the integrated density of states and give a formula for it.

scholarly journals Dependence of the Density of States on the Probability Distribution for Discrete Random Schrödinger Operators

2018 ◽  
Vol 2020 (17) ◽  
pp. 5279-5341 ◽  
Author(s):  
Peter D Hislop ◽  
Christoph A Marx
Keyword(s):  
Probability Measure ◽  
Density Of States ◽  
Anderson Model ◽  
Schrödinger Operators ◽  
Bethe Lattice ◽  
Single Site ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
Wide Range

Abstract We prove that the density of states measure (DOSm) for random Schrödinger operators on $\mathbb{Z}^d$ is weak-$^{\ast }$ Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general compactly supported single-site probability measure, without any further assumptions. The few previously available results for the disorder dependence of the IDS valid for dimensions $d \geqslant 2$ assumed absolute continuity of the single-site measure and thus excluded the Bernoulli–Anderson model. As a further application of our main result, we establish quantitative continuity results for the Lyapunov exponent of random Schrödinger operators for $d=1$ in the probability measure with respect to the weak-$^{\ast }$ topology.

scholarly journals On the Continuity of the Integrated Density of States in the Disorder

2019 ◽  
Author(s):  
Mira Shamis
Keyword(s):  
Density Of States ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
Quantitative Form ◽  
Alternative Approach ◽  
Ky Fan Inequalities ◽  
Ky Fan ◽  
Sharp Version

Abstract Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\mathbb{R}^d$.

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.

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