scholarly journals Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators

2017 ◽  
Vol 7 (2) ◽  
pp. 361-386 ◽  
Author(s):  
Kai Tao ◽  
Mircea Voda
Keyword(s):  
Density Of States ◽  
Hölder Continuity ◽  
Integrated Density Of States ◽  
Holder Continuity ◽  
Jacobi Operators

scholarly journals HÖLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR MATRIX-VALUED ANDERSON MODELS

2008 ◽  
Vol 20 (07) ◽  
pp. 873-900 ◽  
Author(s):  
HAKIM BOUMAZA
Keyword(s):  
Density Of States ◽  
Anderson Model ◽  
Final Section ◽  
Hölder Continuity ◽  
Regularity Result ◽  
Previous Article ◽  
Integrated Density Of States ◽  
Holder Continuity ◽  
Transfer Matrices ◽  
Anderson Models

We study a class of continuous matrix-valued Anderson models acting on L2(ℝd) ⊗ ℂN. We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then, for d = 1 and for arbitrary N, we prove the Hölder continuity of the Integrated Density of States under some assumption on the group GμE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group GμE is verified. Therefore, the general results developed here can be applied to this model.

Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycle with weak Liouville frequency

2013 ◽  
Vol 34 (4) ◽  
pp. 1395-1408 ◽  
Author(s):  
JIANGONG YOU ◽  
SHIWEN ZHANG
Keyword(s):  
Lyapunov Exponent ◽  
Density Of States ◽  
Large Deviation ◽  
Hölder Continuity ◽  
Integrated Density Of States ◽  
Diophantine Condition ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Subharmonic Functions ◽  
Large Deviation Theorem

AbstractFor analytic quasiperiodic Schrödinger cocycles, Goldshtein and Schlag [Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154 (2001), 155–203] proved that the Lyapunov exponent is Hölder continuous provided that the base frequency $\omega $ satisfies a strong Diophantine condition. In this paper, we give a refined large deviation theorem, which implies the Hölder continuity of the Lyapunov exponent for all Diophantine frequencies $\omega $, even for weak Liouville $\omega $, which improves the result of Goldshtein and Schlag.

scholarly journals Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

2020 ◽  
Author(s):  
Rui Han ◽  
Shiwen Zhang
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Density Of States ◽  
Large Deviation ◽  
Schrödinger Operators ◽  
Positive Lyapunov Exponent ◽  
Integrated Density Of States ◽  
Holder Continuity ◽  
One Dimensional ◽  
Large Coupling

Abstract We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.

scholarly journals Oscillation Theory for the Density of States of High Dimensional Random Operators

2017 ◽  
Vol 2019 (15) ◽  
pp. 4579-4602
Author(s):  
Julian Groß mann ◽  
Hermann Schulz-Baldes ◽  
Carlos Villegas-Blas
Keyword(s):  
Density Of States ◽  
Von Neumann Algebra ◽  
Jacobi Operator ◽  
Oscillation Theory ◽  
Integrated Density Of States ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Sturm Liouville ◽  
Finite Trace

Abstract Sturm–Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

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