Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.     

Bounds on the Spectral Shift Function and the Density of States

We study spectra of Schrödinger operators on ? ^d . First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values μ _n of the difference of the semigroups as n→∞ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger operators. The single site potential u is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.     

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically Hölder observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.     

Non-lacunary Gibbs Measures for Certain Fractal Repellers

In this paper, we study non-uniformly expanding repellers constructed as the limit sets for a non-uniformly expanding dynamical systems. We prove that given a Hölder continuous potential φ satisfying a summability condition, there exists non-lacunary Gibbs measure for φ, with positive Lyapunov exponents and infinitely many hyperbolic times almost everywhere. Moreover, this non-lacunary Gibbs measure is an equilibrium measure for φ.     

Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian

We obtain the asymptotics of the optimal global Hölder exponent of the integrated density of states of the Fibonacci Hamiltonian for large and small couplings.     

Localization for a Matrix-valued Anderson Model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrödinger operators, acting on $L^2(\mathbb R)\otimes \mathbb C^NWe study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr?dinger operators, acting on , for arbitrary N ≥ 1. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval , they exhibit localization properties on I, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Fürstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.      

Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.     

The Random Phase Property and the Lyapunov Spectrum for Disordered Multi-channel Systems

A random phase property establishing in the weak coupling limit a link between quasi-one-dimensional random Schrödinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-Ando model on a tubular geometry with magnetic field and spin-orbit coupling, the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the Lyapunov spectrum.     

Cauchy Problem for Dissipative Hölder Solutions to the Incompressible Euler Equations

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent ${\theta < \frac{1}{16}}$ . Using the techniques introduced in De Lellis and Székelyhidi (Inventiones Mathematicae 9:377–407, 2013; Dissipative Euler flows and Onsager’s conjecture, 2012), we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.     

Correlation Decay and Recurrence Asymptotics for Some Robust Nonuniformly Hyperbolic Maps

We study a robust class of multidimensional non-uniformly hyperbolic transformations considered by Oliveira and Viana (Ergod. Theory Dyn. Syst. 28:501–533, 2008). For an open class of Hölder continuous potentials with small variation we show that the unique equilibrium state has exponential decay of correlations and that the distribution of hitting times is asymptotically exponential. Furthermore, using that the equilibrium states satisfy a weak Gibbs property we also prove log-normal fluctuations of the return times around their average.     

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr?dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr?dinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr?dinger operators with Neumann boundary conditions. Received: 20 October 2000 / Accepted: 8 March 2001     

Continuity of the Measure of the Spectrum for Quasiperiodic Schrödinger Operators with Rough Potentials

We study discrete quasiperiodic Schrödinger operators on ${\ell^2(\mathbb{Z})}$ with potentials defined by γ-Hölder functions. We prove a general statement that for γ > 1/2 and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of convergence from above in the subadditive ergodic theorem for strictly ergodic cocycles.     

Hölder Continuity of Absolutely Continuous Spectral Measures for One-Frequency Schrödinger Operators

We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrödinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Hölder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.     

Phase Transitions for Uniformly Expanding Maps

Given a uniformly expanding map of two intervals we describe a large class of potentials admitting unique equilibrium measures. This class includes all Hölder continuous potentials but goes far beyond them. We also construct a family of continuous but not Hölder continuous potentials for which we observe phase transitions. This provides a version of the example in (9) for uniformly expanding maps.     

A Hilbert space approach to the Lippmann-Schwinger equation in momentum representation

We study the Lippmann-Schwinger equation for the quantum mechanical two-body scattering problem. We propose a Hilbert space approach in momentum-angular momentum representation. Imposing a Hölder integrability condition on the potential, the kernel of the integral equation becomes a compact operator in an adequate Hilbert space H⁰. We show that expansion into orthogonal polynomials becomes very simple, and we give an application to the three-particle problem.     

Electronic properties of quasiperiodic Fibonacci chain including second-neighbor hopping in the tight-binding model

We present an exact real-space renormalization group (RSRG) scheme for the electronic Green's functions of one-dimensional tight-binding systems having both nearest-neighbor and next-nearest-neighbor hopping integrals, and determine the electronic density of states for the quasiperiodic Fibonacci chain. This RSRG method also gives the Lyapunov exponents for the eigenstates. The Lyapunov exponents and the analysis of the flow pattern of hopping integrals under renormalization provide information about the nature of the eigenstates. Next we develop a transfer matrix formalism for this generalized tight-binding system, which enables us to determine the wave function amplitudes. Interestingly, we observe that like the nearest-neighbor tight-binding Fibonacci chain, the present generalized tight-binding system also have critical eigenstates, Cantor-set energy spectrum and highly fragmented density of states. It indicates that these exotic physical properties are really the characteristics of the underlying quasiperiodic structure. Received 5 April 1999     

Generalized Hölder Continuity and Oscillation Functions

We study a notion of generalized Hölder continuity for functions on ?^d. We show that for any bounded function f of bounded support and any r >?0, the r-oscillation of f defined as \(osc_{r} f (x):= \sup _{B_{r}(x)} f - \inf _{B_{r}(x)} f\) is automatically generalized Hölder continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.     

Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, “hyperbolic”. We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Hölder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Hölder continuous functions. Another consequence is that every Hölder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.     

Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < h top (f)

We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials φ with the ‘bounded range’ condition sup φ ? inf φ < h _top (f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of Perron-Frobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context.     

Uniform Spectral Properties of One-Dimensional Quasicrystals,II. The Lyapunov Exponent

In this Letter we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrödinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.