Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications

Abstract

Dedicated to Barry Simon on the occasion of his 60th birthday.Abstract. The absolutely continuous spectrum of an ergodic family of onedimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur.Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum.We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolutely continuous spectrum for potentials defined by the doubling map on the circle, and the absence of singular spectrum for the subcritical almost Mathieu operator.

Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling

“…We refer the reader to [11,19] for more information on the theory leading to the formulae (14) and (15); see especially [11,Theorem 5] and [19,Theorem 4.8].…”

“…A beautiful result of Kotani [19], which has not yet received the attention and exposure it deserves, shows that if the Lyapunov exponent vanishes in the spectrum, then absolute continuity of the IDS is equivalent to absolute continuity of the spectral measures for almost every θ; see also the survey [11] of Kotani theory and its applications. By [9], if the coupling is subcritical, the Lyapunov exponent is zero on the spectrum.…”

Expansivity and shadowing in linear dynamics

“…Proof. By (13), for each x ∈ X, there are unique x (1) ∈ M and x (2) ∈ N satisfying x = x (1) + x (2) . Moreover, there is a constant β > 0 such that…”

Positive Lyapunov exponents and localization bounds for strongly mixing potentials

“…If P = p ⊗Z is a product measure of a compactly supported probability measure p on Σ so that the random variables of the sequence (V (S n ω)) n∈Z of potential values are independent, the model exhibits the so-called Anderson localization, namely the spectrum of H λ,ω is P-almost surely pure-point with exponentially localized eigenstates [14], and the induced quantum dynamics is bounded in time (in the precise sense given below). The question considered in this work (and many others, see the reviews [5,9] and references therein) concerns the spectral properties as well as the quantum dynamics in situations where P is not a product measure so that the random variables (V (S n ω)) n∈Z are correlated. This situation typically arises when the dynamical system (Ω, S, P) is the symbolic dynamics associated to a (possibly weakly) hyperbolic discrete time dynamics; then Σ is the Markov partition.…”

“…This positivity can either be established by Kotani theory [5], a version of Furstenberg's theorem for correlated random matrices (work by Avila and Damanik cited in [5]) or by a perturbative calculation (for small λ) of the Lyapunov exponent. This latter calculation was first done by Chulaevsky and Spencer [4] by carrying over the argument of Thouless [18], in a version given by Pastur and Figotin [14], to the case of correlated potential values.…”