[Paper Review] Multiscale Analysis for Ergodic Schrödinger operators and positivity of Lyapunov exponents

Helge Krueger|ArXiv.org|May 12, 2009

Spectral Theory in Mathematical Physics16 references4 citations

TL;DR

This paper develops a multiscale analysis framework for ergodic Schrödinger operators to prove positivity of Lyapunov exponents under initial scale estimates and a Wegner estimate. It establishes positive Lyapunov exponents for high-dimensional skew-shifts at small coupling and for potentials generated by the doubling map, except on a superpolynomially small set, using the Pastur-Figotin formalism and non-degeneracy conditions on the potential function.

ABSTRACT

A variant of multiscale analysis for ergodic Schrödinger operators is developed. This enables us to prove positivity of Lyapunov exponents given initial scale estimates and an initial Wegner estimate. This is then applied to high dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur--Figotin formalism. Furthermore, it is shown that for potentials generated by the doubling map one has positive Lyapunov exponent except in a superpolynomially small set.

Motivation & Objective

• To develop a variant of multiscale analysis tailored for ergodic Schrödinger operators to establish positivity of Lyapunov exponents.
• To address open problems posed by Schlag and Bourgain concerning positivity of Lyapunov exponents for skew-shift and doubling map potentials.
• To extend known results from small coupling regimes to large coupling regimes for the doubling map model.
• To prove that for non-degenerate potentials generated by the doubling map, Lyapunov exponents are positive except on a superpolynomially small set of energies.

Proposed method

• A new multiscale analysis framework is constructed that relies on initial scale estimates and an initial Wegner estimate to propagate positivity of Lyapunov exponents.
• The method uses a non-degeneracy condition on the potential function f, ensuring that level sets have controlled measure via the inequality μ(|f(ω)−E|≤ε)≤Fε^α.
• The analysis applies the Pastur-Figotin formalism to verify initial conditions for the skew-shift model, particularly for small coupling.
• A key technical tool is the control of the derivative of the trace of spectral projections with respect to the dynamical variable, using measure-preserving properties of the shift.
• The proof leverages the fact that perturbations in the potential lead to rank-one changes in the Hamiltonian, enabling bounds on spectral shift functions.
• The method is general and applies independently of the underlying ergodic transformation, with specific applications to the doubling map and skew-shift systems.

Experimental results

Research questions

• RQ1 Can positivity of Lyapunov exponents be established for ergodic Schrödinger operators using a refined multiscale analysis framework?
• RQ2 Does the skew-shift model with small coupling exhibit positive Lyapunov exponents for all energies, as conjectured by Schlag and Bourgain?
• RQ3 For the doubling map potential, is the Lyapunov exponent positive outside a set of superpolynomially small measure, even at large coupling?
• RQ4 Can the range of coupling constants for which positive Lyapunov exponents hold be extended beyond the small coupling regime for the doubling map?
• RQ5 What conditions on the potential function ensure that the initial scale estimates and Wegner estimate are satisfied, enabling the multiscale argument?

Key findings

• For the doubling map potential with a non-degenerate function f, there exist λ₀(f) > 0 and κ(f) > 0 such that for λ > λ₀, the Lyapunov exponent Lλ(E) is positive for all E outside a set E_b of measure at most exp(−λ^{α/2}).
• For the skew-shift model at small coupling, the paper proves positivity of the Lyapunov exponent for all energies E ∈ [−2+δ, −δ] ∪ [δ, 2−δ], extending previous results.
• The method allows extending the range of coupling constants for which positive Lyapunov exponents are known to hold, including large coupling for the doubling map model.
• The paper establishes a general multiscale framework that proves positivity of Lyapunov exponents under initial scale estimates and a Wegner estimate, applicable to a broad class of ergodic potentials.
• For potentials generated by the doubling map, the set of energies with zero Lyapunov exponent has measure at most exp(−λ^{α/2}), indicating superpolynomially small exceptional set.
• The analysis confirms that the skew-shift model satisfies the required initial conditions for multiscale analysis via the Pastur-Figotin formalism, enabling the proof of positivity of Lyapunov exponents.