[Paper Review] Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential

Jean Bourgain, Svetlana Jitomirskaya|ArXiv.org|Oct 31, 2001

Spectral Theory in Mathematical Physics10 references19 citations

TL;DR

This paper establishes the continuity of the Lyapunov exponent for one-dimensional quasiperiodic Schrödinger operators with analytic potentials, both in energy E and jointly in (E, ω) when ω is irrational. Using large deviation estimates and the avalanche principle, the authors prove uniform convergence of Lyapunov exponent approximations, leading to continuity results that resolve long-standing regularity questions in spectral theory of quasiperiodic systems.

ABSTRACT

We study regularity properties of the Lyapunov exponent L of quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove that L is jointly continuous, in frequency and energy, at every irrational frequency.

Motivation & Objective

• To establish the continuity of the Lyapunov exponent L(E, ω) for 1D quasiperiodic operators with real analytic potentials.
• To resolve the long-standing open problem of whether L(E, ω) is continuous in E, especially when L(E) > 0.
• To prove joint continuity in (E, ω) at irrational frequencies, despite known discontinuities in ω at rational points.
• To provide uniform estimates on transfer matrix norms that are crucial for non-perturbative analysis and dynamical localization.

Proposed method

• Derives a large deviation theorem (Lemma 4) for the growth of transfer matrices, showing that typical deviations from the Lyapunov exponent are exponentially small in q.
• Applies the avalanche principle to control the logarithmic norm of products of SL2(R) matrices under suitable hyperbolicity and correlation conditions.
• Constructs a sequence of approximants {qs}, {Ns} to the frequency ω, satisfying recursive inequalities that ensure exponential decay of error terms.
• Uses the avalanche principle and large deviation estimates to bound the difference |L(E) + LN0(E) - 2L2N0(E)| by e^{-cκ q}, enabling uniform approximation.
• Employs subharmonicity and compactness arguments to control the convergence of L(E) along sequences Eα → E.
• Establishes joint continuity in (E, ω) by controlling the dependence on both variables simultaneously through uniform bounds in the approximation sequence.

Experimental results

Research questions

• RQ1 Is the Lyapunov exponent L(E, ω) continuous in energy E for quasiperiodic operators with analytic potentials?
• RQ2 Is the Lyapunov exponent jointly continuous in (E, ω) at irrational frequencies ω?
• RQ3 Can uniform estimates on the growth of transfer matrices be established that are independent of x and E on compact sets?
• RQ4 Does the continuity of L(E, ω) hold even when the potential is analytic and the frequency is Liouville?
• RQ5 Can the uniformity of the large deviation estimate be leveraged to extend results on dynamical localization beyond Diophantine frequencies?

Key findings

• The Lyapunov exponent L(E, ω) is continuous in E for all E and all analytic potentials v.
• For any irrational ω0, the Lyapunov exponent L(E, ω) is jointly continuous in (E, ω) at (E, ω0).
• The estimate |L(E) + LN0(E) - 2L2N0(E)| < e^{-cκ q} holds uniformly for all E in a compact set and all ω close to a rational approximant a/q.
• The uniform bound (1.4) holds: lim sup_{N→∞} (1/N) log ||MN(E, x, ω)|| ≤ L(E) uniformly in x and E on compact sets.
• The result implies strong dynamical localization for the almost Mathieu operator at all λ > 2 and all irrational ω, without requiring strong Diophantine conditions.
• The continuity result is sharp: L(E, ω) may be discontinuous in ω at every rational ω, but is continuous at all irrational frequencies.