[Paper Review] Subcritical behavior for quasi-periodic Schrödinger cocycles with trigonometric potentials

Christoph A. Marx, Li Shou+1 more|arXiv (Cornell University)|Sep 17, 2015

Spectral Theory in Mathematical Physics26 references3 citations

TL;DR

This paper establishes a criterion for subcritical behavior in quasi-periodic Schrödinger cocycles with trigonometric polynomial potentials using complexified Lyapunov exponent analysis. By analyzing the analytic continuation of the potential and applying Herman's bounds on the Lyapunov exponent, the authors prove that positive Lyapunov exponent implies purely absolutely continuous spectrum for all irrational frequencies and phases, extending Avila's global theory to non-AMO trigonometric potentials.

ABSTRACT

We give a criterion implying subcritical behavior for quasi-periodic Schrödinger operators where the potential sampling function is given by a trigonometric polynomial. Subcritical behavior, in the sense of Avila's global theory, is known to imply purely absolutely continuous spectrum for all irrational frequencies and all phases.

Motivation & Objective

• To extend Avila's global theory of one-frequency operators to quasi-periodic Schrödinger operators with trigonometric polynomial potentials.
• To provide a criterion for subcritical behavior that implies purely absolutely continuous spectrum for all irrational frequencies and phases.
• To analyze the complexified Lyapunov exponent and derive lower bounds on the real Lyapunov exponent using Herman's radius and root localization techniques.
• To demonstrate that positivity of the Lyapunov exponent—implying subcriticality—can be established for non-AMO potentials through numerical and analytical methods.

Proposed method

• The authors analyze the complexified Lyapunov exponent $ L(\epsilon; E) $ of the Schrödinger cocycle $ (\alpha, B^E) $, using analytic continuation of the potential $ v(x) $ into the complex strip.
• They apply Herman's lower bound on the Lyapunov exponent, expressed as $ L(\epsilon; E) \geq \log|\lambda_M| + 2\pi\epsilon M $, where $ \lambda_M $ is the dominant Fourier coefficient and $ M $ is the number of terms in the trigonometric polynomial.
• A key step involves estimating the infimum of $ |v(x + i\epsilon)| $ over $ x \in \mathbb{T} $, denoted $ \widetilde{m}(\epsilon; E) $, to ensure it exceeds a threshold that enables application of the Herman bound.
• The method uses root localization in the complex plane to compute the spectral radius $ \gamma $, defined as $ \gamma = -\log|z_0| - \log 2 $, where $ z_0 $ is the unique root inside the unit disk.
• The uniform Herman radius $ \epsilon_{H;\text{unif}} $ is computed numerically to derive a uniform lower bound on the Lyapunov exponent across the spectrum.
• Numerical validation is performed using Mathematica for a specific potential $ v(x) = 18\cos(2\pi x) + 1.6\cos(4\pi x) $, confirming $ L(0; E) > 0 $ for $ E \in [-21.6, 21.6] $.

Experimental results

Research questions

• RQ1 Can subcritical behavior be established for quasi-periodic Schrödinger operators with general trigonometric polynomial potentials beyond the almost Mathieu operator?
• RQ2 Does the positivity of the Lyapunov exponent—implying purely absolutely continuous spectrum—follow from a criterion based on complexified potential growth and Herman's bound?
• RQ3 Can the uniform Herman radius be computed numerically to yield a uniform lower bound on the Lyapunov exponent across the spectrum?
• RQ4 How does the root distribution of the associated Laurent polynomial relate to the spectral radius and Lyapunov exponent?

Key findings

• For the potential $ v(x) = 18\cos(2\pi x) + 1.6\cos(4\pi x) $, the Lyapunov exponent satisfies $ L(0; E) \geq 0.727 > 0 $ at $ E = -2 $, confirming subcritical behavior.
• The uniform Herman radius was numerically computed as $ \epsilon_{H;\text{unif}} = 0.4142 $, enabling a uniform lower bound on the Lyapunov exponent across the spectrum.
• The spectral radius $ \gamma $ was computed as $ \gamma = 0.458 $, derived from the unique root $ z_0 = -0.3161 $ inside the unit disk.
• The complexified potential magnitude satisfies $ \min_{x \in \mathbb{T}} |v(x + i0.2)| \approx 24.242 > 23.6 = 4 + 2\lambda_1 + 2\lambda_2 $, validating the applicability of the criterion.
• The lower bound $ L(0; E) \geq \log|\lambda_2| + \gamma \frac{\epsilon_{H;\text{unif}}}{\epsilon_{H;\text{unif}} - \epsilon_1} $ yields $ L^{-}(E) > 0 $ for all $ E \in [-21.6, 21.6] $, confirming positivity of the Lyapunov exponent.
• The proof establishes that for any irrational frequency $ \alpha $, the spectrum is purely absolutely continuous for all phases when the Lyapunov exponent is positive, extending Avila's theory beyond the almost Mathieu operator.