Multiplicative Jensen's formula and quantitative global theory of one-frequency Schrödinger operators

2023/06/28 by Lingrui Ge, Svetlana Jitomirskaya, Ge, Lingrui +5 · 1 citation
Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Nonlinear Photonic Systems #Mechanical and Optical Resonators

paper · pdf · doi:10.48550/arxiv.2306.16387

Abstract

We introduce the concept of dual Lyapunov exponents, leading to a multiplicative version of the classical Jensen's formula for one-frequency analytic Schrödinger cocycles. This formula, in particular, gives a new proof and a quantitative version of the fundamentals of Avila's global theory \citeavila, fully explaining the behavior of complexified Lyapunov exponent through the dynamics of the dual cocycle. In particular, concepts of (sub/super) critical regimes and acceleration are all explained (in a quantitative way) through the duality approach. This leads to a number of powerful spectral and physics applications

Cited by

• Cantor spectrum for multidimensional quasi-periodic Schrödinger operators

Related

• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity
• KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-frequency Schrodinger operators
• What are Lyapunov exponents, and why are they interesting?
• Effective multi-scale approach to the Schrödinger cocycle over a skew shift base
• Lyapunov exponents in complex dynamics and potential theory