Sets of non-Lyapunov behaviour for scalar and matrix Schr\"odinger cocycles

TL;DR

This paper investigates the growth behavior of singular values in symplectic transfer matrices for ergodic Schrödinger operators, revealing non-Lyapunov behaviors and their implications for spectral analysis.

Contribution

It provides new results on the exceptional sets where singular value growth deviates from Lyapunov exponents, including measure-theoretic properties and generalizations of the Thouless formula.

Findings

Residual set of energies with slow singular value growth for any square-summable sequence

Set of energies with non-Lyapunov growth has zero Hausdorff measure under certain conditions

Generalization of the Thouless formula relating Lyapunov exponents to logarithmic potentials

Abstract

We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schr\"odinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_n$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_n^{-1}$. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $\rho(t)$ such that $\rho(t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure.