Discontinuity of the Lyapunov Exponent

TL;DR

This paper investigates the discontinuity phenomena of the Lyapunov exponent in limit-periodic Schrödinger operators, demonstrating the existence and density of potentials with discontinuous Lyapunov exponents.

Contribution

It constructs a specific limit-periodic Schrödinger operator with a positive measure set of discontinuities in its Lyapunov exponent and proves the density of such potentials.

Findings

Lyapunov exponent can be discontinuous on a positive measure set

Discontinuous Lyapunov exponents are dense among limit-periodic potentials

Constructed explicit examples of operators with discontinuous Lyapunov exponents

Abstract

We study discontinuity of the Lyapunov exponent. We construct a limit-periodic Schr\"odinger operator, of which the Lyapunov exponent has a positive measure set of discontinuities. We also show that the limit-periodic potentials, whose Lyapunov exponent is discontinuous, are dense in the space of limit-periodic potentials.