# Ergodic Schr\"odinger Operators in the Infinite Measure Setting

## TL;DR

This paper extends the theory of ergodic Schr"odinger operators to infinite measure spaces, establishing key spectral properties and providing counterexamples that highlight differences from the finite measure case.

## Contribution

It develops foundational results for ergodic Schr"odinger operators in infinite measure settings, including spectral constancy, density of states, and Lyapunov exponents, with new counterexamples.

## Key findings

- Spectral and spectral type constancy almost surely
- Definition and analysis of density of states measure and Lyapunov exponent in infinite measure setting
- Counterexamples showing some finite measure results do not extend to infinite measure case

## Abstract

We develop the basic theory of ergodic Schr\"odinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur--Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.