Ergodic Schr\"odinger Operators in the Infinite Measure Setting
Spectral Theory
2019-07-30 v1 Mathematical Physics
Dynamical Systems
math.MP
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.
Cite
@article{arxiv.1907.12471,
title = {Ergodic Schr\"odinger Operators in the Infinite Measure Setting},
author = {Michael Boshernitzan and David Damanik and Jake Fillman and Milivoje Lukić},
journal= {arXiv preprint arXiv:1907.12471},
year = {2019}
}
Comments
23 pages