Ergodic Hankel operators

Abstract

We introduce a new class of operators: ergodic families of self-adjoint Hankel operators realised as integral operators on the half-line. Inspired by the spectral theory of differential and finite-difference operators with ergodic coefficients, we develop a spectral theory of ergodic Hankel operators. For these operators, we define the Integrated Density of States (IDS) measure and establish its fundamental properties. In particular, we determine the total mass of the IDS measure in the positive semi-definite case. We also consider in more detail two classes of ergodic Hankel operators for which we are able to make further progress: periodic Hankel operators and the random Kronig--Penney--Hankel (rKPH) model. For periodic Hankel operators, we prove that the IDS measure is a sum of a pure point and absolutely continuous components, and describe the structure of both components. For the rKPH model, we prove the counterparts of the cornerstone results of the spectral theory of random Schr\"odinger operators: Lifshitz tails at the edges of the spectrum, the Wegner bound and Anderson localisation in a natural asymptotic regime. We conclude with some open problems.