Continuity properties of the integrated density of states on manifolds

TL;DR

This paper investigates the continuity of the integrated density of states (IDS) for periodic Schrödinger operators on manifolds, providing criteria for continuity, examples of discontinuities, and showing how random perturbations can regularize the IDS.

Contribution

It offers a new criterion for IDS continuity on manifolds and demonstrates how randomness can smooth out discontinuities in the IDS.

Findings

Criteria for IDS continuity at specific energies

Examples of operators with both continuous and discontinuous IDS

Random perturbations can regularize discontinuous IDS

Abstract

We first analyze the integrated density of states (IDS) of periodic Schr\"odinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS'. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous "periodic" IDS which is regularized by a random perturbation.