Density of States for Random Contractions

TL;DR

This paper introduces a new density of states functional for random contraction operators, establishing its integral representations and convergence properties, extending spectral analysis techniques to non-normal operators.

Contribution

It defines the DOS functional for random contractions, proves its integral representations, and demonstrates its convergence and relation to spectral measures in the infinite volume limit.

Findings

DOS functional has natural integral representations on the unit circle and disk

Proven to be the almost sure limit of finite volume trace measures

Established connection between the DOS functional and spectral measures

Abstract

We define a linear functional, the DOS functional, on spaces of holomorphic functions on the unit disk which is associated with random ergodic contraction operators on a Hilbert space, in analogy with the density of state functional for random self-adjoint operators. The DOS functional is shown to enjoy natural integral representations on the unit circle and on the unit disk. For random contractions with suitable finite volume approximations, the DOS functional is proven to be the almost sure infinite volume limit of the trace per unit volume of functions of the finite volume restrictions. Finally, in case the normalised counting measure of the spectrum of the finite volume restrictions converges in the infinite volume limit, the DOS functional is shown admit an integral representation on the disk in terms of the limiting measure, despite the discrepancy between the spectra of non normal operators and their finite volume restrictions. Moreover, the integral representation of the DOS functional on the unit circle is related to the Borel transform of the limiting measure.