Full Szeg\H{o}-type trace asymptotics for ergodic operators on large boxes

TL;DR

This paper establishes comprehensive Szeg"H{o}-type asymptotic formulas for the trace of functions of ergodic operators on large finite boxes, providing detailed expansion in terms of the box size.

Contribution

It proves the full asymptotic expansion of the trace for ergodic operators on large boxes, extending Szeg"H{o}-type results to a broad class of operators.

Findings

Full asymptotic expansion of the trace in terms of box size L

Applicable to a wide class of ergodic operators with decaying kernels

Generalizes classical Szeg"H{o} asymptotics to higher dimensions and ergodic settings

Abstract

We prove full Szeg\H{o}-type large-box trace asymptotics for selfadjoint $\mathbb{Z}^d$-ergodic operators $\Omega\ni \omega\mapsto H_\omega$ acting on $L^2(\mathbb{R}^d)$. More precisely, let $g$ be a bounded, compactly supported and real-valued function such that the (averaged) operator kernel of $g(H_\omega)$ decays sufficiently fast, and let $h$ be a sufficiently smooth compactly supported function. We then prove a full asymptotic expansion of the averaged trace of the operator $h(g(H_\omega)_{[-L,L]^d})$ in terms of the length-scale $L$.