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

We prove full Szeg\H{o}-type large-box trace asymptotics for selfadjoint ${\mathbb{Z}}^d$-ergodic operators $\mathrm{\Omega }\ni \omega \mapsto H_{\omega }$ 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 $L$.