Full SzegH{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 $\Omega\ni \omega\mapsto H_\omega$ 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 $L$.