  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 $L$.
