Quasi-classical asymptotics for functions of Wiener-Hopf operators: smooth vs non-smooth symbols

We consider functions of Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. It has been known for a long time that the quasi-classical asymptotics for traces of resulting operators strongly depend on the smoothness of the symbol: for smooth symbols the expansion is power-like, whereas discontinuous symbols (e.g. indicator functions) produce an extra logarithmic factor. We investigate the transition regime by studying symbols depending on an extra parameter $T\ge 0$ in such a way that the symbol tends to a discontinuous one as $T\to 0$. The main result is two-parameter asymptotics (in the quasi-classical parameter and in $T$), describing a transition from the smooth case to the discontinuous one. The obtained asymptotic formulas are used to analyse the low-temperature scaling limit of the spatially bipartite entanglement entropy of thermal equilibrium states of non-interacting fermions.