Leschke, H; Sobolev, AV; Spitzer, W; (2017) Trace formulas for Wiener-Hopf operators with applications to entropies of free fermionic equilibrium states. Journal of Functional Analysis , 273 (3) pp. 1049-1094. 10.1016/j.jfa.2017.04.005.

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Abstract

We consider non-smooth functions of (truncated) Wiener–Hopf type operators on the Hilbert space L2(Rd) . Our main results are uniform estimates for trace norms ( d≥1 ) and quasiclassical asymptotic formulas for traces of the resulting operators ( d=1 ). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space Rd ( d≥1 ) at positive temperature, T>0 . In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small T and large scaling parameter α>0 provided that the product Tα remains bounded from below. Here α is the reciprocal quasiclassical parameter. For d=1 we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at T=0 ), previously established by the authors for d≥1 .

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