Abstract
We consider functions of Wiener–Hopf type operators on the Hilbert space \({\mathsf{L}^2(\mathbb{R}^d)}\). 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.
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Sobolev, A.V. Quasi-classical asymptotics for functions of Wiener–Hopf operators: smooth versus non-smooth symbols. Geom. Funct. Anal. 27, 676–725 (2017). https://doi.org/10.1007/s00039-017-0408-9
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DOI: https://doi.org/10.1007/s00039-017-0408-9