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

2016/09/07 by Sobolev, Alexander V. · 1 citation
#35S05 (Primary) #45M05 #47B10 #47B35 (Secondary) #47G30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1609.02068

Abstract

We consider functions of Wiener--Hopf type operators on the Hilbert space L²(\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≥ 0 in such a way that the symbol tends to a discontinuous one as T→ 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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• Area Law for the entanglement entropy of free fermions in nonrandom ergodic field

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