Szeg? limit theorem for operators with discontinuous symbols and applications to entanglement entropy

摘要

The main result in this paper is a one-term Szegö-type asymptotic formula with a sharp remainder estimate for a class of integral operators of the pseudodifferential type with symbols which are allowed to be nonsmooth or discontinuous in both position and momentum. The simplest example of such symbols is the product of the characteristic functions of two compact sets, one in real space and the other in momentum space. The results of this paper are used in a study of the violation of the area-entropy law for free fermions by Gioev and Klich (2006). This work also provides evidence towards a conjecture due to Harold Widom.