Two-term Szeg\H{o} theorem for generalised anti-Wick operators

Abstract

This article concerns the asymptotics of pseudodifferential operators whose Weyl symbol is the convolution of a discontinuous function dilated by a large scaling parameter with a smooth function of constant scale. These operators include as a special case generalised anti-Wick operators, also known as Gabor-Toeplitz operators, with smooth windows and dilated discontinuous symbol. The main result is a two-term Szeg\H{o} theorem, that is, the asymptotics of the trace of a function of the operator. A special case of this is the asymptotic terms of the eigenvalue counting function. In both cases, previously only the first term in the asymptotic expansion was known explicitly.