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.
Cite
@article{arxiv.1404.2256,
title = {Two-term Szeg\H{o} theorem for generalised anti-Wick operators},
author = {J. P. Oldfield},
journal= {arXiv preprint arXiv:1404.2256},
year = {2014}
}
Comments
v2 slightly clarifies the proof of Lemma 3.4 (the statement remains the same)