The Widom-Sobolev formula for discontinuous matrix-valued symbols
Abstract
We prove the Widom-Sobolev formula for the asymptotic behaviour of truncated Wiener-Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with H\"older singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.
Cite
@article{arxiv.2311.06036,
title = {The Widom-Sobolev formula for discontinuous matrix-valued symbols},
author = {Leon Bollmann and Peter Müller},
journal= {arXiv preprint arXiv:2311.06036},
year = {2025}
}
Comments
41 pages; v3: as published; changes in v2: Introduction changed. New Remark 3.4 adds information on the scaling properties of the constants in Lemma 3.3. Proof of Theorem 3.9 corrected and expanded. Proof of Theorem 4.22 reorganised with parts of it now contained in the new Lemma 4.19