Volume estimates and spectral asymptotics for a class of pseudo-differential operators

Two asymptotic expansions, known as the heat and Szegö expansions, are studied for a class of operators defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega )$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a compact region in Euclidean space with smooth boundary. Pseudo-differential operator methods are combined with a version of a theorem originally due to H. Weyl on the volume of certain tubular neighborhoods to obtain significant new information about the higher order terms in the expansions.