A short proof of Weyl's law for fractional differential operators

Abstract

We study spectral asymptotics for a large class of differential operators on an open subset of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with non-homogeneous symbols. Based on a sharp estimate for the sum of the eigenvalues we establish the first term of the semiclassical asymptotics. This generalizes Weyl's law for the Laplace operator.