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

摘要

We study spectral asymptotics for a large class of differential operators on an open subset of R~d with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operatorswith 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.