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

TL;DR

This paper establishes a generalized Weyl's law for a broad class of differential operators, including fractional and non-homogeneous operators, by deriving spectral asymptotics and sharp eigenvalue estimates.

Contribution

It extends Weyl's law to fractional and non-homogeneous differential operators, providing the first semiclassical asymptotic term for these operators.

Findings

Derived sharp eigenvalue sum estimates.

Established first term of spectral asymptotics for fractional operators.

Generalized Weyl's law to non-homogeneous symbols.

Abstract

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 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.