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 ${\mathbb{R}}^d$ 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.

7 pages

References in corpus (3)

• Two dimensional Berezin-Li-Yau inequalities with a correction term
• Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
• Spectral Properties of the Dirichlet Operator ${\sum }_{i=1}^d(-{\partial }_i^2{)}^s$ on Domains in d-Dimensional Euclidean Space

Cited by in corpus (4)

• Spectral results for mixed problems and fractional elliptic operators
• On the spectral asymptotics for the buckling problem
• Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets
• Weyl asymptotics for functional difference operators with power to quadratic exponential potential