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.

arXiv comment7 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