Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator

TL;DR

This paper derives a precise asymptotic formula for the eigenvalues of the fractional Laplacian on domains, including leading and subleading terms, under minimal boundary regularity assumptions.

Contribution

It extends semi-classical analysis techniques to fractional Laplacians, providing a two-term eigenvalue asymptotic formula with weak boundary regularity conditions.

Findings

Established a two-term asymptotic formula for fractional Laplacian eigenvalues

The leading term depends on the volume of the domain

The subleading term depends on the surface area of the boundary

Abstract

We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for the sum of eigenvalues with the leading (Weyl) term given by the volume and the subleading term by the surface area. Our result is valid under very weak assumptions on the regularity of the boundary.