Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator.

link: http://arxiv.org/abs/1105.5181
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.