Leander Geisinger: Refined semi-classical asymptotics for the Dirichlet Laplacian and the Fractional Laplacian

We consider the Dirichlet Laplacian and the Fractional Laplacian on a domain and investigate the asymptotic behaviour of its eigenvalues. Extending methods from semi-classical analysis we prove a two-term formula for the sum of eigenvalues with the leading (Weyl) term given by the volume and the sub-leading term by the surface area. The result is valid under weak assumptions on the regularity of the boundary. We also discuss relations of the asymptotic results to improved uniform estimates for eigenvalue sums and eigenvalue means.