Dedicated to Ari Laptev on the occasion of his 60th birthday.Let −∆ denote the Dirichlet Laplace operator on a bounded open set in R d . We study the sum of the negative eigenvalues of the operator −h 2 ∆ − 1 in the semiclassical limit h → 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on 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.

Abstract. We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set inIn particular, we derive upper bounds on Riesz means of order 3=2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.Mathematics Subject Classification (2010). Primary 35P15; Secondary 47A75.

We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this reformulation it is natural to allow the function describing the boundary conditions to depend on the semi-classical parameter and we identify and analyze three different regimes for this dependence.