Two-Term Spectral Asymptotics for the Dirichlet Laplacian on a Bounded Domain

Abstract

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.

Semi-classical analysis of the Laplace operator with Robin boundary conditions

“…Hence, this result is independent of the boundary coefficient b and the proof of Proposition 2.2 is the same as in [4].…”

“…We obtain these results by further extending the approach developed in [4,5], where we treated the Dirichlet Laplacian and the fractional Laplacian on a domain. One virtue of this approach is that it requires only rather weak regularity assumptions on ∂ and b.…”

“…In order to prove (6.5) we introduce local coordinates similarly as in Sect. 4. In this way we are reduced to the situation whereÑ is a subset of R d−1 satisfying a uniform interior ball condition (with a possibly smaller radius).…”

Improved Berezin—Li—Yau inequalities with magnetic field

“…Recently, (2.8) was extended to all domains with C 1,α boundary (with α > 0) by Frank and Geisinger [6].…”

Heat trace of non-local operators

“…These results are in parallel with the results of van den Berg [7] and Brown [9] for the Laplacian. For further recent work on the Dirichlet case in domains of Euclidean space, see Frank and Geisinger [22,23]. For other results related to the spectral theory of these operators, see [2,3] and references therein.…”