Spectral theory for perturbed Krein Laplacians in nonsmooth domains

Abstract

We study spectral properties for HK, Ω, the Krein-von Neumann extension of the perturbed Laplacian - Δ + V defined on C0∞ (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1, r, r > 1 / 2. In particular, in the aforementioned context we establish the Weyl asymptotic formula# {j ∈ N | λK, Ω, j ≤ λ} = (2 π)- n vn | Ω | λn / 2 + O (λ(n - (1 / 2)) / 2) as λ → ∞, where vn = πn / 2 / Γ ((n / 2) + 1) denotes the volume of the unit ball in Rn, and λK, Ω, j, j ∈ N, are the non-zero eigenvalues of HK, Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein-von Neumann extension of - Δ + V defined on C0∞ (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exterior-type domains Ω = Rn {set minus} K, n ≥ 3, with K ⊂ Rn compact and vanishing Bessel capacity B2, 2 (K) = 0, to prove equality of Friedrichs and Krein Laplacians in L2 (Ω ; dn x), that is, - Δ |C0∞ (Ω) has a unique nonnegative self-adjoint extension in L2 (Ω ; dn x). © 2009 Elsevier Inc. All rights reserved.