A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains

Abstract

This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian − C ∞ 0 ( ) in L 2 ( ; d n x). Here, the domain belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C 1,r , for r > 1/2. Second, we establish Kreȋn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

“…Moreover, the spectral properties of A δ,α can be described with the help of the perturbation term (Θ δ,α − M (λ)) −1 . We mention that in the context of the more general notion of quasi boundary triples and their Weyl functions from [5,7] a similar approach as in this note and closely related results can be found in [8,9]; we also refer to [26,27,29,31,35,[38][39][40] for other methods in extension theory of elliptic differential operators.…”

Generalized interactions supported on hypersurfaces

“…Note that ran (−∆ free − λ) [18,Section 3] for the required properties of trace maps on Lipschitz domains. Since the embeddings of…”

Spectral asymptotics for Robin problems with a discontinuous coefficient

“…Let us also mention that we do not here address the question of nonsmooth domains, as e.g. in Gesztesy and Mitrea [12], [13], [14] and Abels, Grubb, and Wood [1], [21], and their references.…”