NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (2), P. 315–323
Laplacians with singular perturbations supported on hypersurfaces
A. Mantile – Laboratoire de Mathématiques de Reims, EA4535 URCA, Fédération de Recherche ARC Mathématiques, FR 3399 CNRS, France; firstname.lastname@example.org
A. Posilicano – DiSAT, Sezione di Matematica, Università dell’Insubria, via Valleggio 11, I22100 Como, Italy; email@example.com
We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the n-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This allows one to obtain Krein-like resolvent formulae where the reference operator coincides with the free self-adjoint Laplacian in Rn, providing in this way with an useful tool for the scattering problem from a hypersurface. As examples of this construction, we consider the cases of Dirichlet and Neumann boundary conditions assigned on an unclosed hypersurface.
Keywords: Krein’s resolvent formula, boundary conditions, self-adjoint extensions.
PACS 02.30.Tb, 02.30.Jr