Geodesic
Laplacians with singular perturbations supported on hypersurfaces
Nanosistemy: fizika, himiâ, matematika, Tome 7 (2016) no. 2, pp. 315-323

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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 $\mathbb{R}^n$, 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. 
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     author = {A. Mantile and A. Posilicano},
     title = {Laplacians with singular perturbations supported on hypersurfaces},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
     pages = {315--323},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/NANO_2016_7_2_a3/}
}
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A. Mantile; A. Posilicano. Laplacians with singular perturbations supported on hypersurfaces. Nanosistemy: fizika, himiâ, matematika, Tome 7 (2016) no. 2, pp. 315-323. http://geodesic.mathdoc.fr/item/NANO_2016_7_2_a3/