Scattering theory with both regular and singular perturbations

  • Andrea Mantile

    Université de Reims Champagne-Ardenne, Reims Cedex 2, France
  • Andrea Posilicano

    Università dell'Insubria, Como, Italy
Scattering theory with both regular and singular perturbations cover
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We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple , where both and are self-adjoint operator and formally corresponds to adding to two terms, one regular and the other singular. In particular, our abstract results apply to the couple , where is the free self-adjoint Laplacian in and is a self-adjoint operator in a class of Laplacians with both a regular perturbation, given by a short-range potential, and a singular one describing boundary conditions (like Dirichlet, Neumann and semi-transparent and ones) at the boundary of an open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for and a Kreĭn-like formula for the resolvent difference which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato–Rellich potential suffices) and the singular perturbations.

Cite this article

Andrea Mantile, Andrea Posilicano, Scattering theory with both regular and singular perturbations. J. Spectr. Theory 13 (2023), no. 3, pp. 1057–1109

DOI 10.4171/JST/467