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Asymptotic completeness and S-matrix for singular perturbations. (English. French summary) Zbl 07107299

Summary: We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple \((A_0,A)\), where \(A_0\) and \(A\) are semi-bounded self-adjoint operators in \(L^2(M,\mathcal{B},m)\) such that the set \(\{u\in\operatorname{dom}(A_0)\cap\operatorname{dom}(A):A_0u=Au\}\) is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which \(A_0\) corresponds to the free Laplacian in \(L^2(\mathbb{R}^n)\) and \(A\) describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

MSC:

47A40 Scattering theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
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