Asymptotic Completeness and S-Matrix for Singular Perturbations

Abstract

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 $\{u\in D(A_0)\cap D(A):A_{0}u=Au\}$ 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.

Final version, to appear in Journal de Mathématiques Pures et Appliquées

References in corpus (4)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Generalized interactions supported on hypersurfaces
• Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces
• Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

Cited by in corpus (8)

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• Optimization of the lowest eigenvalue of a soft quantum ring
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• Limiting absorption principle and scattering matrix for Dirac operators with $\delta$-shell interactions
• Scattering of particles bounded to an infinite planar curve
• On quasi-Herglotz functions in one variable
• Inverse wave scattering in the time domain for point scatterers
• Inverse wave scattering in the Laplace domain: a factorization method approach