Asymptotic completeness and S-matrix for singular perturbations

Journal of Mathematical Physics

et al. 2020

Self Cite

Section: The Scattering Matrixcontrasting

confidence: 40%

“…(iv) The claim is a consequence of the limiting absorption principle for G z . It follows from the estimate (3.16) in [30] and (2.6).…”

Section: Auxiliary Maps and Estimatesmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

Limiting absorption principle and scattering matrix for Dirac operators with δ-shell interactions

Behrndt

Holzmann

Journal of Mathematical Physics

et al. 2020

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“…Ω in/ex be the self-adjoint operators in L 2 (Ω in/ex ) corresponding to the Laplace operator with Dirichlet boundary conditions. One has ∆ D [15,Section 5.2]).…”

Section: The Dirichlel Laplacian Let ∆ Dmentioning

confidence: 99%

Inverse wave scattering in the Laplace domain: A factorization method approach

Posilicano

2020

Proc. Amer. Math. Soc.

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Let ∆ Λ ≤ λ Λ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle Ω. Let u Λ f and u 0 f denote the solutions of the wave equations corresponding to ∆ Λ and to the free Laplacian ∆ respectively, with a source term f concentrated at time t = 0 (a pulse). We show that for any fixed λ > λ Λ ≥ 0 and any fixed B ⊂⊂ R n \Ω, the obstacle Ω can be reconstructed by the dataA similar result holds in the case of screens reconstruction, when the boundary conditions are assigned only on a part of the boundary. Our method exploits the factorized form of the resolvent difference (−∆ Λ + λ) −1 − (−∆ + λ) −1 .

“…Here Λ : z → Λ z is an operator-valued map which univocally defines ∆ Λ and fixes the boundary conditions realized by the corresponding operator (see Sections 4.1 and 5.1 below for various explicit examples). Our representation formula gives S Λ λ = 1 − 2πiL λ Λ + λ L * λ , where Λ + λ is the limit of Λ λ+iǫ as ǫ ↓ 0 (which, under suitable hypotheses, exists in operator norm through a Limiting Absorption Principle, see [22]), and L λ is defined in term of the trace (either Dirichlet or Neumann or both) at the boundary Γ of the free waves with wavenumber |λ| 1/2 . Introducing the Far-Field operator F Λ λ := 1 2πi (1 − S Λ λ ) (see [18, relation (1.31)]), one gets F Λ λ = L λ Λ + λ L * λ ; such a factorized form suggests to study the inverse scattering problem (concerning the reconstruction of the shape of Ω by the knowledge of the scattering data at a fixed frequency) by means of Kirsch's Factorization Method (see [18] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Inverse scattering for the Laplace operator with boundary conditions on Lipschitz surfaces

Posilicano

2019

Inverse Problems

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We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of selfadjoint operators ( ∆, ∆), where ∆ is the free Laplacian in L 2 (R 3 ) and ∆ is one of its singular perturbations, i.e., such that the set {u ∈ H 2 (R 3 ) ∩ dom( ∆) : ∆u = ∆u} is dense. Typically ∆ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at Γ = ∂Ω, where Ω ⊂ R 3 is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on Σ ⊂ Γ, a relatively open subset with a Lipschitz boundary. We show that either the obstacle Ω or the screen Σ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.