Scattering theory with both regular and singular perturbations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914938994294784 |
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| author | Mantile, Andrea Posilicano, Andrea |
| author_facet | Mantile, Andrea Posilicano, Andrea |
| contents | We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(Δ_B,Δ)$, where $Δ$ is the free self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $Δ_B$ 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 $δ$ and $δ'$ ones) at the boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_B$ and a Krein-like formula for the resolvent difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices) and the singular perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_03106 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Scattering theory with both regular and singular perturbations Mantile, Andrea Posilicano, Andrea Mathematical Physics Analysis of PDEs We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(Δ_B,Δ)$, where $Δ$ is the free self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $Δ_B$ 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 $δ$ and $δ'$ ones) at the boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_B$ and a Krein-like formula for the resolvent difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices) and the singular perturbations. |
| title | Scattering theory with both regular and singular perturbations |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2208.03106 |