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Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces
Analysis of PDEs · Submitted 17 Feb 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary.
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We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schr\"odinger operators with singular $\delta$-type potential supported on the surface $\Gamma$ and of strength $\alpha\in L^{p}(\Gamma)$, $p>2$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. | result 0.70 | 0.95 | 0.95 | asserted 1.00 |
| s1 | This follows from a uniqueness result in inverse scattering for Schr\"odinger operators with singular $\delta$-type potential supported on the surface $\Gamma$ and of strength $\alpha\in L^{p}(\Gamma)$, $p>2$. | method 0.75 | 0.56 | 0.11 | asserted 1.00 |
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Original abstract
We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schr\"odinger operators with singular $\delta$-type potential supported on the surface $\Gamma$ and of strength $\alpha\in L^{p}(\Gamma)$, $p>2$.
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The Schrödinger approach allows the use of techniques from quantum mechanical stationary scattering theory, in particular, by extending some results provided in [ref] , we get a limiting absorption principle (LAP for short in the following) for our class of Schrödinger operators; as a consequence, the scattering amplitude is derived and used to define the acoustic far-field patterns.
Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces · Introduction · 0.83
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