High frequency asymptotics of the filtered scattering amplitudes and the inverse scattering problem for dissipative hyperbolic systems. I. (English) Zbl 0581.35059

Georgiev, Vladimir S.

Math. Nachr. 117, 111-128 (1984).

Scattering by a compact obstacle K for dissipative hyperbolic systems is considered in the spirit of Lax and Phillips scattering theory. If the obstacle K has a smooth boundary in \({\mathbb{R}}^ n\), the kernel of the scattering operator is represented by a distribution \[ S(\sigma,\theta,\omega)\in {\mathcal D}' ({\mathbb{R}}\times S^{n-1}\times S^{n-1}) \] where \(S^{n-1}\) is the unit sphere

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in \({\mathbb{R}}^ n\). Multiplying by suitable \({\mathcal C}_ 0^{\infty}({\mathbb{R}})\) functions \(\rho(\sigma)\) and taking the Fourier transform with respect to \(\sigma\) yields the so called ”filtered scattering amplitude” \(a_{\rho}(\lambda,\theta,\omega)\). Its asymptotic behavior when \(\lambda\to \infty\) and \(\theta\) is close to \(\omega\) is studied under a strict convexity condition on K and suitable conditions on the dissipative hyperbolic pseudo-differential operator. This ”high frequency” asymptotics aims at solving some inverse scattering problems.

Reviewer: M.Combescure

MSC:

35P25	Scattering theory for PDEs
35B40	Asymptotic behavior of solutions to PDEs
35L40	First-order hyperbolic systems
47A40	Scattering theory of linear operators

Keywords:

high frequency asymptotics; Scattering; compact obstacle; dissipative hyperbolic systems; scattering operator; Fourier transform; filtered scattering amplitude; inverse scattering

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