Singularities of the scattering kernel related to trapping rays. (English) Zbl 1197.35183

Petkov, Vesselin; Stoyanov, Luchezar

Bove, Antonio (ed.) et al., Advances in phase space analysis of partial differential equations. In Honor of Ferruccio Colombini’s 60th birthday. Selected papers based on the workshop, Siena, Italy, October 2007. Boston, MA: Birkhäuser (ISBN 978-0-8176-4860-2/hbk; 978-0-8176-4861-9/ebook). Progress in Nonlinear Differential Equations and Their Applications 78, 235-251 (2009).

Summary: An obstacle \(K\subset \mathbb R^n\), \(n\geq 3\), \(n\) odd, is called trapping if there exists at least one generalized bicharacteristic \(\gamma(t)\) of the wave equation staying in a neighborhood of \(K\) for all \(t\geq 0\). We examine the singularities of the scattering kernel \(s(t,\theta,\omega)\) defined as the Fourier transform of tlie scattering amplitude \(a(\lambda,\theta,\omega)\) related to the Dirichlet problem for the wave equation in \(\Omega=\mathbb R^n\setminus K\). We prove that if \(K\) is trapping and \(\gamma(t)\) is nondegenerate, then there exist reflecting \((\omega_m,\theta_m)\)-rays \(\delta_m\), \(m\in\mathbb N\), with sojourn times \(T_m\to +\infty\) as \(m\to\infty\), so that \(-T)_m\in \text{sing\, supp}\,s(t,\theta_m,\omega_m)\), \(\forall m\in\mathbb N\). We apply this property to study the behavior of the scattering amplitude in \(\mathbb C\).
For the entire collection see [Zbl 1187.35004].

MSC:

35P25	Scattering theory for PDEs
47A40	Scattering theory of linear operators
35L05	Wave equation

Keywords:

scattering amplitude; reflecting rays; trapping trajectories; sojourn time