×

Dynamics and Lax–Phillips scattering for generalized Lamb models. (English) Zbl 1109.81081

The authors treat the dynamics and scattering of a model of coupled oscillating systems, a finite dimensional one and a wave field on the half line. The authors define rigorously the coupled system through self-adjoint extensions of suitable restrictions of the uncoupled operators. They study the spectral theory of the family, and construct the associated quadratic forms. The dynamics turns out to be Hamiltonian and the Hamiltonian is described, including the case in which the finite-dimensional systems comprise nonlinear oscillators; in this case, the dynamics is shown to exist as well. The authors provide classes of examples like the Lamb model or the Pauli-Fierz model and show how they fit into the general set-up. In the linear case, the system is equivalent, on a dense subspace, to a wave equation on the half line with higher order boundary conditions, described by a differential polynomial \(p(\partial _x)\) explicitly related to the model parameters. In terms of such structure, the Lax–Phillips scattering of the system is studied. In particular, the authors determine the scattering operator, which turns out to be unitarily equivalent to the multiplication operator given by the rational function \(-p(\mathrm i\kappa )*/p(\mathrm i\kappa)\), the incoming and outgoing translation representations and the Lax–Phillips semigroup, which describes the evolution of the states which are neither incoming in the past nor outgoing in the future.
Reviewer: Olaf Post (Berlin)

MSC:

81U05 \(2\)-body potential quantum scattering theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P25 Scattering theory for PDEs
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
PDFBibTeX XMLCite
Full Text: DOI arXiv