Dynamics and Lax-Phillips scattering for generalized Lamb models

TL;DR

This paper analyzes the dynamics and scattering of a coupled oscillator-wave system, revealing Hamiltonian structure, spectral properties, and explicit scattering operators, including nonlinear cases, using Lax-Phillips theory.

Contribution

It introduces a novel framework for studying coupled oscillators and wave fields via selfadjoint extensions and Lax-Phillips scattering, including nonlinear dynamics.

Findings

Spectral theory of the coupled system is developed.

Explicit form of the scattering operator is derived.

Nonlinear oscillators are incorporated into the Hamiltonian framework.

Abstract

This paper treats the dynamics and scattering of a model of coupled oscillating systems, a finite dimensional one and a wave field on the half line. The coupling is realized producing the family of selfadjoint extensions of the suitably restricted self-adjoint operator describing the uncoupled dynamics. The spectral theory of the family is studied and the associated quadratic forms constructed. The dynamics turns out to be Hamiltonian and the Hamiltonian is described, including the case in which the finite dimensional systems comprises nonlinear oscillators; in this case the dynamics is shown to exist as well. 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 $p(\partial_x)$ explicitely related to the model parameters. In terms of such structure the Lax-Phillips scattering of the system is studied. In particular we determine the incoming and outgoing translation representations, the scattering operator, which turns out to be unitarily equivalent to the multiplication operator given by the rational function $-p(i\kappa)^*/p(i\kappa)$, and the Lax-Phillips semigroup, which describes the evolution of the states which are neither incoming in the past nor outgoing in the future.