广义Lamb模型的动力学和Lax-Phillips散射
Dynamics and Lax-Phillips scattering for generalized Lamb models
摘要：
本文讨论了耦合振动系统模型的动力学和散射，一个有限维的模型和半直线上的波场。耦合实现生产家庭的自伴扩展的适当限制的自伴算子描述的非耦合动力学。研究了该族的谱理论，并构造了相应的二次型。的动力学原来是哈密顿和描述的哈密顿，包括有限维系统包括非线性振子的情况下，在这种情况下，动态是存在的。在线性的情况下，该系统是等价的，在一个稠密的子空间，半线上的波动方程与高阶边界条件，描述了一个微分多项式$p(\partial_x)$明确相关的模型参数。在此基础上研究了系统的Lax-Phillips散射。特别是，我们确定传入和传出的平移表示，散射算子，这原来是酉等价的乘法运算符给出的有理函数$-p(iκ)^*/p(iκ)$，和Lax-Phillips半群，它描述的状态，既不是传入的过去，也不是传出的未来的演变。
英文摘要：
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 $-p(iκ)^*/p(iκ)$, and the Lax-Phillips semigroup, which describes the evolution of the states which are neither incoming in the past nor outgoing in the future.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
