Chapter 6.1.4 - Scattering and Semigroup Theory

Abstract

This chapter provides a description of the state of art of the interface of the spectral theory of semigroups in terms of the functional model and scattering theory. Using the language of the spectral theory of analytic functions, the basic ideas of the Lax–Phillips approach to scattering theory and connections of it to the functional model of non-self-adjoint operators are presented. The essential features of the behavior of a linear dynamical system are encoded inspectral characteristics of the generator of the corresponding evolution group or semigroup. The spectral properties of the operator of multiplication by the independent variable x in the Hilbert space of all square-integrable complex functions L2(R, μ) with a finite positive measure μ are directly defined by the structure of the measure—the corresponding spectral function. This multiplication operator displays a sort of universality—each self-adjoint operator with simple spectrum is unitary equivalent to some operator of this type with properly chosen functional parameter μ.