# A Remark on Spectral Meaning of the Symmetric Functional Model

## Abstract

The imaginary part of a dissipative operator *L* is weak if it is pre-sented by a positive operator *T* such that the square *T* ^{2} of it is a product of an operator with a finite trace and an operator from Macaev class. For a dissipative operator with a weak imaginary part the families of incoming and outgoing scattered waves form a non-orthogonal and often even over-complete system {Ψ_{in}, Ψ_{out}}
of eigenfunctions of the corresponding self-adjoint dilation *L*. The rescription of *L* in the spectral representation associated with {Ψ_{in}, Ψ_{out}}
gives the Symmetric Functional Model of *L*, and the characteristic function *S* of *L* coincides with the transmission coefficient of the outgoing waves. A general construction based on the self-adjoint delation and an example of the Lax-Phillips Semigroup for the 1-D wave equation on the infinite string with a bounded non-negative potential supported by semi-axis are considered.

## Keywords

Symmetric functional model## Preview

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