A Remark on Spectral Meaning of the Symmetric Functional Model

  • Boris Pavlov
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 154)


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 {Ψin, Ψout} of eigenfunctions of the corresponding self-adjoint dilation L. The rescription of L in the spectral representation associated with {Ψin, Ψout} 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.


Symmetric functional model 


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© Springer Basel AG 2004

Authors and Affiliations

  • Boris Pavlov
    • 1
  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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