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 {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.
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Pavlov, B. (2004). A Remark on Spectral Meaning of the Symmetric Functional Model. In: Janas, J., Kurasov, P., Naboko, S. (eds) Spectral Methods for Operators of Mathematical Physics. Operator Theory: Advances and Applications, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7947-7_11
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DOI: https://doi.org/10.1007/978-3-0348-7947-7_11
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