Stable Dissipative Linear Stationary Dynamical Scattering Systems

  • D. Z. Arov
  • J. Rovnyak
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)


In the theory of passive linear electrical networks, the Darlington method is well known as a realization of a finite ideal passive 1-port with losses via a finite ideal passive lossless 2-port closed by one resistance [9]. The reflection coefficient Θ of such a 1-port is an element of the scattering matrix \(\tilde \Theta\) of a corresponding loss-less 2-port; the lossless behavior is indicated in the property that \(\tilde \Theta\) has unitary values on the boundary of the physical domain (in the right or upper half-plane, or inside the unit disk). The consideration of scattering matrices allowed Belevich to generalize Darlington’s result on finite ideal n-ports with losses [16]. Darlington himself did not consider Θ and \(\tilde \Theta\) but other frequency characteristics: the impedance Zof 1-ports and the transmission matrix à of 2-ports (Z and à have simple representations by means of Θ and \(\tilde \Theta\)). In this way, the Darlington result was generalized to finite ideal n-ports with losses by V.P. Potapov [14] and his student E.Ya. Melamud [12].


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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • D. Z. Arov
    • 1
  • J. Rovnyak
    • 2
  1. 1.Department of Physics and Mathematics Division of Mathematical AnalysisSouth-Ukrainian Pedagogical UniversityOdessaUkraine
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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