Nagy-Foias type functional models of nondissipative operators in   parabolic domains
  A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space X is constructed. This model is analogous to the Nagy--Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-plane. The transformation map from X to the model space and the analogue of the characteristic function are given explicitly.   All usual consequences of the Nagy--Foias construction (the H-infty calculus, the commutant lifting, etc.) hold true in our context. 
Dmitry V. Yakubovich