Study for spectrum of dissipative operators and classification for the solutions of dissipative equations

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1. One dimensional Scheodinger and wave equations with dissipative perturbation of rank oneBy using scattering theory, we deal with Scheodinger equations with delta function as dissipative perturbation and wave equations with some dissipative term. Our assumptions of perturbation are special or artificial. However, to characterize the spectrum of generator (dissipative operator ) of equation these assumptions are need. Using the spectrum of generator obtained we construct Parseval formula and classify the behavior of the solutions. Concretely, initial data associated with real continuous spectrum and non-real spectrum are evolved scattering state and dissipative state, respectively. Therefore any solutions are represented as linear combination of these states2. Spectral structure of generator of wave equations with some dissipations and exponential decay solutionsFor wave equations with Coulomb type dissipative term or dissipative boundary condition at origin dissipations, we show existence of exponential decay or disappearing solutions. Especially we show that the spectrum of generator associated with wave equations with Coulomb type dissipative term consists with complex lower half-plain. However we do not characterize the relation between the spectrum and exponential or disappearing solution.3. Eigenfunction expansion of solutions for wave equations with dissipative boundary condition on finite intervalWe show that any solutions are represented by using eigenfunction for the generator of wave equations. The proof is done by using the separation of variable and usual Fourier series. However the solutions are represented in the energy space not usual LA 2-space4. Out-going and In-coming subspaces of Lax-Phillips type and the proof for asymptotic completeness of wave operatorsWe give new definition of Out-going and In-coming spaces by using the idea of Lax-Phillips(1967). Using these subspaces and combining the proof of Perry(1980) we show asymptotic completeness.