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

1. 利用散射理论，讨论了具有1阶耗散微扰的一维Scheodinger方程和具有若干耗散项的波动方程。我们对扰动的假设是特殊的或人为的。然而，为了表征方程的产生子（耗散算子）的谱，需要这些假设。利用得到的生成子谱构造了Parseval公式，并对解的性质进行了分类。具体而言，实连续谱和非实连续谱的初始数据分别为演化的散射态和耗散态。因此，任何解都被表示为这些状态的线性组合。具有某些耗散和指数衰减解的波动方程的谱结构对于具有库仑型耗散项或初始耗散边界条件的波动方程，给出了指数衰减解或消失解的存在性。特别地，我们证明了与具有库仑型耗散项的波动方程相关的发生器谱由复下半平原组成。然而，我们没有描述谱与指数解或消失解之间的关系。有限区间上具有耗散边界条件的波动方程解的本征函数展开式，证明了任意解都可以用波动方程发生器的本征函数表示。证明是通过分离变量和通常的傅里叶级数来完成的。然而，解是在能量空间中表示的，而不是通常的la2 -空间。Lax-Phillips型的出、入子空间及波算子渐近完备性的证明利用Lax-Phillips（1967）的思想给出了出、入子空间的新定义。利用这些子空间并结合Perry（1980）的证明，我们证明了渐近完备性。

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.