On some properties of abstract wave equation. (English) Zbl 0940.47007
The author considers the abstract wave equation \(u_{tt}=-Lu\), where \(L\) is such a self-adjoint operator on a Hilbert space that \(L\) is an extension of the square \(B^2\) of a simple maximal symmetric operator \(B\), and \((Lu,u)=\|B^*u\|^2\) for any \(u\in D(L)\).
Under these assumptions an analogue of the Lax-Phillips scattering scheme is developed. In particular, the structure of the incoming and outgoing subspaces is described. The translation representation of the corresponding group of operators is obtained.
Under these assumptions an analogue of the Lax-Phillips scattering scheme is developed. In particular, the structure of the incoming and outgoing subspaces is described. The translation representation of the corresponding group of operators is obtained.
Reviewer: A.N.Kochubei (Kyïv)
MSC:
47A40 | Scattering theory of linear operators |
47B25 | Linear symmetric and selfadjoint operators (unbounded) |
35L05 | Wave equation |