Spectral analysis of the direct sum Hamiltonian operators

Abstract

In this paper we investigate the deciency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foias characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces.

Keywords: Hamiltonian system, dissipative operator, characteristic function, scattering matrix, completeness theorem