SPECTRAL PROBLEMS OF NONSELFADJOINT 1D SINGULAR HAMILTONIAN SYSTEMS

Abstract

In this paper, the maximal dissipativeone dimensional singular Hamiltonian operators (in limit-circle case atsingular end point $b$) are considered in the Hilbert space $\mathcal{L}_{W}^{2}(\left[ a,b\right) ;\mathbb{C}^{2}) (-\infty\lt a\lt b\leq \infty ).$ The maximal dissipative operators with general boundary conditions are investigated. A selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations are constructed. These representations allows us to determine the scattering matrix of the dilation. Further a functional model of the dissipative operator is constructed and its characteristic function in terms of the scattering matrix of dilation is considered. Finally, the theorem on completeness of the system of root vectors of the dissipative operators is proved.