Non-self-adjoint singular matrix Sturm-Liouville operators with general boundary conditions

Abstract

In the Hilbert space L2 A(I;E) (I := [a;b); 1 < a < b +1; dimE = m <+1; A>0), the maximal dissipative singular matrix-valued Sturm-Liouville operators that the extensions of a minimal symmetric operator with maximal de ciency indices (2m;2m) (in limit-circle case at singular endpoint b) are studied. The maximal dissipative operators with general (for example coupled or separated) boundary conditions are investigated. A self-adjoint dilation is constructed for dissi pative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or in terms of the Weyl function of self-adjoint operator). Moreover a theorem on completeness of the system of eigen vectors and associated vectors (or root vectors) of the dissipative operators proved.