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

Abstract

In the Hilbert space L 2 A (I; E) (I := [a, b), −∞ < a < b ≤ +∞, dim E = m < +∞, A > 0), the maximal dissipative singular matrix-valued Sturm–Liouville operators that the extensions of a minimal symmetric operator with maximal deficiency 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 dissipative 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 eigenvectors and associated vectors (or root vectors) of the dissipative operators proved.