Dilations, models and spectral problems of non-self-adjoint Sturm-Liouville operators

Bilender P. Allahverdiev;


In this study, we investigate the maximal dissipative singular Sturm-Liouville operators acting in the Hilbert space L_{r}²(a,b) (-∞≤a<b≤∞), that the extensions of a minimal symmetric operator with defect index (2,2) (in limit-circle case at singular end points a and b). We examine two classes of dissipative operators with separated boundary conditions and we establish, for each case, a self-adjoint dilation of the dissipative operator as well as its incoming and outgoing spectral representations, which enables us to define the scattering matrix of the dilation. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl function of a self-adjoint operator. We present several theorems on completeness of the system of root functions of the dissipative operators and verify them.

Vol. 22 (2021), No. 1, pp. 17-32

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