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

Miskolc Mathematical Notes

article

Bilender P. Allahverdiev¹ 

[1] Suleyman Demirel University, Department of Mathematics

关键词: symmetric Sturm-Liouville operator;    maximal dissipative operator;    self-adjoint dilation;    scattering matrix;    functional model;    characteristic function;    completeness of the root functions;   

DOI  :  10.18514/MMN.2021.2007

学科分类：数学（综合）

来源: Miskolci Egyetem

【 摘 要 】

In this study, we investigate the maximal dissipative singular Sturm-Liouville operators acting in the Hilbert space L 2 r (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.

【 授权许可】

CC BY