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Spectral Analysis of Singular Matrix-Valued Sturm–Liouville Operators

  • Bilender P. AllahverdievEmail author
Article
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Abstract

In the Hilbert space \(L_{W}^{2}([a,b);E)\) (\(-\infty<a<b\le +\infty ,\) \(\dim E=N<+\infty ,\) \(W>0\)) a space of boundary values of the symmetric singular matrix-valued Sturm–Liouville operator with maximal deficiency indices (2N, 2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called ‘dissipative at a’ and ‘self-adjoint at b’ are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm–Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.

Keywords

Singular matrix-valued Sturm–Liouville equation symmetric operator space of boundary values self-adjoint and nonself-adjoint extensions self-adjoint dilation scattering matrix functional model characteristic function completeness of the system of root vectors 

Mathematics Subject Classification

Primary 34B40 34B24 34L10 47A20 Secondary 47A40 47A45 47E05 47B25 47B44 

Notes

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Authors and Affiliations

  1. 1.Department of MathematicsSuleyman Demirel UniversityIspartaTurkey

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