SPECTRAL PROBLEMS OF NONSELFADJOINT 1D SINGULAR HAMILTONIAN SYSTEMS

In this paper, the maximal dissipative one dimensional singular Hamiltonian operators (in limit-circle case at singular end pointb) are considered in the Hilbert space ℒ W 2 ( [ a ,   b ) ; ℂ 2 ) ( − ∞ < a < b ≤ ∞ ) . The maximal dissipative operators with general boundary conditions are inves...

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Bibliographic Details
Published inTaiwanese journal of mathematics Vol. 17; no. 5; pp. 1487 - 1502
Main Author Allahverdiev, Bilender P.
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.10.2013
Subjects
Boundary conditions
Eigenfunctions
Hilbert spaces
Mathematical vectors
S matrix theory
Spectroscopy
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Summary:In this paper, the maximal dissipative one dimensional singular Hamiltonian operators (in limit-circle case at singular end pointb) are considered in the Hilbert space ℒ W 2 ( [ a ,   b ) ; ℂ 2 ) ( − ∞ < a < b ≤ ∞ ) . The maximal dissipative operators with general boundary conditions are investigated. A selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations are constructed. These representations allows us to determine the scattering matrix of the dilation. Further a functional model of the dissipative operator is constructed and its characteristic function in terms of the scattering matrix of dilation is considered. Finally, the theorem on completeness of the system of root vectors of the dissipative operators is proved. 2010Mathematics Subject Classification: 47A20, 47A40, 47A75, 47B44, 34L10, 34L40, 34B40, 34L25, 47A45. Key words and phrases: 1D singular Hamiltonian system, Maximal dissipative operator, Selfadjoint dilation, Scattering matrix, Functional model, Characteristic function, Completeness of the system of root vectors.
ISSN:1027-5487
2224-6851
DOI:10.11650/tjm.17.2013.2734