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Extensions, dilations, and spectral problems of singular Hamiltonian systems
Author(s) -
Allahverdiev Bilender P.
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4703
Subject(s) - dissipative operator , dissipative system , mathematics , hilbert space , dilation (metric space) , self adjoint operator , operator (biology) , spectral theory , mathematical analysis , pure mathematics , hamiltonian (control theory) , multiplication operator , hamiltonian system , boundary value problem , quantum mechanics , physics , combinatorics , mathematical optimization , biochemistry , chemistry , repressor , transcription factor , gene
In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a ( b ) and limit‐circle case at b ( a )) acting in the Hilbert spaceL Ω 2 ( ( a , b ) ; C 2 ) . In terms of boundary conditions at a and b , all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a ” and “dissipative at b .” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.