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Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems of arbitrary order
Author(s) -
Muzzulini M.
Publication year - 2011
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq079
Subject(s) - mathematics , hamiltonian (control theory) , spectral theory , operator (biology) , mathematical physics , pure mathematics , order (exchange) , scalar (mathematics) , hamiltonian system , realization (probability) , algebra over a field , geometry , mathematical optimization , hilbert space , biochemistry , chemistry , statistics , repressor , transcription factor , gene , finance , economics
The main object of this paper is to extend the non‐self‐adjoint matrix‐valued Titchmarsh–Sims–Weyl theory, which was studied by Brown, Evans, and Plum in [‘Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems’, Proc. London Math. Soc. 87 (2003) 419–450] in the even‐order case, to the case of arbitrary order. In this way it also extends the self‐adjoint arbitrary‐order theory which was studied by Hinton and Schneider in [‘Titchmarsh–Weyl coefficients for odd‐order linear Hamiltonian systems’, J. Spectral Math. Appl. 1 (2005)]. The system case also includes the well‐known theory of higher‐order scalar problems. Corresponding Weyl‐type circles are constructed, then an analytic M function, and finally an operator realization of the given problem, the spectrum of which is carefully investigated. This paper contributes a more general application of Titchmarsh–Sims–Weyl theory to non‐self‐adjoint operator theory, which appears to be of increasing interest.