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Descriptions of spectra of infinite dimensional Hamiltonian operators and their applications
Author(s) -
Huang Junjie,
Wu Hongyou
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710076
Subject(s) - mathematics , hamiltonian (control theory) , continuous spectrum , spectrum (functional analysis) , essential spectrum , operator (biology) , pure mathematics , complex plane , residual , spectral line , mathematical analysis , mathematical physics , quantum mechanics , physics , algorithm , mathematical optimization , biochemistry , chemistry , repressor , transcription factor , gene
This paper deals with a class of infinite dimensional Hamiltonian operators. Explicit descriptions of their spectrum, point spectrum, residual spectrum and continuous spectrum are obtained, in terms of those of the compositions of two block operators in the Hamiltonian operators. Using these descriptions, it is shown that for two arbitrary closed sets on any given horizontal line and the imaginary axis, respectively, subject to a symmetry restriction, there is a Hamiltonian operator whose spectrum equals the union of these two sets. We also construct a family of explicit Hamiltonian operators whose residual spectrum consists of a single point, and this single point can be an arbitrary point in the complex plane other than the always impossible choices – the points on the imaginary axis (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)