Abstract
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)
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