Abstract
This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators. It is shown that the spectrum, the union of the point spectrum and residual spectrum, and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover, it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis; and a complete characterization of the residual spectrum in terms of the point spectrum is then given. As applications of these structure results, we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.
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