Symmetry Of The Spectrum Of Infinite Dimensional Hamiltonian Operator

The spectral analysis of infinite dimensional Hamiltonian operators is the theoretic foundation of the method of separation of variables based on Hamiltonian systems,and plays a important role in applied mechanics and related fields.The present thesis considers the symmetry of the spectrum of infinite dimensional Hamiltonian operators.We study the symmetry with respect to the imaginary axis and the symmetry with respect to the real axis, respectively.For the diagonal infinite dimensional Hamiltonian operators,the characterization of the symmetry of the point spectrum with respect to the imaginary axis is obtained by using the spectrum of its diagonal elements;and for general infinite dimensional Hamiltonian operators H,a necessary and sufficient condition is given that the spectrum,point spectrum, residual spectrum and continuous spectrum is symmetric with respect to the real axis, respectively.To illustrate the results of the thesis,several examples are presented.