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. Keywords: non-self-adjoint operator, infinite dimensional Hamiltonian operator, structure of spectrum MSC(2000): 47A10, 47B99

Huang Jun-Jie(黄俊杰) a) , Alatancang(阿拉坦仓) a) , and Wang Hua(王 华) a)b) †

The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion. Keywords: infinite dimensional Hamiltonian operator k-compact operator, eigenvalue, eigenfunction system, Cauchy principal value, completeness MSC(2000): 47A75

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp(A) ∪ σ 1 p (−A * ). Using the characteristic of the set σ 1 p (−A * ), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σ 1 p (−A * ) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.

The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.