Symmetry of the point spectrum of infinite dimensional hamiltonian operators and its applications

Abstract

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) ∪ σp1(−A*). Using the characteristic of the set σp1(−A*), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(−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.