Eigenvalue Problem Of A Class Of Infinite Dimensional Hamiltonian Operators

In this thesis, the eigenvalue problem of a class of infinite dimensional Hamiltonian operators is studied, including the geometric multiplicity, algebraic index and algebraic multiplicity of eigenvalues, and the completeness of eigen and root vector systems. Ac-cording to the relationship between the operators HA and A, the eigenvalues and the eigen and root vectors of HA are obtained, and then we prove by their properties that the multiplicity of eigenvalues is2or4. Based on the above analysis, the necessary and sufficient condition for the eigen or root vector systems of HA to be complete is given, which depends on the first components of the eigenvectors of A. The result provides many conveniences for us to determine the completeness of eigen and root vector systems of this kind of Hamiltonian operators. Finally, some examples are presented to illustrate the validity and efficiency of the results.