The Inverse Problem For The Symplectic Orthogonality Of Eigenfunction System Of Infinite Dimensional Hamiltonian Operator

In this thesis, based on an example of non-Hamiltonian operator with the normalized and symplectic orthogonal eigenfunction system, it is shown that whether an operator is a Hamiltonian operator is not equivalent to whether the operator's eigenfunction system is normalized and symplectic orthogonal, and a Hamiltonian operator is constructed whose eigenfunction system is not normalized and symplectic orthogonal, also necessary and sufficient conditions that a class of non-Hamiltonian operators' eigenfunction system is normalized and symplectic orthogonal is shown , in addition, an concrete example is given in order to show the validity of the results. Similarly, this paper discussed the problem whether self-adjoint operators have normalized and orthogonal eigenfunction system, and proved it is not equivalent that whether an operator is a symmetric operator and whether the operator's eigenfunction system is normalized and orthogonal, further, the necessary and sufficient conditions for a class of non-symmetric operators whose eigenfunction systems are normalized and orthogonal is obtained. This paper also investigated the algebra ascent problem of infinite dimensional Hamiltonian operators, and an example is cited to illustrate the effectiveness of the results.