Eigenvalue Problem Of Infinite Dimensional Hamiltonian Operators

Combining the elasticity with infinite dimensional Hamiltonian operators, aca-demician Zhong extended the classical method of separation of variables-the method of separation of variables based on Hamiltonian systems, and a new systematic method-ology for theory of elasticity is further established. The mathematical foundation of this method is the eigenvalue problem of infinite dimensional Hamiltonian operators, and the completeness of the eigenvector system plays the most important role. This pa-per is devoted to the eigenvalue problem of infinite dimensional Hamiltonian operators, including the symmetry of eigenvalues (point spectrum), the geometric multiplicity, al-gebraic index and algebraic multiplicity of eigenvalues, the symplectic orthogonality of eigen and root vectors, and the completeness of eigen and root vector systems.As is well known, the union of the point spectrum and residual spectrum of infinite-dimensional Hamiltonian operators is symmetric with respect to the imaginary axis, but the symmetry of the point spectrum is unknown. In view of the symplectic or-thogonality of eigen and root vectors, the symmetry of the point spectrum should be investigated before studying the completeness of eigen and root vector systems. To this end, the symmetry of the point spectrum of upper triangular infinite dimensional Hamiltonian operators is investigated, and we obtain some necessary and sufficient conditions on the symmetry of the point spectrum with respect to the imaginary and real axis, respectively. Moreover, using the spectral structure of infinite dimensional Hamiltonian operators, the description of the residual spectrum of upper triangular infinite dimensional Hamiltonian operators is given.Up to now, the discussions about the completeness of the eigenvector system of infinite dimensional Hamiltonian operators are restricted to the cases with real eigenval-ues and pure imaginary eigenvalues. Also, the completeness of the root vector system of the operators is not mentioned in mathematics, and there is no any material on the geometric multiplicity and algebraic multiplicity of the eigenvalues of infinite dimen-sional Hamiltonian operators. On the other hand, the completeness in Cauchy principal value of a vector system was introduced in investigating the completeness of the eigen-vector system of infinite dimensional Hamiltonian operators with real eigenvalues and pure imaginary eigenvalues. This concept fails to work for infinite dimensional Hamil-tonian operators with general eigenvalues (not necessarily real eigenvalues and pure imaginary eigenvalues, or others). So, we propose the notion of the completeness in the sense of generalized Cauchy principal value of a vector system, which lays a foun-dation for further research on the completeness of the eigen and root vector system of infinite dimensional Hamiltonian operators.In practical applications, a problem can be equivalently written as various Hamil-tonian forms, and so we obtain various infinite dimensional Hamiltonian operators. Among them, upper triangular infinite dimensional Hamiltonian systems have certain advantages in calculations. For example, the eigen equation of the corresponding upper triangular infinite dimensional Hamiltonian operators is not coupled, then we can con-veniently calculate the eigenvalues, the eigen and root vectors. To this end, we study the completeness of eigen and root vector systems of upper triangular infinite dimen-sional Hamiltonian operators. We discuss from three ideas:for 2×2 upper triangular infinite dimensional Hamiltonian operators and forth-order upper triangular infinite dimensional Hamiltonian operators arising from mechanics, the geometric multiplicity, algebraic index and algebraic multiplicity of eigenvalues, and the completeness in the sense of Cauchy principal value of the eigen and root vector systems are investigated; we propose a new method to solve upper triangular matrix differential systems arising from mechanics-the double eigenfunction expansion method, which need study the completeness of eigenvector systems of off-diagonal operator matrices, so some sufficient and necessary conditions are further given.Moreover, we consider the eigenvalue problem of four block infinite dimensional Hamiltonian operators with at least one of the diagonal elements being invertible and the diagonal elements being constant. The results about the geometric multiplicity, algebraic index and algebraic multiplicity of eigenvalues, and the completeness of the eigen and root vector systems are obtained, and in some degree extend and enrich the present conclusions.As applications, the plane elasticity problem, free vibration problem of rectangular thin plates, bending problem of rectangular plates and Stokes flow are presented to illustrate these results.