Completeness Of The Eigenvector And Root Vector System Of A Class Of Eighth-order Hamiltonian Operator And Its Application

The traditional separate variable method is an e?ective way for solving problems of mathematical physics. However, when the problem is non-self-adjoint operator, the method doesn’t work. In the early 1990 s, the professor Zhong Wanxie introduced the infinite dimensional Hamiltonian operator into elasticity, and created the separation variable method based on Hamiltonian systems. In a certain extent, it break out the limit of self-adjoint operators for the traditional separation variable method. Since then, this method has been successfully applied to the viscoelasticity, fluid mechanics, functionally graded materials, piezoelectric,and fracture problems, and shows its vitality. But, the theoretical basis of this method is the problem of completeness of eigenvector and root vector system of Hamiltonian operators.This paper studies a class of eighth-order Hamiltonian operator. First, we get the expressions of eigenvalues, eigenvectors and root vectors of Hamiltonian operator under certain conditions. Among them, the non-zero eigenvalues have the third-order root vectors.Then. we obtain symplectic orthogonality of eigenvectors and root vectors. Based on this,we discuss the necessary and su?cient conditions of the completeness of eigenvector and root vector systems, and the symplectic eigenvector expansion theorem is obtained for the Hamiltonian operator. It enriches the results of completeness of the Hamiltonian operator in a certain extent. Finally, we applied the results to the plane elasticity problem of quasicrystals with point group 10 mm, which illustrates the correctness and validity of the results.