| The theory of differential operator is a branch of mathematics based on the theory and method of integrated differential equation,functional analysis and operator algebra so on which is a essential mathematical tool to handle math-ematical physics equations.The theory of operator spectrum originates from the solution of differential equations,integral equation and algebra equation or the mathematical description of quantum mechanics.With the development of space technology recently,differential equation plays an significant impact on space technology.In this paper,I mainly combine the knowledge of differ-ential equations,operator spectrum,determinants and analytic functions to study the eigenvalue problem of coupling boundary conditions of n-dimension ordinary differential equations.In this paper,I draw lessons from the method of Hamiltonian system under boundary condition to study the eigenvalue problem of coupling bound-ary conditions of n-dimension ordinary differential equations.According to the relation between the infinite product of the operator eigenvalues and the de-terminant of the matrix under boundary conditions,the Hill-type formula of n-dimension ordinary differential system under coupled boundary conditions is obtained.The main idea of this paper is to transform n-dimension ordinary-differential system into 2n-dimension Hamiltonian system by unitary transfor-mation,and then calculate the spectrum of defined operators.According to the analysis of the operator spectrum,the properties of the solution are obtained.According to the properties of the integral function and the determinant,so I could gain the Hill-type formula of the differential equation by using the Taylor expansion formula of the analytic function. |
