Canonical Forms For Third Order Self-adjoint Boundary Conditions And Dependence Of Eigenvalue On The Problem

In this paper,we study the canonical forms for third order self-adjoint boundary conditions(including regular and singular case)and the dependence of the eigenvalues for third order differential operators with real coupled boundary conditions.Firstly,by classifying the self-adjoint boundary conditions,we obtain all canonical forms for third order regular differential operators.Unlike the even order case,the strictly separated self-adjoint boundary conditions cannot be realized in the third order case.For coupled and mixed cases,there are some different types for the canonical forms:2 for coupled and 4 for mixed self-adjoint boundary conditions.Next,by using the deficit index theory,we generalize the conclusions of the regular case to the singular case.When the deficit index is equal,there are seven different canonical forms for those self-adjoint boundary conditions.When the deficit index is not equal,there are four different canonical forms.Finally,we study the dependence of the eigenvalues of the third order differential operators with real coupled self-adjoint boundary conditions on the problem and obtain the differential expressions for a given parameter.