The Study Of Forth-order Differential Operator With Finite Transmission Conditions

As the study second-order differential operator with discontinuity in theinterior point and eigenparameter-dependent boundary conditions, in the actualphysical problems, in this field many workers find that part of the researchproblems can be transformed for even high-order differential operators,especially the forth-order differential operator. So, the study of the spectrum offorth-order differential operator has a certain practical value, its researchmethods and theoretical results provide a train of thought for even high-orderdifferential operators. In this paper, we investigate the spectrum of forth-orderdifferential operator, it include two parts. In the first part, we investigate thespectrum of forth-order differential operator L which we give generaleigenparameter-dependent boundary conditions and finite transmission condi-tions. The study contains the necessary and sufficient condition of theeigenvalues, the completeness of eigenfunctions and Green’s function. Themain research methods: we define a linear operator T in a suitable Hilbertspace H to make the eigenvalues of the operator T and L same. By thedefinition of unbounded linear operator self-adjointness, we prove the linearoperator T is self-adjoint. We obtain the identifying function of eigenvaluescombining with boundary conditions and transmission conditions, whichprovides the concrete theory basis for numerical of eigenvalues. By the methodof functional analysis, we know that the eigenvalues of self-adjoint operatorT are bounded below and has only point-spectrum. In combination of thespectral theorem for compact operator, we prove the eigenfunctions of T arecompleted on H, which augmented by eigenfunction of the operator L. TheGreen’s function of self-adjoint operatorT is given. In the second part, wediscuss the spectrum problem distribution of second-order and eigenvaluesproblem of forth-order differential operator L with finite transmission conditions and indefinite weight function. Putting this problem into Kreinspace K, we define a new linear operator T in Krein space K to make theeigenvalues of the operator T and L same. We obtain the identifying functionof eigenvalues as the basic of self-adjointness operator L. Furthermore, theeigenvalues distribution of the operator T are discussed. Finally, we obtainedthe identifying function of eigenvalues and proved it has not cluster point oflimited value.