Asymplotic Behavior Of A Second Order Differential Operator Of Discontinuities At Finite Points With Eigenparameter Dependent Boundary Conditions

This article is mainly focused on the asymptotic analysis of eigenvalueand self-adjointness of boundary conditions with eigenparameter andsecond-order differential operator with discontinuous points. Because of theirphysics-related application, the problems of discontinuousSturm-Liouville(S-L) and interior singular points of S-L have attracts moreand more researchers`attention. For example, the mass-heat conversion, thediffraction as well as the string pendulum caused by the mass of a point are allbelong to the Sturm-Liouville problems with transmission conditions.Meanwhile, the distribution of eigenvalue of discontinuous S-L withtransmission conditions and the completeness of eigenfunction have also beenpaid attention by the mathematician. On the basis of these researches, thispaper would carry out the studies of two boundary conditions witheigenparameter and S-L with finite discontinuous points.Firstly, the boundary conditions with eigenparameter andself-adjointness of second-order differential operators with manydiscontinuous points are given. Secondly, the features of eigenvalue havebeen discussed and relevant Wronski determinant has been formed inaccordance with the boundary conditions of fundamental solution which is setup for the problems. Thirdly, the eigenvalue problem has been changed intothe zero of integral function by getting the Wronski determinat. Finally, bycombining the features of eigenvalue and Rouché in complex function, theasymptotic formula of eigenvalue has been gotten.