Inverse Problem Of Non-Selfadjoint Discontinuous Sturm-Liouville Operators With Boundary Conditions Dependent On Spectral Parameter

The research of ordinary differential operator theory can be traced back to the Sturm-Liouville problem of solid heat conduction in the middle of the 19th century.Its application has been widely involved in many fields,such as mathematical physics,geophysics.It forms an important theoretical research branch.In 1929,Russian astrophysicist V.A.Ambartsum-ian first proposed the Sturm-Liouville inverse spectrum problem.Generally speaking,the solution of the classical Sturm-Liouville problem and its quasi derivative are required to be absolutely continuous in the defined interval,but some practical problems in the application fields do not meet this requirement,such as heat conduction equation,diffraction of light,mass transfer problem,etc.We call this kind of problem"internal discontinuity problem".This means that the solution of the equation and its quasi derivative may be discontinuous at points within the interval defined by some differential operators.In order to deal with the discontinuity of the problem,the usual method is adding discontinuous conditions on the discontinuous points to describe the relationship of solution at such points between adjacent intervals.In recent years,with many problems arising in the application field,researchers and scholars pay more attention to the boundary value problems of differential equations with internal discontinuities and boundary conditions dependent on spectral parameters.In this paper,we consider the following non-selfadjoint boundary value problem for the equation:Ly:=-y"(x)+q(x)y(x)=?y(x),x ?[0,?],where ? is a spectrum parameter,q(x)? L2[0,?]is a complex-valued function.In 2019,Shi Guoliang et al.[8]extended the Sturm-Liouville inverse problem with non-selfadjoint to the case with discontinuous condition.On the basis of reference[8],we naturally consider whether the initial conditions of Sturm-Liouville problem can be extended to boundary conditions with spectral parameter.We obtain the asymptotic estimates,generalized gauge constants and generalized spec-tral data of the two fundamental solutions.Furthermore,the definition and expression of Weyl function are given,and the uniqueness theorem of inverse problem is proved by the relation between Weyl function and spectral data.Finally,the reconstruction algorithm of potential function,boundary condition coefficient and transfer condition coefficient is given.This thesis is divided into four chapters according to contents:In chapter 1,we introduce the background of considered problems and the main results of this paper.In chapter 2,the necessary preliminaries and lemmas are given in this paper,and the definitions of generalized normal constant and generalized spectral data are given.In chapter 3,the expression of Weyl function and its corresponding uniqueness theorems are proved;then,the inverse problem are considered,and the reconstruction algorithms of the potential function,boundary condition parameters and discontinuous condition coefficients are given.In chapter 4,this chapter summarizes and looks forward to this article.