Study On Eigenvalue Of A Class Of Thirdorder Differential Operators With Spectral Parameters In Boundary Conditions

Differential operators are a kind of basic unbounded linear operators,which have extensive and direct applications,for example,in the fields of natural sciences such as mechanics,physics,mathematics,engineering and meteorology,many problems can be summarized as the solution of differential operators.The spectral problem and the inverse spectral problems are two important basic topics in the study of differential operators.The spectral problem refers to the spectral analysis of operators,which is including solving the eigenvalues,eigenfunctions and any function expansion,according to the series or integral of the characteristic function,etc.So as to propose appropriate solutions to related problems.The study of the inverse spectral problem describes and constructs the original system and infers a property of the system itself by analyzing the measured or given spectral data.This paper mainly studies the spectral of several types of third-order differential operators with different boundary conditions and transmission conditions.The denseness and self-adjoint of the operator are respectively proved and the dependence of the eigenvalues on the coefficients of the equation and the parameters of the boundary conditions is obtained.The specific research contents are as follows:Firstly,the research background and the main contents of this paper are introduced.Secondly,the eigenvalue problem of a third-order differential operator with a spectral parameter at one end of a class of boundary condition is studied.Through the analysis method,the denseness and self-adjoint of the operator are obtained,and the continuous dependence of the eigenvalue on some parameters is further obtained.Thirdly,the eigenvalue problem of a class of third-order differential operators with transmission conditions and a boundary condition containing spectral parameters is studied.Through the pure analysis method,the denseness and self-adjoint of the operators are obtained,and the properties of the eigenvalues are further obtained.At last,the eigenvalue problem of a class of third-order differential operators with transmission conditions and two boundary conditions containing spectral parameters is studied.Through analysis,the denseness and self-adjoint of the operators are obtained,and the properties of the eigenvalues are further obtained.