Spectral Problems For Two Kinds Of Higher Order Regular Ordinary Differential Operators

Due to the fact that spectral problems of ordinary differential operators have been applied to many disciplines and many engineering and technical fields,an increasing num-ber of mathematicians are devoting themselves to study such problems.The dependence of eigenvalues on parameters and inverse spectral problems are two important subjects in spectral problems.They have practical applications in the electronics,the quantum me-chanics and other fields.They also play an important role in the numerical calculation of eigenvalues and solving the nonlinear evolution equations in mathematical physics.The main object of this thesis is to study a complex third order measure differ-ential equation and a non-selfadjoint Sturm-Liouville operator with discontinuity condi-tions inside a finite interval,including the dependence of solutions and eigenvalues of a complex third order measure differential equation with coupled boundary conditions on the measure coefficients with different topologies,the Ambarzumyan-type theorem for a complex third order measure differential equation with coupled boundary conditions,and the uniqueness theorem for a non-selfadjoint Sturm-Liouville operator with discontinuity conditions inside a finite interval and the algorithm for reconstructing the operator.This thesis is divided into six chapters.In the first chapter,the background,the research advances and the significance of the issues are introduced,and we also give the main results of this paper.In the second chapter,some basic definitions and relative properties of measures,Lebesgue-Stieltjes integral and weak~*topology are presented.We also introduce the defi-nitions and essential properties for solutions of a complex third order measure differential equation.In the third chapter,we systematically investigate the dependence of solutions of a complex third order measure differential equation on measure coefficients with differen-t topologies.We find that the solutions are continuous in measure coefficients with the weak~*topology,and continuously differentiable in measure coefficients under the norm topology of total variation.Besides,the estimates and analyticity with respect to pa-rameter?of basic solutions of a complex third order measure differential equation are discussed.In the forth chapter,we consider a complex third order measure differential equa-tion with coupled boundary conditions.We discuss the dependence of eigenvalues of this boundary value problem on measure coefficients with different topologies and the dependence of eigenvalues on boundary conditions.Based on the relations between the multiplicities of eigenvalues and the boundary conditions,the dependence of solutions on measure coefficients with different topologies and the implicit function theorem,we obtain that the eigenvalues are continuous in measure coefficients with the weak~*topol-ogy,and continuously differentiable in measure coefficients under the norm topology of total variation.The eigenvalues are also continuously differentiable in the boundary con-ditions.Assume that the parameter in the boundary condition is of special value,then by using the estimates of solutions and the counting lemma of eigenvalues,we prove that the n-th eigenvalue is continuous in measure coefficients when the norm topology of total variation and the weak~*topology are considered.Moreover,it is also investigated that the n-th eigenvalue is differentiable in measure coefficients with the norm topology of total variation.The fifth chapter is devoted to prove the Ambarzumyan-type theorem for a complex third order measure differential equation with coupled boundary conditions.By virtue of the estimates of solutions,the distribution of eigenvalues,and the distributions of zeros of the eigenfunctions,we prove that one spectrum can uniquely determine the measure coefficients and the boundary conditions.In the sixth chapter,we consider the inverse problem for a non-selfadjoint Sturm-Liouville operator with discontinuity conditions inside a finite interval.First,we give the definitions of the generalized spectral data for the operator which may have the multi-ple spectrum.Second,we derive that the generalized spectral data determines the Weyl function uniquely,and the Weyl function uniquely determines the potential,the boundary conditions and the discontinuity conditions.Finally,in view of the method of spectral mappings,we give an algorithm for reconstructing this operator.