Spectral Properties Of Two Classes Sturm-Liouville Operators With Spectral Parameters In Boundary Conditions

The spectral theory of linear operators not only has rich connotation in basic research,but also occupies an important position in application fields.Firstly,the eigenvalue problems of linear operators with spectral parameters in boundary conditions can be derived from the research of physics,engineering and other disciplines.Secondly,the spectral theory of linear operators is an important basic theory and tool to study the corresponding nonlinear problems.Therefore,the study of spectral properties of differential operators with spectral parameters in boundary conditions has important theoretical significance and application value.Based on this,this thesis will focus on two kinds of eigenvalue problems of Sturm-Liouville operators with spectral parameters in boundary conditions.The main contents include:In Chapter 1,the physical background of Sturm-Liouville operator and the research status of eigenvalue problems of Sturm-Liouville operator with spectral parameters in boundary conditions are summarized.In Chapter 2,we study the spectral properties of a class of discontinuous SturmLiouville operators.In particular,this operator not only contains linear spectral parameters in boundary conditions,but also the discontinuities depend on NevanlinnaHerglotz function.By equivalent alternation of Herglotz condition,we transform the considered problem into a self-adjoint operator eigenvalue problem in Hilbert space.Furthermore,the characteristic determinant and the Green function of the operator are obtained,and then the norm of its resolvent operator is estimated.In Chapter 3,we study the continuous dependence of eigenvalues of a class of fourth-order Sturm-Liouville differential operators.By establishing boundary condition spaces and constructing embedding mappings,we prove the continuous dependence of eigenvalue branch,and give the differential expression of the eigenvalue branch with respect to all parameters in the sense of Fr’echet derivative.