Research On Eigenvalues Of Some Ordinary Differential Operators

In this paper, we study an important problem in the field of differential operators: spectral analysis. First we analyze a class of right-definite Sturm-Liouville operators with general separated boundary conditions. With the different method, we obtain the different asymptotic expansion. Using a modification of Prüfer's substitution and the Fréchet derivative technology, we give two more sophisticated analyses for the eigenvalues than the conclusion in book [1], which revealed clearly the explicit effects of the equation coefficient q (x) and the boundary condition. When the weight function satisfies w≠1, we firstly use the Green-Liouville transformation to work out the corresponding solution of Cauchy question, and then utilize the corresponding method to decide their eigenvalues and eigenfunctions.In the second part of this paper, we investigate a class of differential operators with"discontunity", i.e., Sturm-Liouville problems with transmission conditions at an interior point, which are concerned by many mathematical and physical researchers. For Sturm-Liouville operators with separated boundary conditions and transmission conditions, we discuss their eigenvalues and eigenfunctions by the theory of function. In particular, we study Sturm-Liouville operators with eigenparameter dependent boundary conditions and transmission conditions, by establishing a new operator associated with the problem, prove that the operator is self-adjoint in an appropriate space and discuss the asymptotic expansion of its eigenvalues and eigenfunctions.Then we consider a kind of singular perturbation eigenvalue problem with high order turning points. Using the Langer transformation, the uniformly valid asymptotic solution, expressed by Bessel Function, of the equation is given and the eigenvalues of the problem with one turning point and two turning points are considered, which generalize the known results. Finally, the inequalities among the eigenvalues of two left-definite Sturm-Liouville problems with separated boundary conditions are established. The methods are based on some conclusions and monotonicity of eigenvalue onα,βin Sturm-Liouville operators with general separated boundary conditions.This paper contains six parts. The first part: an introduction of the background of the problems we investigate and main results we obtain in this paper. The second part: asymptotic behavior of eigenvalues for a right-definite Sturm-Liouville problem. The third part: asymptotic behavior of a Sturm-Liouville problem with separated boundary conditions and transmission conditions. The fourth part: asymptotic behavior of a Sturm-Liouville problem with eigenparameter-dependent boundary conditions and transmission conditions. The fifth part: a singularly perturbed eigenvalue problem with high order turning points. The last part: inequalities among eigenvalues of two left-definite Sturm-Liouville problems.