The Spectrum Of Higher Order Differential Operators With Periodic

The theory of differential operators can be supplied in modern physics, classical physics, differential equations and other fields of engineering technology. It is an important research content of functional analysis, and is the basic theory of modern mathematics. It consists of many important branches, such as self-adjoint extensions, spectral theory, numerical computation, deficiency index theory, eigenfunction expansion and so on. It is not only an important theoretical basis for many mathematical problems, but also a support of the engineering fields. The theory of differential operators has been the focus of many excellent mathematical workers. Many Chinese and foreign mathematics workers dedicated to this area.Professor Eastham M. S. P, a British mathematician, studies the second-order differential operators with real periodic coefficients(i.e. the Hill’s operators). He proved that the gaps in the spectrum of the Hill’s operators are equivalent to the union of the instability intervals of corresponding differential equations. He obtained that the instability intervals are constituted by some open intervals and estimated the lengths of the gaps to a class of Hill’s operators. At the same time, he also analyzes and discusses its least eigenvalues. What are these when it extending to the case of higher-order differential operators? In this thesis, we study the gaps in the spectrum and the least eigenvalues in the field of higher-order differential operators. The gaps in the spectrum are equivalent to the union of the instability interval of the corresponding the higher-order differential equation will be proved, the lengths of the gaps under special cases and the least eigenvalues will be estimated.This thesis include following four chapters. The first chapter is the introduction, gives the background of the problems and practical significance and main results in this paper. The second chapter is preliminary knowledge, gives the concept, lemma and some important conclusions, including the definition of the self-adjoint operator, the concept of the Parseval formula and so on. The third chapter studies the gaps in the spectrum of 2n-th order differential operators. This chapter consists of three parts. The first is the introduction. The second discusses the gaps in the spectrum of 2n-th order with periodic real coefficients. And the third discusses the estimation of the lengths of the gaps in the special cases. The fourth chapter studies the least eigenvalues of 2n-th order differential operators. The first section in this chapter is the introduction. The second section gives some important conclusions about the least eigenvalues. The third chapter and the fourth chapter are the core content of this thesis.