Discontinuous Sturm Liouville Problem Of Inverse Spectrum Theory

Sturm-Liouville spectrum theory has been known for a long time, which origi-nated in processing for the mathematical method of heat conduction of solid. The theory is used in a variety of theoretical science and applied science widely, which make it to be a study field of experts on related fields. The paper studies by the appli-cation of system eigenvalue, eigenfunction, asymptotic of solutions to a holomorphic function, based on Gelfand-levitan integral equation. We consided non continuous Sturm-Liouville inverse problem with two discontinuous notes, and given uniqueness theorema of two discontinuous notes.The arrangement of this thesis is as follows:Chapter One: Analysis of Sturm-Liouville spectrum. In this chapter, the basic concepts and existing results of the theories of spectral for Sturm-Liouville problem is given.Chapter Two:On the determination of Sturm-Liouville operator from two par-tial spectra. The Sturm-Liouville problem is considered on a closed interval in this chapter. we give the Sturm-Liouville operator from two partial spectra, meanwhile, extend the Levitan conclusion.Chapter Three:Unique definition of Sturm-Liouville inverse nodal problems. The inverse Sturm-Liouville problem with separated boundary value conditions is considered on a finite interval. Through analysing the asymptotic of eigenvalues and the nodals of eigenfunction, we give the conditions that density function and potential function can be uniquely determined by the nodals. It is detailed proved that density function can be uniquely determined up to one multiplicative constant by a dense subset of nodal positions as potential function is given; And potential function can be uniquely determined by the same nodal positions of systems as density function is given.Chapter Four:Inverse Sturm-Liouville problems with discontinuous boundary value problems. The inverse Sturm-Liouville problems are considered on a finite in-terval with two discontinuous points in the interval. Uniqueness theorems are proved by Volterra integral equation and densities of zeros of a class of entire functions.