The Asymptotic Behavior Of The Eigenvalues ​​of Sturm-liouville Problems And Their Application Of The Inverse Problem

Sturm-Liouville theory has been known for a long time, which originated in processing for heat conduction of solid with mathematical method. The theory is widely used in physics, numerical methods and a variety of theoretical science and applied science, which make it a study field of experts on related fields. In 1946, Borg proved that the only known two spectra determine potential function, which created a precedent on inverse spectral theory. So the idea that eigenvalues can determine potential function and boundary conditions has attracted widespread attention and many new ideas and approaches have been introduced.In this paper, on the basis of solutions for differential equation, the high-order asymptotic expressions for eigenvalues and norming constants of S-L problem with smooth potentials are studied. Asymptotic behavior for diference of eigenvalues and norming constants of two S-L problems with smooth potentials are proved. The high-order estimate of the entire functions associated with S-L problems are given.The arrangement of this thesis is as follows:Chapter One:Preliminaries. The basic concepts and existing results of the theories of spectral for S-L problem which will be used throughout the thesis are given and the high-order expressions for fundamental solutions are established.Chapter Two:Asymptotic expressions for eigenvalues of S-L problem. In this chapter, the high-order asymptotic expressions for eigenvalues and norming con-stants of S-L problem with general boundary conditions and Dirichlet boundary condition are considered. Firstly, applying the high-order expressions for funda-mental solutions and boundary conditions, the asymptotic expressions for eigenval-ues of S-L problem with potentials q from space L1([0.π]) are given. Furthermore, the asymptotic expressions for eigenvalues of S-L problem with potentials q from space W1(m-1)([0,π])(m≥2) are given. Finally, some fundamental properties about norming constant with above boundary conditions are introduced and then with the high-order asymptotic expressions for eigenvalues and solutions, the high-order asymptotic expressions for norming constants are obtained.Chapter Three:Asymptotic behavior for diference of eigenvalues of two S-L problems. Firstly, the recurrence relations of coefficients of high-order expressions for fundamental solutions are established. High-order expressions of characterized functions are proved. Secondly, by means of estimates of the remainders for solution, asymptotic behavior for diference of eigenvalues of two S-L problems which are under given boundary conditions are studied. Finally, with the high-order asymptotic expressions for norming constants and their coefficients characteristics, asymptotic behavior for diference of norming constants of two S-L problems which are under above boundary conditions are given.Chapter Four:The high-order estimate of the entire functions associated with S-L problems. Firstly, the high-order expressions of the product of fundamental solutions for two S-L differential equations are given and the high-order expression for entire function f(z) and its coefficients characteristics are obtained. Secondly, applying the common spectrum of two S-L problems, the high-order estimate of entire function f(z) are obtained and are used in inverse S-L problems, so Amour's open problem is solved.