Three Eigenvalues ​​of Sturm-Liouville Problem

It is well known that the Sturm-Liouville problems are derived in processing for solid heat conduction model. Its theory are applied widely, and mainly used in mathematical physics, engineering and technology, meteorological physics and other types of theory and applications. Therefore, for more than a century, ordinary differential operator has gradually formed into an important research branch in the field of mathematics and physics. In this paper, on the basis of the high order expansion of basic solutions for differential equation, the asymptotic behavior of the eigenvalues of Sturm-Liouville operators with boundary conditions containing spectral parameters are studied. By the asymptotic approximation for the solution of the initial value, restoring a integral identity and the residue method, the trace formula of2×2S-L eigenvalue problem with boundary conditions containing spectral parameters is given.The arrangement of the thesis is as follows:Chapter1:In this chapter, the research development status of the Sturm-Liouville theory and the work of this paper are introduced.Chapter2:In this chapter, the regular Sturm-Liouville problem defined on [0,1] is considered, with the boundary conditions not containing spectral parameters and the potential function q(x)∈W2m-1([0,1])(m∈N). The high order expansions of basic solutions for differential equation are given.Chapter3:In this chapter, the Sturm-Liouville problem defined on [0,1] is considered, with the boundary conditions containing spectral parameters and the potential function q(x)∈W2m-1([0,1])(m∈N). With the help of the high order ex-pansion of basic solutions for differential equation and adopting residual estimation method, the asymptotic expansions of the eigenvalues of the problem are available.Chapter4:In this chapter, the2×2Sturm-Liouville problem defined on [0,π] is considered, with the boundary conditions containing spectral parameter. By the given eigenfunction of the problem, the trace formula of eigenvalues of the problem is given.