Inequalities Among Eigenvalues Of Sturm-liouville Problems

The Sturm-Liouville problem has become a Sturm-Liouville theory from production to development for more than 170 years.It is an extremely active research field with rich research content.It gets extensive attention from many mathematicians,physical workers,engineers and technicians.In the related literature of Sturm-Liouville theory,the study of inequality among eigenvalues are more in-depth.The method of proof is also constantly improved along the path from special to general.The extensive application of the inequalities among eigenvalues has promoted the scholars to study it.And it has a certain theoretical and practical significance for scholars to explore a more simplified method to prove it.In 1999,Q.Kong,H.Wu and A.Zettl gave a manifold structure of Grassman on the boundary condition space.From this geometric structure.a natural geometric structure on the coupled self-adjoint boundary condition space is derived.It is related properties play a decisive role to prove the inequality among eigenvalues of the Sturm-Liouville problem.In the literature of Q.Kong,Qun Lin,Wu and Anton Zettl in 2000,the matrix in the second order linear group SL(2,R)is constructed and by using the increase and decrease of eigenvalue under the separation boundary condition,the proof of the inequality among the eigenvalues on the existing Sturm-Liouville problem is given.There is no use of the related conclusions on the inequality among the eigenvalues under the general boundary conditions already proved.That is the inequality among the eigenvalues under the periodic boundary conditions and the semi-periodic boundary conditions.It also simplify the whole proof process.In this paper,firstly,we give proof of the inequalities among the eigenvalues for the existing Sturm-Liouville problem with positive leading coefficient by using the method of matrix constructing and the increase and decrease of eigenvalue in the separation boundary condition.Secondly,inequality among the eigenvalues is constructed and proved for Sturm-Liouville with sign-changing of leading coefficient.Finally,the existing inequalities are refined and the proof is given.Key and difficult points are how to construct the matrix in the second order linear group,and its relevant conclusions satisfy the increase and decrease of eigenvalues under the separation boundary condition...