The Self-Adjoint Boundary Conditions And Inequalities Among Eigenvalues Of Differential Operators

In this paper, the classifications of self-adjoint boundary conditions for differential operators and inequalities among eigenvalues are studied. We have known,for 2n-order self-adjoint boundary conditions, that the ranks of coefficients matrices are equal and don't less than n,see [60] .In Chapter 1, we will give a new proof for this conclusion. We believe that our proof is essentially different from [60]. So by the rank of coefficients matrices, the 2n-order self-adjoint boundary conditions can be separated n+1 classes at most. In Chapter 1, we will see every class exists. In addition, we will give the classifications of 4-order separated boundary conditions, we find that there are four boundary conditions for 4-order differential operators, which are similar with Dirichlet boundary condition and Neumann boundary condition. In §1.4, for differential equation y⁽⁴⁾= λy, we will research it's eigenvalues for these boundary conditions.For 2-order regular Sturm-Liouville problems, Wujian Peng etc. found some inequalities among eigenvalues Lately, which differ from [26].In Chapter 2, we introduce the theory of double coefficients Sturm-Liouville problems and spectral curves at first. By this theory,we establish some similar inequalities for Left-definite and Right-indefinite problems.Notice that for any fixed LCNO Sturm-Liouville problem, by regularization function we can get a regular Sturm-Liouville problem which has the same eigenvalues with this singular Sturm-Liouville problem. So in Chapter 3, for LCNO Sturm-Liouville problems, we establish a new sequence of inequalities among eigenvalues for different boundary conditions.