Inequalities among eigenvalues of Sturm-Liouville problems

There are well-known inequalities among the eigenvalues of Sturm-Liouville problems with periodic, semi-periodic, Dirichlet and Neumann boundary condi tions. In this paper, for an arbitrary coupled self-adjoint boundary condit ion, we identify two separated boundary conditions corresponding to the Dir ichlet and Neumann conditions in the classical case, and establish analogou s inequalities. It is also well-known that the lowest periodic eigenvalue i s simple; here we prove a similar result for the general case. Moreover, we show that the algebraic and geometric multiplicities of the eigenvalues of self-adjoint regular Sturm-Liouville problems with coupled boundary condit ions are the same. An important step in our approach is to obtain a represe ntation of the fundamental solutions for sufficiently negative values of th e spectral parameter. Our approach yields the existence and boundedness fro m below of the eigenvalues of arbitrary self-adjoint regular Sturm-Liouvill e problems without using operator theory.