Left-Definite Sturm-Liouville Problems

Left-definite regular self-adjoint Sturm-Liouville problems with separated and coupled boundary conditions are studied. A characterization is given in terms of right-definite problems, a concrete and “natural” indexing scheme for the eigenvalues is proposed. Pruefer transformation techniques can be used to establish the existence of and to give a characterization for the eigenvalues in the case of separated boundary conditions. Here we give an elementary proof of the existence of the eigenvalues for the coupled case. Furthermore we study the continuous and differentiable dependence of the eigenvalues on all parameters of the problem. For a fixed equation we find the range of each eigenvalue as a function of the boundary conditions and inequalities among the eigenvalues as the boundary conditions vary. These extend the classical inequalities among the Neumann, Dirichlet, Periodic, and Semi-Periodic eigenvalues and our recent generalizations for the right-definite case. Some of our results here yield an algorithm for the numerical computation of the eigenvalues of left-definite problems with coupled boundary conditions.