Left-definite Sturm-Liouville problems

Left-definite regular self-adjoint Sturm Liouville problems, with either se parated or coupled boundary conditions, are studied. We give an elementary proof of the existence of eigenvalues for these problems. For any fixed equ ation, we establish a sequence of inequalities among the eigenvalues for di fferent boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions. Some of our results here yiel d an algorithm for numerically computing the eigenvalues of a left-definite problem with an arbitrary coupled boundary condition. Our inequalities imp ly that the well-known asymptotic formula for the eigenvalues in the separa ted case also holds in the coupled case. Moreover, we study the continuous and differentiable dependence of the eigenvalues of the general left-defini te problem on all the parameters in its differential equation and boundary condition. (C) 2001 Academic Press.