The geometric multiplicity of each eigenvalue of a self-adjoint Sturm-Liouville problem is equal to its algebraic multiplicity. This is true for regular problems and for singular problems with limit-circle endpoints, including the case when the leading coefficient changes sign.

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...

We consider a Sturm-Liouville boundary value problem in a bounded domain D of Rn. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on ∂D. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the me...