REGULAR STURM-LIOUVILLE PROBLEMS WHOSE COEFFICIENTS DEPEND RATIONALLYON THE EIGENVALUE PARAMETER

In this note we consider regular Sturm-Liouville equations with a floa ting singularity of a special type: the coefficient of the second orde r derivative contains the eigenvalue parameter. We determine the form of the boundary conditions which make the problem selfadjoint after li nearizing. In general the boundary conditions for the linearized syste m give rise to boundary conditions which involve the eigenvalue parame ter in the original, non-linearized, problem. The boundary conditions give rise to a 2 x 2 matrix function, the so - called Titchmarsh-Weyl coefficient. The characteristic properties of this function are studie d. The formal aspects of the theory of this class of equations turn ou t to be quite parallel to those for the usual situation when there is no floating singularity.