Geometric aspects of Sturm—Liouville problems I. Structures on spaces of boundary conditions

Abstract

We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.