GEOMETRIC ASPECTS OF HIGH-ORDER EIGENVALUE PROBLEMS I. STRUCTURES ON SPACES OF BOUNDARY CONDITIONS

We consider some geometric aspects of regular eigenvalue problems of an arbitrary order. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of eigenvalues on the boundary condition involved, and reveals new properties of these eigenvalues. Then, we solve the selfadjointness condition explicitly and obtain a manifold structure on the space of selfadjoint boundary conditions and several other consequences. Moreover, we give complete characterizations of several subsets of boundary conditions such as the set of all complex boundary conditions having a given complex number as an eigenvalue, and describe some of them topologically. The shapes of some of these subsets are shown to be independent of the quasidifferential equation in question.