Geometric and function analytic approaches to the study of Sturm-Liouville theory

Abstract

Ordinary second order linear di erential boundary value problems are of great interest in the study of physical systems, despite this and the length of their history they have many aspects which have only recently been understood. The aim of this work is to examine the geometric and functional analytical approaches to the study of Sturm-Liouville eigenvalue problem with separated boundary conditions. In particular, the following sub-problems will be investigated: Self-adjointness, oscillation theory, variational formulations and the completeness of eigenfunction expansions. The construction of Self-adjoint operators will be studies through de ciency indicies and through symplectic geometry. After the classical variational problem is identi- ed and studied, Lusternik-Schnirelmann theory which is a topologically invariant variational principle will be examined