Survey Article: Self-adjoint ordinary differential operators and their spectrum

Abstract

We survey the theory of ordinary self-adjoint differential operators in Hilbert space and their spectrum. Such an operator is generated by a symmetric differential expression and a boundary condition. We discuss the very general modern theory of these symmetric expressions which enlarges the class of these expressions by many dimensions and eliminates the smoothness assumptions required in the classical case as given, e.g., in the celebrated books by Coddington and Levinson and Dunford and Schwartz. The boundary conditions are characterized in terms of square-integrable solutions for a real value of the spectral parameter, and this characterization is used to obtain information about the spectrum. Many of these characterizations are quite recent and widely scattered in the literature, some are new. A comprehensive review of the deficiency index (which determines the number of independent boundary conditions required in the singular case) is also given for an expression $M$ and for its powers. Using the modern theory mentioned above, these powers can be constructed without any smoothness conditions on the coefficients.