Real-parameter square-integrable solutions and the spectrum of differential operators

Abstract

We continue to investigate the connection between the spectrum of self-adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square-integrable solutions for real values of the spectral parameter λ. We show that if, for all λ in an open interval I, there are d linearly independent square-integrable solutions, then there is no continuous spectrum in I. This for any self-adjoint realization with boundary conditions which may be separated, coupled, or mixed. The proof is based on a new characterization of self-adjoint domains and on limit-point (LP) and limit-circle (LC) solutions established in an earlier paper.