Continuous spectrum and square-integrable solutions of differential operators with intermediate deficiency index

Abstract

We explore the connection between square-integrable solutions for real-values of the spectral parameter λ and the continuous spectrum of self-adjoint ordinary differential operators with arbitrary deficiency index d. We show that if, for all λ in an open interval I, there are d of linearly independent square-integrable solutions, then for every extension of D min the point spectrum is nowhere dense in I, and there is a self-adjoint extension of S min which has no continuous spectrum in I. This analysis is based on our construction of limit-point (LP) and limit-circle (LC) solutions obtained recently in an earlier paper.