Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

D. J. Gilberta1

a1 Department of Applied Mathematics, University of Hull, Hull HU6 7RX, U.K.


The theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expression L = – d2/(dr2) + V(r) is in the limit point case at both ends of an interval (a, b), with V(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of Lu = xu, x εℝ, as ra and rb. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.

(Received January 22 1988)

(Revised January 31 1989)