Deficiency indices and spectral theory of third order differential operators on the half line

Abstract

We investigate the spectral theory of a general third order formally symmetric differential expression of the form L[y] = 1/w {-i(q(0)(q(0)y')')' i (q(1)y' (q(1)y)') - (p(0)y')' p(1)y} acting in the Hilbert space L-w(2) (a, infinity). A Kummer-Liouville transformation is introduced which produces a differential operator unitarily equivalent to L. By means of the Kummer-Liouville transformation and asymptotic integration, the asymptotic solutions of L[y] = zy are found. From the asymptotic integration, the deficiency indices are found for the minimal operator associated with L. For a class of operators with deficiency index (2, 2), it is further proved that almost all selfadjoint extensions of the minimal operator have a discrete spectrum which is necessarily unbounded below. There are however also operators with continuous spectrum. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.