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

We investigate the spectral theory of a general third order formally symmetric differential expression of the form$$ L[y] = { 1 \over w} \left \{ - i (q_{0} (q_{0} y' ) ' ) ' + i (q_{1}y' + ( q_{1} y ) ' ) - (p_{0}y' ) ' + p_{1}y \right \} $$acting in the Hilbert space ℒ 2 w ( a ,∞). 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. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)