Abstract
AbstractWe investigate the spectral theory of a general third order formally symmetric differential expression of the form acting in the Hilbert space ℒ︁2w (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)
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