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

✍ Scribed by Horst Behncke; Don Hinton

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 341 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310314

✦ Synopsis

Abstract

We investigate the spectral theory of a general third order formally symmetric differential expression of the form

acting in the Hilbert space ℒ︁^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)