Premium
Deficiency indices and spectral theory of third order differential operators on the half line
Author(s) -
Behncke Horst,
Hinton Don
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310314
Subject(s) - mathematics , differential operator , hilbert space , spectrum (functional analysis) , operator (biology) , transformation (genetics) , pure mathematics , spectral theory , order (exchange) , mathematical analysis , class (philosophy) , biochemistry , chemistry , physics , finance , repressor , quantum mechanics , artificial intelligence , computer science , transcription factor , economics , gene
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)