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

Autor(en): | Behncke, H Hinton, D |

Stichwörter: | ASYMPTOTIC INTEGRATION; asymptotic solutions; deficiency indices; EQUATIONS; Mathematics; spectral theory |

Erscheinungsdatum: | 2005 |

Herausgeber: | WILEY-V C H VERLAG GMBH |

Journal: | MATHEMATISCHE NACHRICHTEN |

Volumen: | 278 |

Ausgabe: | 12-13 |

Startseite: | 1430 |

Seitenende: | 1457 |

Zusammenfassung: | 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. |

ISSN: | 0025584X |

DOI: | 10.1002/mana.200310314 |

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