## Abstract

The spectral problem for the Laguerre equation on (0, infinity) with real parameter a in the case 0 </alpha/ <1 is closely related to the Nevanlinna function

Q(alpha)(z) -piGamma(-z)/(sinpialpha)Gamma(-z -alpha ).

If /alpha/ > 1 and /alpha/ not equal 2, 3,..., this function belongs to the generalized Nevanlinna class N-m, m = [/alpha/+1/2]. A natural question appears: to what spectral problem does this function correspond? For alpha <-1, alpha not equal -2, -3...., an answer was given by Derkach [D]. He obtained an operator representation for the function m(alpha)(Z) = -Q(alpha)(-z)/Gamma(2)(1 + alpha) in terms of a self-adjoint operator in a Pontyragin space. and an interpretation, of m. (z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That ail indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for alpha > 1, alpha not equal 2, 3... we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Q (z) and compare them with the models for -alpha. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Q(alpha)(z) and -Q(alpha)(z)(-1). Finally, we describe the connection between the self-adjoint linear relations in the representations of Q(alpha)(z) and Q(-alpha)(z+alpha) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions.

Original language | English |
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Title of host publication | OPERATOR METHODS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS |

Editors | S Albeverio, N Elander, WN Everitt, P Kurasov |

Place of Publication | BASEL |

Publisher | Birkhauser |

Pages | 141-181 |

Number of pages | 41 |

ISBN (Print) | 3-7643-6790-3 |

Publication status | Published - 2002 |

Event | Marcus Wallenberg Symposium held in Memory of Sonja Kovalevsky - , Sweden Duration: 18-Jun-2000 → 22-Jun-2000 |

### Publication series

Name | OPERATOR THEORY : ADVANCES AND APPLICATIONS |
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Publisher | BIRKHAUSER VERLAG AG |

Volume | 132 |

### Other

Other | Marcus Wallenberg Symposium held in Memory of Sonja Kovalevsky |
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Country/Territory | Sweden |

Period | 18/06/2000 → 22/06/2000 |

## Keywords

- RANK-ONE PERTURBATIONS
- SELF-ADJOINT OPERATORS
- IIX