### Abstract

Original language | English |
---|---|

Pages (from-to) | 257-300 |

Number of pages | 44 |

Journal | Integral Equations and Operator Theory |

Volume | 75 |

Issue number | 2 |

Early online date | 1 Dec 2012 |

DOIs | |

Publication status | Published - Feb 2013 |

### Fingerprint

### Keywords

- Bessel-type operators
- singularity
- Hilbert space treatment
- inner singularity
- primary 34B30
- 34B20
- secondary 47E05
- 34L40
- 47B50
- 46C20
- bessel equation
- singular potential
- symmetric operators in pontryagin spaces
- self-adjoint extensions
- weyl function
- generalized nevanlinna function

### Cite this

*Integral Equations and Operator Theory*,

*75*(2), 257-300. https://doi.org/10.1007/s00020-012-2023-3

}

*Integral Equations and Operator Theory*, vol. 75, no. 2, pp. 257-300. https://doi.org/10.1007/s00020-012-2023-3

**Bessel-type operators with an inner singularity.** / Brown, B. Malcolm; Langer, Heinz; Langer, Matthias.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Bessel-type operators with an inner singularity

AU - Brown, B. Malcolm

AU - Langer, Heinz

AU - Langer, Matthias

PY - 2013/2

Y1 - 2013/2

N2 - We consider a Bessel-type differential expression on [0,a], a>1, with the singularity at the inner point x=1, see (1.2) below. This singularity is in the limit point case from both sides. Therefore in a Hilbert space treatment in L²(0,a), e.g. for Dirichlet boundary conditions at x=0 and x=a, a unique self-adjoint operator is associated with this differential expression. However, in papers by J. F. van Diejen and A. Tip, Yu. Shondin, A. Dijksma, P. Kurasov and others, in more general situations, self-adjoint operators in some Pontryagin space were connected with this kind of singular equations; for (1.2) this connection appeared also in the study of a continuation problem for a hermitian function by H. Langer, M. Langer and Z. Sasvári. In the present paper we give an explicit construction of this Pontryagin space for the Bessel-type equation (1.2) and a description of the self-adjoint operators which can be associated with it.

AB - We consider a Bessel-type differential expression on [0,a], a>1, with the singularity at the inner point x=1, see (1.2) below. This singularity is in the limit point case from both sides. Therefore in a Hilbert space treatment in L²(0,a), e.g. for Dirichlet boundary conditions at x=0 and x=a, a unique self-adjoint operator is associated with this differential expression. However, in papers by J. F. van Diejen and A. Tip, Yu. Shondin, A. Dijksma, P. Kurasov and others, in more general situations, self-adjoint operators in some Pontryagin space were connected with this kind of singular equations; for (1.2) this connection appeared also in the study of a continuation problem for a hermitian function by H. Langer, M. Langer and Z. Sasvári. In the present paper we give an explicit construction of this Pontryagin space for the Bessel-type equation (1.2) and a description of the self-adjoint operators which can be associated with it.

KW - Bessel-type operators

KW - singularity

KW - Hilbert space treatment

KW - inner singularity

KW - primary 34B30

KW - 34B20

KW - secondary 47E05

KW - 34L40

KW - 47B50

KW - 46C20

KW - bessel equation

KW - singular potential

KW - symmetric operators in pontryagin spaces

KW - self-adjoint extensions

KW - weyl function

KW - generalized nevanlinna function

UR - http://www.scopus.com/inward/record.url?scp=84872608669&partnerID=8YFLogxK

UR - http://link.springer.com/journal/volumesAndIssues/20

U2 - 10.1007/s00020-012-2023-3

DO - 10.1007/s00020-012-2023-3

M3 - Article

VL - 75

SP - 257

EP - 300

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 2

ER -