# Resonances of a λ-rational Sturm–Liouville problem

Langer, Matthias
(2001)
*Resonances of a λ-rational Sturm–Liouville problem.*
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131 (3).
pp. 709-720.
ISSN 0308-2105

## Abstract

We consider a family of self-adjoint 2 × 2-block operator matrices Ã_\theta in the space L_2(0,1) \oplus L_2(0,1), depending on the real parameter \theta. If Ã_0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for \theta ≠ 0 this eigenvalue in general disappears, but the resolvent of Ã_\theta has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane C^+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small \theta is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.

Item type: | Article |
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ID code: | 35432 |

Keywords: | Sturm–Liouville problem, Titchmarsh–Weyl coefficient, analytic continuation, Mathematics, Mathematics(all) |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Pure Administrator |

Date Deposited: | 04 Nov 2011 15:24 |

Last modified: | 27 May 2016 03:49 |

URI: | http://strathprints.strath.ac.uk/id/eprint/35432 |