### Abstract

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

Pages (from-to) | 709-720 |

Number of pages | 12 |

Journal | Proceedings of the Royal Society of Edinburgh: Section A Mathematics |

Volume | 131 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 2001 |

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### Keywords

- Sturm–Liouville problem
- Titchmarsh–Weyl coefficient
- analytic continuation

### Cite this

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*Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, vol. 131, no. 3, pp. 709-720. https://doi.org/10.1017/S0308210500001062

**Resonances of a λ-rational Sturm–Liouville problem.** / Langer, Matthias.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Resonances of a λ-rational Sturm–Liouville problem

AU - Langer, Matthias

PY - 2001/6

Y1 - 2001/6

N2 - 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.

AB - 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.

KW - Sturm–Liouville problem

KW - Titchmarsh–Weyl coefficient

KW - analytic continuation

U2 - 10.1017/S0308210500001062

DO - 10.1017/S0308210500001062

M3 - Article

VL - 131

SP - 709

EP - 720

JO - Proceedings of the Royal Society of Edinburgh: Section A Mathematics

JF - Proceedings of the Royal Society of Edinburgh: Section A Mathematics

SN - 0308-2105

IS - 3

ER -