# Resonances of a λ-rational Sturm–Liouville problem

Research output: Contribution to journalArticle

3 Citations (Scopus)

### 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.
Language English 709-720 12 Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131 3 10.1017/S0308210500001062 Published - Jun 2001

### Fingerprint

Sturm-Liouville Problem
Pole
Essential Spectrum
Eigenvalue
Operator Matrix
Block Matrix
L-space
Analytic Continuation
Resolvent
Half-plane
Riemann Surface
Asymptotic Behavior
Coefficient

### Keywords

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

### Cite this

title = "Resonances of a λ-rational Sturm–Liouville problem",
abstract = "We consider a family of self-adjoint 2 × 2-block operator matrices {\~A}_\theta in the space L_2(0,1) \oplus L_2(0,1), depending on the real parameter \theta. If {\~A}_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 {\~A}_\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.",
keywords = "Sturm–Liouville problem, Titchmarsh–Weyl coefficient, analytic continuation",
author = "Matthias Langer",
year = "2001",
month = "6",
doi = "10.1017/S0308210500001062",
language = "English",
volume = "131",
pages = "709--720",
journal = "Proceedings of the Royal Society of Edinburgh: Section A Mathematics",
issn = "0308-2105",
number = "3",

}

In: Proceedings of the Royal Society of Edinburgh: Section A Mathematics , Vol. 131, No. 3, 06.2001, p. 709-720.

Research output: Contribution to journalArticle

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

T2 - 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 -