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.
LanguageEnglish
Pages709-720
Number of pages12
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume131
Issue number3
DOIs
Publication statusPublished - Jun 2001

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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

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

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

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