Resonances of a λ-rational Sturm–Liouville problem

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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.
Original languageEnglish
Pages (from-to)709-720
Number of pages12
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Issue number3
Publication statusPublished - Jun 2001


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


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