Resonances of a λ-rational Sturm–Liouville problem

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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.
Original language English 709-720 12 Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131 3 https://doi.org/10.1017/S0308210500001062 Published - Jun 2001

Keywords

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

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