M. Langer,
Resonances of a λ-rational Sturm–Liouville problem,
Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 709–720

Abstract:
We consider a family of self-adjoint 2 × 2-block operator matrices à in the space L2(0,1) ⊕ L2(0,1), depending on the real parameter ϑ. If Ã0 has an eigenvalue which is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears but the resolvent of Ãϑ 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 ϑ is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.

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