Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Resonances of a λ-rational Sturm–Liouville problem

Matthias Langera1

a1 Department of Analysis and Technical Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/114, A-1040 Wien, Austria ([email protected])

Abstract

We consider a family of self-adjoint 2 × 2-block operator matrices Ãxs03D1 in the space L2(0, 1) xs2295 L2(0, 1), depending on the real parameter xs03D1. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for xs03D1 ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãxs03D1 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 xs03D1 is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.

(Received November 15 1999)

(Accepted May 18 2000)