We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that 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.