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.