Analytic continuation and resonance-free regions for Sturm-Liouville   pote ntials with power decay
  We are concerned with the Sturm-Liouville problem on the half line. We show that when the potential $q$ is subject only to power decay at infinity the $L^2$ solution may be continued into a sector of the so-called un-physical sheet. This gives rise to resonance free regions and the numerical estimation of resonances outside these regions. 
B. M. Brown, M. S. P. Eastham