Resonances and analytic continuation for exponentially decaying Sturm–Liouville potentials

The Sturm–Liouville equation on the half-line [0,∞) is considered with a potential which decays exponentially at infinity. It is known that Greenʹs function has an analytic continuation into part of the unphysical sheet im λ<0. Here it is shown that analytic continuation into the whole of this sheet is possible for certain types of decay. The methods are constructive, and they also provide a new and rigorous way of locating resonances.