Resonance Selection Principle and Low Energy Resonances for a Radial Schrödinger Operator with Nearly Coulomb Potential

We study resonances for the radial Schrödinger operator with Coulomb potential perturbed by a compactly supported function. Resonances are defined as poles of an analytic continuation of the quadratic form of the resolvent to the second Riemann sheet through the continuous spectrum. We show that, like in the non-Coulomb case, resonances can be described as roots of the Jost function. Using this result we prove that zero cannot be a point of accumulation of resonances, i.e., for a given value of an angular momentum l there exists a disk on the second Riemann sheet centered at the origin which is free of resonances. The radii of these disks may tend to zero when l → ∞. In our next paper we show that these radii do tend to zero for a nonnegative perturbation with finite positive first moment. This means that the three-dimensional Schrödinger operator with Coulomb potential perturbed by a complactly supported spherically symmetric function of the above type has a sequence of resonances accumulating to zero.