LOW-ENERGY CHAIN OF RESONANCES FOR 3-DIMENSIONAL SCHRODINGER OPERATORWITH NEARLY COULOMB POTENTIAL

We study resonances for a three-dimensional Schrodinger operator with Coulomb potential perturbed by a spherically symmetric compactly suppo rted function. Resonances are defined as poles of an analytical contin uation of the resolvent to the second Riemann sheet through the contin uous spectrum. It is proved that for a nonnegative perturbation with f inite positive first moment there exists a chain of resonances accumul ating to zero. It is known that in the non-Coulomb case of a rapidly d ecreasing potential the perturbation can produce only ''high-energy'' series of resonances converging to infinity. The above result shows th at, in contrast with the non-Coulomb case, a small perturbation of the Coulomb potential can produce also, a ''low-energy'' sequence of reso nances. The latter means that zero becomes a ''triple singular'' point of the spectrum, being the point of accumulation of the discrete, con tinuous and ''resonance'' spectra. It was shown in our previous paper that for the radial Schrodinger operator with perturbed Coulomb potent ial there exists a disk centered at the origin on the second sheet whi ch is free of resonances. We prove that the radius r(l) of the maximal resonance-free disk corresponding to the angular momentum I has an es timate: r(l)less than or equal to Cl-1. The results are obtained based on a detailed analysis of an asymptotic behavior at low energies of t he Jest functions corresponding to the different values of an angular momentum. (C) 1994 Academic Press, Inc.