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Low-Energy Chain of Resonances for Three-Dimensional Schrödinger Operator with Nearly Coulomb Potential

Journal of Differential Equations
Publication Date
DOI: 10.1006/jdeq.1994.1145


Abstract We study resonances for a three-dimensional Schrödinger operator with Coulomb potential perturbed by a spherically symmetric compactly supported function. Resonances are defined as poles of an analytical continuation of the resolvent to the second Riemann sheet through the continuous spectrum. It is proved that for a nonnegative perturbation with finite positive first moment there exists a chain of resonances accumulating to zero. It is known that in the non-Coulomb case of a rapidly decreasing potential the perturbation can produce only "high-energy" series of resonances converging to infinity. The above result shows that, in contrast with the non-Coulomb case, a small perturbation of the Coulomb potential can produce also, a "low-energy" sequence of resonances. The latter means that zero becomes a "triple singular" point of the spectrum, being the point of accumulation of the discrete, continuous and "resonance" spectra. It was shown in our previous paper that for the radial Schrödinger operator with perturbed Coulomb potential there exists a disk centered at the origin on the second sheet which is free of resonances. We prove that the radius r( l) of the maximal resonance-free disk corresponding to the angular momentum l has an estimate: r( l) ≤ Cl −1. The results are obtained based on a detailed analysis of an asymptotic behavior at low energies of the Jest functions corresponding to the different values of an angular momentum.

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