Absolute determination of zero-energy phase shifts for multiparticle single-channel scattering: Generalized Levinson theorem

Levinson{close_quote}s theorem relates the zero-energy phase shift {delta} for potential scattering in a given partial wave {ital l}, by a spherically symmetric potential that falls off sufficiently rapidly, to the number of bound states of that {ital l} supported by the potential. An extension of this theorem is presented that applies to single-channel scattering by a compound system initially in its ground state. As suggested by Swan [Proc. R. Soc. London Ser. A {bold 228}, 10 (1955)], the extended theorem differs from that derived for potential scattering; even in the absence of composite bound states {delta} may differ from zero as a consequence of the Pauli principle. The derivation given here is based on the introduction of a continuous auxiliary {open_quote}{open_quote}length phase{close_quote}{close_quote} {eta}, defined modulo {pi} for {ital l}=0 by expressing the scattering length as {ital A}={ital a}cot{eta}, where {ital a} is a characteristic length of the target. Application of the minimum principle for the scattering length determines the branch of the cotangent curve on which {eta} lies and, by relating {eta} to {delta}, an absolute determination of {delta} is made. The theorem is applicable, in principle, to single-channel scattering in any partial wave for {ital e}{sup {plus_minus}}-atom and nucleon-nucleus systems. In addition to a knowledge of the number of composite bound states, information (which can be rather incomplete) concerning the structure of the target ground-state wave function is required for an explicit, absolute, determination of the phase shift {delta}. As for Levinson{close_quote}s original theorem for potential scattering, {ital no} {ital additional} {ital information} {ital concerning} {ital the} {ital scattering} {ital wave} {ital function} {ital or} {ital scattering} {ital dynamics} {ital is} {ital required}. {copyright} {ital 1996 The American Physical Society.}