Levinson’s theorem and the second virial coefficient in one, two, and three dimensions

The second virial coefficient B(T) can be expressed in terms of two-body phase shifts and bound-state energies. The fact that B(T) must remain continuous as the potential strength is varied is used to deduce the relationship between the phase shifts at zero energy and the number of bound states supported by the potential (Levinson’s theorem). There still remains an ambiguity due to the possibility of zero-energy resonances; in one and three dimensions this can be removed by the additional information of whether the tangent of the phase shift goes to zero or to infinity at zero energy. In two dimensions, the problem is more subtle and a difficulty is encountered if the potential strength is such that a p-wave zero-energy resonance is present; the expression for B(T) as the inverse Laplace transform of the Jost function gives the correct contribution from this resonance, but the phase-shift formula derived using asymptotic wave functions fails to include it, and hence is in disagreement with Levinson’s theorem. The origin of this disagreement is traced to a noncommutivity in the wave function between the limits of large distance and low energy; this occurs only in the two-dimensional case, and needs to be handled carefully.