Nodal structure of zero-energy wave functions: New approach to Levinson’s theorem

It is well known that minimum principles for energy eigenvalues can be used not only for calculational purposes but also formally as the basis for studies of the nodal structure of the associated eigenfunctions. This formal feature is shown here to have a natural generalization to the zero-energy scattering problem, the scattering length playing the role of the energy eigenvalue in the formulation of the minimum principle. The number of nodes in the zero-energy wave function in a given partial wave for nonrelativistic potential scattering is shown to be equal to the number of negative-energy bound states of the same angular momentum L. Potentials with Coulomb tails are excluded from the present analysis, but will be treated elsewhere. (The connection with classical Sturm-Liouville theory comes perhaps as less of a surprise if one views the zero-energy state as being at the top of the discrete spectrum as well as at the bottom of the continuum.) The result derived here, when combined with the nodal definition of the phase shift along with some information on its threshold behavior, provides us with an alternative derivation of Levinson’s theorem relating the zero-energy phase shift for orbital angular momentum L, ${\delta }_L$(0), to the number of bound states of the same L. (Very similar results can be derived for potential scattering as described by the Dirac and Klein-Gordon equations.) It may well be possible to extend some of the results obtained here to a number of single-channel multiparticle scattering problems, but this will be discussed elsewhere.