Derivation of Nonrelativistic Sum Rules from the Causality Condition of Wigner and Van Kampen

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Abstract
Nonrelativistic sum rules previously obtained for each phase shift in the framework of elastic scattering by a local central potential of finite radius are shown to follow from the causality principle of Wigner and Van Kampen, under the assumptions that the phase shift satisfies the Levinson theorem and that, at high energies, the integrability condition holds. In fact, it is shown that under these assumptions, any interaction of finite radius which is causal in the sense of Wigner and Van Kampen is equivalent to a local potential of the same radius. First we recall the sum rules and mention some of their applications. We give a brief survey of their proof in potential scattering, which is based essentially on the analytic and asymptotic properties of the Jost function (or the S matrix). Then we show that, under the assumptions mentioned above on the phase shifts, the properties of the R matrix derived by Wigner and Van Kampen from causality lead to the same analytic and asymptotic properties of the Jost function (defined now directly from the S matrix) as in potential scattering, providing, therefore, sufficient information for the direct derivation of the sum rules. Finally, using the Gel'fand‐Levitan and Marchenko integral equations of the inverse‐scattering problem, we show that, in fact, this information is sufficient to entail that the causal interaction of Wigner and Van Kampen is equivalent to a local potential of the same radius.

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