Contour and surface integrals in potential scattering


When the Schrödinger equation for stationary states is studied for a system described by a central potential in n-dimensional Euclidean space, the radial part of stationary states is an even function of a parameter \(\lambda \) which is a linear combination of angular momentum quantum number l and dimension n, i.e., \(\lambda =l+{(n-2)\over 2}\). Thus, without setting a priori \(n=3\), complex values of \(\lambda \) can be achieved, in particular, by keeping l real and complexifying n. For suitable values of such an auxiliary complexified dimension, it is therefore possible to obtain results on scattering amplitude and phase shift that are completely equivalent to the results obtained in the sixties for Yukawian potentials in \(\mathbf{R}^{3}\). Moreover, if both l and n are complexified, the possibility arises of recovering the partial wave amplitude from residues of a function of two complex variables. Thus, the complex angular momentum formalism can be imbedded into a broader framework, where a correspondence exists between the scattering amplitude and a skew curve in \(\mathbf{R}^{3}\).

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  1. 1.

    Recall from Ref. [3] that the potential V(z) is Yukawian if one can express it in the form

    $$\begin{aligned} V(z)=\int _{m}^{\infty }C(\mu )e^{-\mu z}\mathrm {d}\mu , \end{aligned}$$

    where \(C(\mu )\) is a distribution. Other relevant cases occur when \(C(\mu )\) is a function of bounded variation, for which the potential reads as

    $$\begin{aligned} V(z)=\int _{m}^{\infty }{e^{-\mu z}\over z}\mathrm {d}C_{\mu }, \end{aligned}$$

    or when C is an absolutely continuous function with first derivative \(C'(\mu )=\sigma (\mu )\), for which V(z) takes the form

    $$\begin{aligned} V(z)=\int _{m}^{\infty }{e^{-\mu z}\over z}\sigma (\mu )\mathrm {d}\mu . \end{aligned}$$

    The Yukawa potential is recovered from the first formula when \(C(\mu )\) is just a constant.

  2. 2.

    The assumption of Poincaré is not however too restrictive, since a theorem holds, according to which a function that is meromorphic over the whole of \(\mathbf{C}^{2}\), including the point at infinity, is a rational function [14].


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The author is grateful to the Dipartimento di Fisica “Ettore Pancini” for hospitality and support. The preprint version of this paper was arXiv:2001.11217 [hep-th].

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Correspondence to Giampiero Esposito.

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Esposito, G. Contour and surface integrals in potential scattering. Eur. Phys. J. Plus 135, 501 (2020).

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