Nonlocal potentials and complex angular momentum theory

TL;DR

This paper investigates the mathematical properties of scattering amplitudes for nonlocal potentials, establishing their meromorphic structure in complex angular momentum and momentum planes, and explores implications for resonance and antiresonance phenomena.

Contribution

It provides a rigorous analysis of the meromorphy of partial scattering amplitudes for nonlocal potentials, extending complex angular momentum theory beyond local potential frameworks.

Findings

Meromorphic structure of T(lambda,k) established in complex domains.

Representation of T as a quotient of holomorphic functions using Fredholm-Smithies theory.

Unified description of resonances and antiresonances in the complex angular momentum plane.

Abstract

The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda,k) associated with physically relevant classes N_{w,alpha}^gamma of nonlocal potentials in corresponding domains D_{gamma,alpha}^delta of the space C^2 of the complex angular momentum lambda and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Theta(lambda,k)/sigma(lambda,k) of two holomorphic functions in D_{gamma,alpha}^delta is obtained by using the Fredholm-Smithies theory for complex k, at first for lambda=l integer, and in a second step for lambda complex (Real(lambda)>-1/2). Finally, we justify the "Watson resummation" of the partial wave amplitudes in an angular sector of the lambda-plane in terms of the various components of the polar manifold of T with equation sigma(lambda,k)=0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of lambda, this unified representation of the singularities of T also provides an attractive possible description of antiresonances in the lower half-plane of lambda. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of antiresonances.