On how Jost can help Breit and Wigner to better describe a resonance

R. O. BARRACHINA; P. A. MACRI

Lanzhou

Conferencia; XXVII International Conference on the Physics of Electronic and Atomic Collisions; 2013

While the transparency of an elastic cross section at low energies is usually explained by means of a partial wave decomposition ~cite{Bohm1951}, the zero-energy resonance relies on an analysis of the poles of Scattering operator~cite{Newton1966}. In this communication we demonstrate that in spite of this dissimilar explanations, a unified description of both effect is possible. It is based on the use of Jost functions~cite{Jost1951} and the distribution of their zeros on the plane of the complex valued impulse $k$. We start by generalizing the standard effective range expansion in terms of only two relevant coefficients (the scattering length $a_ell$ and effective range $r_ell$), by incorporating up to four real parameters. At first sight this generalization might seem unnecessary, since it leads to similar results than the standard one when dealing with partial-wave cross sections $sigma_ell$. However we show that it provides the correct starting point for the unified description of zero-energy resonance and transparency effects~cite{Macri2013}.A single zero of the s-wave Jost function located near the origin of the complex $k$-plane is responsible for the zero-energy resonance~cite{Taylor1972}. However, it can be demonstrated that, contrary to the usual expectation, this single zero does not necessarily guarantee that a resonant will actually occur. In other words, the presence of an isolated low-lying bound or virtual state seems to be necessary but not sufficient for the occurrence of a zero-energy resonance, since the ``collective´´ contribution of all the other zeros of the Jost function can not always be discarded as irrelevant. Actually, the zero-energy transparency is shown to be a collective effect, depending on the sum of all the individual scattering lengths instead of a single one. Within this context, it can be demonstrated that in the zero-energy resonance and transparency effects, the $k^{4 ell}$ dependence of the $ell$-wave cross section is changed into $k^{4 (ell-1)}$ ($k^{-2}$ for $ell = 0$) or $k^{4 (ell+1)}$, respectively~cite{Macri2013}.Finally, we discuss the relevance of these results in relation with the stability of Bose-Einstein condensates and the validity~cite{Macri2003} of Wigner´s threshold law~cite{Wigner1948} at the opening of inelastic channels.