Resurgence in quasi-classical scattering

In quasi-classical studies of closed systems, e.g. a billiard, resurgence means that the contribution of long periodic orbits to the spectral determinant can be expressed in terms of composites of short orbits, and the resulting expression for the determinant is manifestly real. The question has thus long been posed whether something like resurgence applies to a scattering system with its resonances. We find here a resurgent expression for Wigner's R-matrix (which gives the S-matrix by a Cayley transform) in which long scattering pseudo-orbits are expressed in terms of composites of short pseudo-orbits, both scattering and periodic, and the result is manifestly Hermitian, giving a unitary expression for the S-matrix. This is particularly useful in the case that the resonance width is comparable with the resonance spacing. The pseudo-orbits are defined in terms of a fictitious and to some extent arbitrary closed reference system. We relate the results to other formulations. We give a simple but non-trivial approximation for a particular example which illustrates the phenomenon of 'resonance trapping'