  In this paper we show how rotational bands of resonances can be described by
using trajectories of poles of the scattering amplitude in the complex angular
momentum plane: each band of resonances is represented by the evolution of a
single pole lying in the first quadrant of the plane. The main result of the
paper consists in showing that also the antiresonances (or echoes) can be
described by trajectories of the scattering amplitude poles, instead of using
the hard-sphere potential scattering as prescribed by the classical
Breit-Wigner theory. The antiresonance poles lie in the fourth quadrant of the
complex angular momentum plane, and are associated with non-local potentials
which take into account the exchange forces; it derives a clear-cut separation
between resonance and antiresonance poles. The evolution of these latter poles
describes the passage from quantum to semi-classical physics. The theory is
tested on the rotational band produced by $\alpha$-$\alpha$ elastic scattering
and on the hadronic rotational bands in $\pi^+$-p elastic scattering.

Nuclear Theory