  Quantum scattering in the presence of a potential valley followed by a
barrier is examined for the case of a Morse potential, for which exact analytic
solutions to the Schr\UNICODE{0xf6}dinger equation are known in terms of
confluent hypergeometric functions. For our application the potential is
characterized by three parameters: the height of the barrier, the distance of
the barrier from the origin of the radial variable $r,$ and a diffuseness
parameter. The wave function, defined in the interval $0\leq r<\infty ,$ is
required to vanish at $r=0,$ and hence represents a radial partial wave for
zero angular momentum. The vanishing at $r=0$ requires a special combination of
hypergeometric functions, and can lead to resonances for incident energies
which occur below the top of the barrier. Numerical values for the analytical
phase shifts are presented in and outside the resonant regions, and the
corresponding properties of the scattering S-matrix are examined in the complex
momentum plane, mainly for pedagogical reasons. The validity of the
Breit-Wigner approximation to the resonant phase shifts is tested, and the
motion of a ''resonant'' wave packet slowly leaking out of the valley region is
also displayed.
