Quantum scattering in the presence of a potential valley followed by a barrier is examined for a Morse potential, for which exact analytical solutions are known. For our application the sign of the potential is reversed, and the wave function is required to vanish at the origin. This condition requires a special combination of hypergeometric functions, and can lead to resonances for incident energies 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 matrix are examined in the complex momentum plane. The validity of the Breit–Wigner approximation to the resonant part of the phase shifts is tested, and a new method for finding the location of narrow resonances is described. The time decay of a resonant wave packet slowly leaking out of the valley region (on a time scale proportional to the inverse of the width of the resonance) is compared with theoretical predictions, and complete agreement is not found.
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