Quantum Reflection of S-wave Unstable States

The phase time in quantum tunneling can be disentangled into a dwell time plus a term arising due to the interference of the reflected and incident waves in front of the barrier. The interference term dominates at low energies and as E → 0, this term and hence the phase time becomes singular. With the s-wave motion in three dimensions being equivalent to that of a one-dimensional motion in the radial coordinate, a similar singularity shows up in the phase time delay of s-wave resonances. Relating the scattering matrix in three dimensional collisions to the reflection amplitude, the interference term in tunneling can be identified as a term given in terms of the transition matrix in scattering. Subtraction of this term from the phase time delay gives the dwell time delay which is finite at all energies and is useful in characterizing s-wave resonances such as the σ meson and mesic nuclei near threshold.