Quantum mechanical threshold resonances for unsymmetric potential energy barriers

We consider quantum mechanical scattering by unsymmetric one-dimensional potential energy functions, and we locate poles of the scattering matrix in the complex energy plane for potential functions that exhibit two local maxima. For such cases, by using especially stable techniques for integrating the Schrodinger equation at complex energies, two quantum mechanical resonances are located, and the real parts of the resonance energies correlate well with the energies of the potential maxima. As the potential function is continuously transformed into one having a single maximum, the real parts of the two resonance energies approach each other, and the nearest pole to the real axis makes the dominant contribution to the observable collisional delay time. In the limit where a double barrier is transformed into a single barrier, the real parts of the two barrier poles become nearly equal, and the imaginary parts tend toward a ratio of about 3. These barrier resonances are further characterized by considering the relationship between the resonance width and the reactive delay time.