Phase Shifts, the Friedel Sum Rule, Resonant Tunneling through Asymmetric Potentials

Phase shifts are defined for the scattering off an asymmetric potential in a one dimensional space. The Friedel sum rule is derived for such a system, and the resonant tunneling through the asymmetric double-barrier potential is analysed. When the potential goes to symmetric, the newly defined phase shifts do not tend to those defined by Kawabata for the symmetric potential, although the physical quantities expressed in terms of the phase shifts are the same in this limit.