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The Woods-Saxon potential in the Dirac equation. (English) Zbl 1041.81100
Summary: The two-component approach to the one-dimensional Dirac equation is applied to the Woods-Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at $E=-m$) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero momentum when the corresponding well supports a half-bound state at $E=-m$ is demonstrated.
##### MSC:
 81U05 2-body potential scattering theory (quantum theory) 35Q40 PDEs in connection with quantum mechanics