Supercriticality and Transmission Resonances in the Dirac equation

TL;DR

This paper demonstrates that a Dirac particle with very small momentum can tunnel through a finite-range potential barrier without reflection if the potential well supports a bound state at energy -m, indicating supercriticality.

Contribution

It introduces the concept of supercritical potential wells in the Dirac equation and analyzes their transmission resonance properties.

Findings

Dirac particles tunnel without reflection through supercritical barriers.

Supercritical potential wells support bound states at energy -m.

Transmission resonances occur at specific supercritical conditions.

Abstract

It is shown that a Dirac particle of mass $m$ and arbitrarily small momentum will tunnel without reflection through a potential barrier $V=U_c(x)$ of finite range provided that the potential well $V=-U_c(x)$ supports a bound state of energy $E=-m.$ This is called a supercritical potential well.