Solutions of the Dirac equation in one-dimensional variable width potential well

The Fermi acceleration mechanism is a significant source of cosmic rays. When the width of a potential well changes over time, the velocity of particles within the well also changes. For quantum systems, such dynamics should be described by the Schr\"odinger, Klein-Gordon, and Dirac equations. Previous studies have solved the Schr\"odinger and Klein-Gordon equations under these conditions, but no research has addressed the Dirac equation for spin-$\frac{1}{2}$ particles like electrons. This paper investigates the solutions of the Dirac equation in a dynamically varying potential well and demonstrates that Dirac particles can exhibit complex-valued momentum states via the Fermi acceleration mechanism, enabling Tachyon-like states preparation.