Dirac equation with complex potentials

TL;DR

This paper investigates the (2+1)-dimensional Dirac equation with complex scalar potentials, demonstrating exact solutions with real eigenvalues and analytically obtaining zero-energy solutions, revealing new solvable cases in relativistic quantum mechanics.

Contribution

It introduces new exactly solvable complex potentials in the Dirac equation and interprets resulting Schrödinger-like equations as energy-dependent solvable models.

Findings

Exact analytical solutions with real eigenvalues for certain complex potentials.

Zero energy solutions can be obtained analytically for specific potentials.

Effective Schrödinger-like equations are energy-dependent and exactly solvable.

Abstract

We study $(2+1)$ dimensional Dirac equation with complex scalar and Lorentz scalar potentials. It is shown that the Dirac equation admits exact analytical solutions with real eigenvalues for certain complex potentials while for another class of potentials zero energy solutions can be obtained analytically. For the scalar potential cases, it has also been shown that the {\it effective} Schr\"odinger-like equations resulting from decoupling the spinor components can be interpreted as exactly solvable energy dependent Schr\"odinger equations.