Dirac equation in low dimensions: The factorization method

• The low-dimensional Dirac equation in the presence of static potentials is solved.
• The factorization method is generalized for energy-dependent Hamiltonians.
• The shape invariance is generalized for energy-dependent Hamiltonians.
• The stability of the Dirac sea is related to the existence of supersymmetric partner Hamiltonians.

We present a general approach to solve the (1+1)(1+1) and (2+1)(2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.