The bound state problem for a Dirac particle in an external static charge distribution in (1+1) dimensions

We study the self-interaction effects for the Dirac particle moving in an external field created by static charges in (1 + 1) dimensions. Assuming that the total electric charge of the system vanishes, we show that the asymptotically linearly rising part of the external potential responsible for the nonexistence of bound states in the external field problem without self-interaction is cancelled by the self-potential of the zero mode of the Dirac particle charge density. We derive the Dirac equation which includes the self-potential of the non-zero modes and is nonlinear. We solve the spectrum problem in the case of two external positive charges of the same value and prove that the Dirac particle and external charges are confined in a stable system.