Particle resonance in the Dirac equation in the presence of a delta interaction and a perturbative hyperbolic potential

We show that the energy spectrum of the one-dimensional Dirac equation, in the presence of an attractive vectorial delta potential, exhibits a resonant behavior when one includes an asymptotically spatially vanishing weak electric field associated with a hyperbolic tangent potential. We solve the Dirac equation in terms of Gauss hyper-geometric functions and show explicitly how the resonant behavior depends on the strength of the electric field evaluated at the support of the point interaction. We derive an approximate expression for the value of the resonances and compare the results calculated for the hyperbolic potential with those obtained for a linear perturbative potential. Finally, we characterize the resonances with the help of the phase shift and the Wigner delay time.