Bound states and resonance analysis of one-dimensional relativistic parity-symmetric two-point interactions

We consider the one-dimensional Dirac equation with the most general relativistic contact interaction supported on two points symmetrically located with respect to the origin. We use a distributional method to determine the shape of the interaction, which, in the present case, is equivalent to the standard method of defining contact interactions by self-adjoint extensions of symmetric operators. The interaction on each of these two points depends on four parameters, each one having a clear physical meaning. We are interested in the scattering and confining properties of this model. We focus our attention on even or odd interactions under parity transformations and investigate the existence of critical and supercritical states, bound states, confinement, and scattering resonances for some particular interactions of special interest.