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Non-self-adjoint relativistic point interaction in one dimension

The one-dimensional Dirac operator with a singular interaction term which is formally given by $A\otimes|δ_0\rangle\langleδ_0|$, where $A$ is an arbitrary $2\times 2$ matrix and $δ_0$ stands for the Dirac distribution, is introduced as a closed not necessarily self-adjoint operator. We study its spectral properties, find its non-relativistic limit and also address the question of regular approximations. In particular, we show that, contrary to the case of local approximations, for non-local approximating potentials, coupling constants are not renormalized in the limit.