Non-local approximations for the Schrödinger operator with a general point interaction via the non-relativistic limit

We consider the one-dimensional Dirac operator $D_{A,c}$ with a general, not necessarily self-adjoint, point interaction at the origin, whose strength is described by an arbitrary complex $2\times 2$ matrix $A$ and suitably rescaled by the speed of light $c$, together with its non-local approximations $D_{A,c}^ε$ obtained by replacing the Dirac distribution by a rank-one projection onto a scaled function $v_ε$. It is known that $D_{A,c}^ε\to D_{A,c}$ as $ε\to 0+$ and that $D_{A,c}-c^2/2$ converges, as $c\to+\infty$, to the Schrödinger operator $H_A$ with the general point interaction, both in the norm resolvent sense. We prove that the first convergence is uniform with respect to $c$ in a neighbourhood of $+\infty$, and we provide explicit rates of convergence. Combining this with an explicit computation of the non-relativistic limit of $D_{A,c}^ε$ for fixed $ε$, we show that the two limits can be interchanged and that the corresponding double limit exists. As a by-product we obtain a family of non-local, rank-two approximations $H_A^ε$ of $H_A$ that converge to $H_A$ in the norm resolvent sense. Contrary to the relativistic setting, the coupling constants of $H_A^ε$ are renormalized; we give an explicit formula for this renormalization.