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The non autonomous Cauchy problem for time dependent 1D point interactions isconsidered. The regularity assumptions for the coupling parameter areaccurately analyzed and show that the general results for non autonomous linearevolution equations in Banach spaces are far from being optimal. In the meantime, this article shows an unexpected application of paraproduct techniques,initiated by J.M. Bony for nonlinear partial differential equations, to aclassical linear problem.

Time-dependent delta-interactions for 1D Schr\"odinger Hamiltonians