One-dimensional Schr\"odinger operators with singular potentials: A Schwartz distributional formulation

Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz} distributions and does not require (as some previous approaches) the use of more general distributions or generalized functions. We determine, in the new formulation, the action and domain of the Schr\"odinger operators with arbitrary singular boundary potentials. We also consider the inverse problem, and obtain a general procedure for constructing the singular (pseudo) potential that imposes a specific set of (local) boundary conditions. This procedure is used to determine the boundary operators for the complete four-parameter family of one-dimensional Schr\"odinger operators with a point interaction. Finally, the $\delta$ and $\delta'$ potentials are studied in detail, and the corresponding Schr\"odinger operators are shown to coincide with the norm resolvent limit of specific sequences of Schr\"odinger operators with regular potentials.