One-dimensional Schrodinger operators with singular potentials: A Schwartz distributional formulation

Using an extension of the Hormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schrodinger operators with singular potentials. This formulation is entirely defined in terms of standard 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 Schrodinger 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 Schrodinger operators with a point interaction. Finally, the delta and delta' potentials are studied in detail, and the corresponding Schrodinger operators are shown to coincide with the norm resolvent limit of specific sequences of Schrodinger operators with regular potentials. (C) 2016 Elsevier Inc. All rights reserved.