Norm resolvent convergence of singularly scaled Schrödinger operators and δ'-potentials

Original · EN

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as εgoes to 0, of a family Sεof one-dimensional Schrödinger operators on the line of the form Sε:= -D² + ε⁻² V(x/ε). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of Sεmight be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ'.