Norm resolvent convergence of singularly scaled Schr\"odinger operators and \delta'-potentials

TL;DR

This paper proves that a family of scaled Schr"odinger operators converges in norm resolvent sense to an operator with a '-potential, providing a rigorous mathematical foundation for this physically motivated singular limit.

Contribution

It establishes the norm resolvent convergence of scaled Schr"odinger operators to a '-potential operator, clarifying the mathematical interpretation of '-potentials.

Findings

Convergence holds for potentials in the Faddeev-Marchenko class.

The scaled operators converge to a '-potential operator.

Provides a rigorous framework for '-potentials in quantum mechanics.

Abstract

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon of one-dimensional Schr\"odinger operators on the line of the form S_\epsilon:= -D^2 + \epsilon^{-2} V(x/\epsilon). 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 S_\epsilon might be considered as a "physically motivated" interpretation of the one-dimensional Schr\"odinger operator with potential \delta'.