Potential approximations to $\delta'$: an inverse Klauder phenomenon with norm-resolvent convergence

TL;DR

This paper demonstrates how a family of scaled Schrödinger operators can approximate the $oldsymbol{ ext{delta}}'$-interaction Hamiltonian in the norm-resolvent sense, revealing unique convergence properties contrary to the Klauder phenomenon.

Contribution

It introduces a novel approximation scheme for the $oldsymbol{ ext{delta}}'$-interaction Hamiltonian using scaled potentials, with proven norm-resolvent convergence.

Findings

Approximation converges in the norm-resolvent sense.

Convergence properties are opposite to the Klauder phenomenon.

Based on a scheme by Cheon and Shigehara.

Abstract

We show that there is a family Schroedinger operators with scaled potentials which approximates the $\delta'$-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon.