Regularization of energy-dependent pointlike interactions in 1D quantum mechanics

TL;DR

This paper develops a new class of hermitian, energy-dependent pointlike potentials in 1D quantum mechanics that extend the standard classification by modifying the inner product in the zero-range limit.

Contribution

It introduces a novel family of hermitian potentials with energy-dependent couplings that differ from the standard pointlike interactions by altering the inner product structure.

Findings

Constructed hermitian potentials converging to energy-dependent delta interactions.

Showed the Hamiltonian's hermiticity depends on the inner product used.

Provided a framework for finite support potentials with energy dependence.

Abstract

We construct a family of hermitian potentials in 1D quantum mechanics that converges in the zero-range limit to a $\delta$ interaction with an energy-dependent coupling. It falls out of the standard four-parameter family of pointlike interactions in 1D. Such classification was made by requiring the pointlike interaction to be hermitian. But we show that although our Hamiltonian is hermitian for the standard inner product when the range of the potential is finite, it becomes hermitian for a different inner product in the zero-range limit. This inner product attributes a finite probability (and not probability density) for the particle to be exactly located at the position of the potential. Such pointlike interactions can then be used to construct potentials with a finite support with an energy-dependent coupling.