The regulated four parameter one dimensional point interaction

TL;DR

This paper introduces a regulated four-parameter point interaction in one-dimensional quantum mechanics, enabling exact solutions and proposing a new asymptotically free formulation that aligns with physical predictions.

Contribution

It presents a novel regulation method for the four-parameter point interaction, allowing both exact and perturbative solutions, and explores the implications of asymptotic freedom in this context.

Findings

Exact solutions for the regulated interaction are obtained.

A new formulation achieves asymptotic freedom while preserving physics.

Perturbative methods now align with the exact solutions.

Abstract

The general four parameter point interaction in one dimensional quantum mechanics is regulated. It allows the exact solution, but not the perturbative one. We conjecture that this is due to the interaction not being asymptotically free. We then propose a different breakup of unperturbed theory and interaction, which now is asymptotically free but leads to the same physics. The corresponding regulated potential can be solved both exactly and perturbatively, in agreement with the conjecture.