Four-Parameter Point-Interaction in 1-D Quantum Systems

TL;DR

This paper develops a four-parameter family of point interactions for a quantum particle on a line, generalizing delta and delta'-interactions, and characterizes the corresponding self-adjoint Hamiltonians.

Contribution

It introduces a new four-parameter point-interaction model as a limit of short-range interactions, extending the class of solvable quantum systems on a line.

Findings

Constructed the four-parameter point-interaction as a zero-range limit.

Identified conditions for delta and delta'-interactions within this framework.

Established the self-adjoint Hamiltonian corresponding to the four-parameter interaction.

Abstract

We construct a four-parameter point-interaction for a non-relativistic particle moving on a line as the limit of a short range interaction with range tending toward zero. For particular choices of the parameters, we can obtain a delta-interaction or the so-called delta'-interaction. The Hamiltonian corresponding to the four-parameter point-interaction is shown to correspond to the four-parameter self-adjoint Hamiltonian of the free particle moving on the line with the origin excluded.