Mathematical justification of the Aharonov-Bohm hamiltonian

TL;DR

This paper rigorously justifies the Aharonov-Bohm Hamiltonian in quantum mechanics by analyzing limits of finite solenoids, ensuring the model's mathematical consistency in both two and three dimensions.

Contribution

It provides a rigorous mathematical derivation of the Aharonov-Bohm Hamiltonian from finite solenoid models using strong resolvent limits, clarifying its theoretical foundation.

Findings

Limits commute in both $ ext{R}^2$ and $ ext{R}^3$ spaces.

The model is justified via increasing sequences of finitely long solenoids.

Rigorous strong resolvent convergence established.

Abstract

It is presented, in the framework of nonrelativistic quantum mechanics, a justification of the usual Aharonov-Bohm hamiltonian (with solenoid of radius greater than zero). This is obtained by way of increasing sequences of finitely long solenoids together with a natural impermeability procedure; further, both limits commute. Such rigorous limits are in the strong resolvent sense and in both $\R^2$ and $\R^3$ spaces.