Two-parameter Asymptotics in Magnetic Weyl Calculus

TL;DR

This paper develops magnetic Weyl calculus for small parameter asymptotics in quantum systems, deriving new expansions and applying them to connect Dirac and Pauli equations with high precision.

Contribution

It introduces a two-parameter magnetic Weyl calculus, deriving novel asymptotic expansions for the magnetic Weyl product, and applies these to quantum equations.

Findings

Derived three new asymptotic expansions for magnetic Weyl products.

Connected magnetic Weyl calculus results with ordinary Weyl calculus.

Obtained a high-order scaling limit from Dirac to Pauli equations.

Abstract

This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter \eps, the case of small coupling $\lambda$ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two H\"ormander class symbols are proven: (i) \eps \ll 1 and \lambda \ll 1, (ii) \eps \ll 1 and \lambda = 1 as well as (iii) \eps = 1 and \lambda \ll 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and versatility of magnetic Weyl calculus, I derive the semirelativistic Pauli equation as a scaling limit from the Dirac equation up to errors of 4th order in 1/c.