Two-parameter Asymptotics in Magnetic Weyl Calculus

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 Hörmander class symbols are proven: (i) \eps \ll 1 and λ\ll 1, (ii) \eps \ll 1 and λ= 1 as well as (iii) \eps = 1 and λ\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.

37 pages (shortened version, fixed typos)

References in corpus (6)

• The Magnetic Weyl Calculus
• Magnetic Pseudodifferential Operators
• Commutator Criteria for Magnetic Pseudodifferential Operators
• Quantum Magnetic Algebra and Magnetic Curvature
• Strict Deformation Quantization for a Particle in a Magnetic Field
• Cotangent bundle quantization: Entangling of metric and magnetic field

Cited by in corpus (5)

• Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
• Dirac's magnetic monopole and the Kontsevich star product
• A Calculus for Magnetic Pseudodifferential Super Operators
• Semi- and Non-relativistic Limit of the Dirac Dynamics with External Fields
• Gradient expansion of the non-Abelian gauge-covariant Moyal star-product