Two-parameter asymptotics in magnetic Weyl calculus

Journal Article · Wed Dec 15 00:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3499660· OSTI ID:21501237

Lein, Max ^[1]

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1. Technische Universitaet Muenchen, Zentrum Mathematik, Boltzmannstrasse 3, 85747 Garching (Germany)

This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter {epsilon}, 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 Hoermander class symbols are proven as (i) {epsilon}<< 1 and {lambda}<< 1, (ii) {epsilon}<< 1 and {lambda}= 1, as well as (iii) {epsilon}= 1 and {lambda}<< 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate the 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 fourth order in 1/c.

• OSTI ID: 21501237
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 12; Other Information: DOI: 10.1063/1.3499660; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

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