TY - JOUR

T1 - Two-parameter asymptotics in magnetic Weyl calculus

AU - Lein, Max

N1 - Funding Information:
This work was supported by a DAAD scholarship. I thank R. Littlejohn and R. Purice for their kind hospitality. I am very grateful for useful discussions, insights, and references from M. Măntoiu, T. Miyao, G. Panati, and H. Spohn.

PY - 2010/12/1

Y1 - 2010/12/1

N2 - This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter ε, the case of small coupling λ 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 as (i) ε ≪ 1 and λ ≪ 1, (ii) ε ≪ 1 and λ = 1, as well as (iii) ε = 1 and λ ≪ 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.

AB - This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter ε, the case of small coupling λ 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 as (i) ε ≪ 1 and λ ≪ 1, (ii) ε ≪ 1 and λ = 1, as well as (iii) ε = 1 and λ ≪ 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.

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U2 - 10.1063/1.3499660

DO - 10.1063/1.3499660

M3 - Article

AN - SCOPUS:78650684705

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

M1 - 123519

ER -