Two-parameter asymptotics in magnetic Weyl calculus

2011

Rev. Math. Phys.

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In this review, we show how advances in the theory of magnetic pseudodifferential operators (magnetic ΨDO) can be put to good use in space-adiabatic perturbation theory (SAPT). As a particular example, we extend results of [24] to a more general class of magnetic fields: we consider a single particle moving in a periodic potential which is subjectd to a weak and slowly-varying electromagnetic field. In addition to the semiclassical parameter ε ≪ 1 which quantifies the separation of spatial scales, we explore the influence of an additional parameter λ that allows us to selectively switch off the magnetic field.We find that even in the case of magnetic fields with components in C ∞ b (R d ), e. g. for constant magnetic fields, the results of Panati, Spohn and Teufel hold, i. e. to each isolated family of Bloch bands, there exists an associated almost invariant subspace of L 2 (R d ) and an effective hamiltonian which generates the dynamics within this almost invariant subspace. In case of an isolated non-degenerate Bloch band, the full quantum dynamics can be approximated by the hamiltonian flow associated to the semiclassical equations of motion found in [24].

“…In [17] it was shown that for Hörmander-class symbols, this product has an asymptotic development in ε and λ,…”

Section: Ordinary Magnetic Weyl Calculusmentioning

confidence: 99%

Section: Assumption 12 (Electromagnetic Fields)mentioning

confidence: 99%

“…contributes only finitely many terms in λ for fixed power of n of ε [17], we can order the terms of the expansion of π and u in powers of λ, e. g.…”

Section: The Dynamics In the Almost Invariant Subspacementioning

confidence: 99%

Section: Equivariant Magnetic Weyl Calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Applications of Magnetic Ψdo Techniques to Sapt

Nittis

2011

Rev. Math. Phys.

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Semi- and Non-Relativistic Limit of the Dirac Dynamics with External Fields

Fürst

2012

Ann. Henri Poincaré

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We show how to approximate Dirac dynamics for electronic initial states by semi-and non-relativistic dynamics. To leading order, these are generated by the semi-and non-relativistic Pauli hamiltonian where the kinetic energy is related to m 2 + ξ 2 and 1 2m ξ 2 , respectively. Higher-order corrections can in principle be computed to any order in the small parameter v /c which is the ratio of typical speeds to the speed of light. Our results imply the dynamics for electronic and positronic states decouple to any order in v /c ≪ 1.To decide whether to get semi-or non-relativistic effective dynamics, one needs to choose a scaling for the kinetic momentum operator. Then the effective dynamics are derived using space-adiabatic perturbation theory by Panati et. al with the novel input of a magnetic pseudodifferential calculus adapted to either the semi-or non-relativistic scaling. Contents Structure of the paperThe paper consists of 6 Sections: first, we discuss the issue of the choice of small parameter and scalings in Section 2. Then we proceed to explain why the c → ∞ limit can be seen as an adiabatic limit. Sections 4 and 5 contain the main results of this work, the derivation of semi-and non-relativistic limits of the dynamics, respectively. In addition, we give some spectral results. Lastly, in Section 6, we compare and contrast our results to previous works. AcknowledgementsThe authors would like to thank Harald Lesch, Radu Purice, Herbert Spohn and Stefan Teufel for useful discussions, comments, references and remarks as well as Friedrich Gesztesy for providing several useful references. M. L. appreciates financial support from the German-Israeli Foundation. M. F. acknowledges financial support from the DFG cluster of excellence »Origin and Structure of the Universe«. Choice of small parameter and scalingsDeciding for a small parameter is crucial since it implicitly contains the physical mechanism that is responsible for the decoupling of electronic and positronic degrees of freedom. Looking at the Dirac hamiltonian,we see there are four potential parameters, the particle mass m, the semiclassical parameter ǫ, the particle's charge e and the speed of light c.

A calculus for magnetic pseudodifferential super operators

Lee

Journal of Mathematical Physics

2022

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This work develops a magnetic pseudodifferential calculus for super operators Op A( F); these map operators onto operators (as opposed to L p functions onto L q functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators [Formula: see text] defined in terms of a magnetic pseudodifferential operator op A( h). Our work combines ideas from the magnetic Weyl calculus developed by Măntoiu and Purice [J. Math. Phys. 45, 1394–1417 (2004)]; Iftimie, Măntoiu, and Purice [Publ. Res. Inst. Math. Sci. 43, 585–623 (2007)]; and Lein (Ph.D. thesis, 2011) and the pseudodifferential calculus on the non-commutative torus from the work of Ha, Lee, and Ponge [Int. J. Math. 30, 1950033 (2019)]. Thus, our calculus is inherently gauge-covariant, which means that all essential properties of Op A( F) are determined by properties of the magnetic field B = d A rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator.