Commutator Criteria for Magnetic Pseudodifferential Operators

Iftimie, Viorel; Măntoiu, Marius; Purice, Radu

doi:10.1080/03605301003717118

Cited by 34 publications

(120 citation statements)

References 18 publications

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“…A recent result by Iftime, Măntoiu and Purice [15] suggests that under these circumstances (H Z is elliptic and selfadjoint operatorvalued) (H Z −z) (−1)B always exists and is a Hörmander symbol even in the presence of a magnetic field. We reckon their result extends to the case of operator-valued symbols, but seeing how tedious the proof is, we simply stick to the procedure used by Panati, Spohn and Teufel [27,Lemma 5.17].…”

Section: The Dynamics In the Almost Invariant Subspacementioning

confidence: 99%

“…Hörmander symbols are preserved under the magnetic Weyl product and quantizations of real-valued, elliptic Hörmander symbols of positive order m define selfadjoint operators on the mth magnetic Sobolev space [14]. A magnetic version of the Caldéron-Vaillancourt theorem [14] and commutator criteria [15] show the interplay between properties of magnetic pseudodifferential operators and their associated symbols.…”

Section: This Is Generically False If We Quantizementioning

confidence: 99%

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Applications of Magnetic Ψdo Techniques to Sapt

Nittis

Lein

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].

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Section: The Dynamics In the Almost Invariant Subspacementioning

confidence: 99%

Section: This Is Generically False If We Quantizementioning

confidence: 99%

Applications of Magnetic Ψdo Techniques to Sapt

Nittis

Lein

2011

Rev. Math. Phys.

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“…In this section, we apply the above abstract results to the magnetic Weyl calculus developed in a series of papers including for instance [1,2,4,10,11,16,17].…”

Section: Applications To the Magnetic Weyl Calculusmentioning

confidence: 97%

Algebras of symbols associated with the Weyl calculus for Lie group representations

Beltiţă

2011

Monatsh Math

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We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we prove that the modulation space of symbols M ∞,1 is an associative Banach algebra and the corresponding operators are bounded. We then apply the abstract results to two classes of representations, namely the unitary irreducible representations of nilpotent Lie groups, and the natural representations of the semidirect product groups that govern the magnetic Weyl calculus. The classical Weyl-Hörmander calculus is obtained for the Schrödinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J. Sjöstrand (Math Res Lett 1(2): [185][186][187][188][189][190][191][192] 1994).

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“…In fact it is well known that the evolution of a particle in a magnetic field that does not vanish at infinity is completely different from the evolution in the absence of the magnetic field. The difficulties generated by the presence of the magnetic factor can be seen already in the magnetic pseudodifferential calculus we have developed [10,11] and in the study of the Bony type of Fourier integral operators we have introduced in [11]. The techniques we use in order to study our FIO are in fact usual in the existing FIO theories: integration by parts for estimation of oscillatory integrals, but the divergent terms produced by the derivatives of the magnetic factor need a special analysis that is one of the main technical points in our paper.…”

Section: Introductionmentioning

confidence: 98%

Magnetic Fourier integral operators

Iftimie

Purice

2011

J. Pseudo-Differ. Oper. Appl.

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In some previous papers we have defined and studied a 'magnetic' pseudodifferential calculus as a gauge covariant generalization of the Weyl calculus when a magnetic field is present. In this paper we extend the standard Fourier integral operators theory to the case with a magnetic field, proving composition theorems, continuity theorems in 'magnetic' Sobolev spaces and Egorov type theorems. The main application is the representation of the evolution group generated by a 1-st order 'magnetic' pseudodifferential operator (in particular the relativistic Schrödinger operator with magnetic field) as such a 'magnetic' Fourier integral operator. As a consequence of this representation we obtain some estimations for the distribution kernel of this evolution group and a result on the propagation of singularities.

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Commutator Criteria for Magnetic Pseudodifferential Operators

Cited by 34 publications

References 18 publications

Applications of Magnetic Ψdo Techniques to Sapt

Applications of Magnetic Ψdo Techniques to Sapt

Algebras of symbols associated with the Weyl calculus for Lie group representations

Magnetic Fourier integral operators

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