The Mathematical Formalism of a Particle in a Magnetic Field

Măntoiu, Marius; Purice, Radu

doi:10.1007/3-540-34273-7_30

Cited by 9 publications

(22 citation statements)

References 21 publications

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“…A usual choice of the vector potential is the transversal gauge: verifying x · A(x) = 0. A Hamiltonian system is described by a smooth function h : Ξ → R, where Ξ := X × X * is the phase space of the system, with X * the dual of X as a finite dimensional real vector space, with the duality map < ·, · >: X * × X → R (see [11] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Magnetic Pseudodifferential Operators

Iftimie¹,

Măntoiu²,

Purice³

2007

Publ. Res. Inst. Math. Sci.

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230

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In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in R n under the influence of a variable magnetic field B. It incorporates phase factors defined by B and reproduces the usual Weyl calculus for B = 0. In the present article we develop the classical pseudodifferential theory of this formalism for the standard symbol classes S m ρ,δ . Among others, we obtain properties and asymptotic developments for the magnetic symbol multiplication, existence of parametrices, boundedness and positivity results, properties of the magnetic Sobolev spaces. In the case when the vector potential A has all the derivatives of order ≥ 1 bounded, we show that the resolvent and the fractional powers of an elliptic magnetic pseudodifferential operator are also pseudodifferential. As an application, we get a limiting absorption principle and detailed spectral results for self-adjoint operators of the form H = h(Q, Π A ), where h is an elliptic symbol, Q denotes multiplication with the variables Π A = D − A, D is the operator of derivation and A is the vector potential corresponding to a short-range magnetic field.

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Section: Introductionmentioning

confidence: 99%

Magnetic Pseudodifferential Operators

Iftimie¹,

Măntoiu²,

Purice³

2007

Publ. Res. Inst. Math. Sci.

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230

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“…We shall work on the phase space Ξ := X ×X * ≡ R n ×R n and use systematically notations of the form X = (x, ξ), Y = (y, η), ... for its points. We shall consider classical Hamiltonians h : Ξ → R (not having a simple specific form), defined on the phase space, smooth magnetic fields B (closed 2-forms with bounded derivatives of any order) and quantum Hamiltonians H A ≡ Op A (h) defined by a choice of a vector potential A (with B = dA) 13,15,16 . Our aim is to study the continuity properties of the spectrum σ(H A ) as a subset of R when both the symbol and the magnetic field B depend on a parameter ǫ belonging to some interval I.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the continuity of spectra for families of magnetic pseudodifferential operators

Athmouni

Măntoiu

Purice

2010

Journal of Mathematical Physics

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“…In this Section we recall the structure of the observable algebras of a particle in a variable magnetic field, both from a classical and a quantum point of view. We follow the references [13,14,15] which contain further details and technical developments. Our main purpose is to introduce the basic objects that will be used subsequently and to give motivations.…”

Section: Quantization Of Observablesmentioning

confidence: 99%

“…The most satisfactory approach to introduce norms would be by using twisted C * -dynamical systems and twisted crossed products, as in [14,15,16]. To spare space and especially to avoid using non-trivial facts about C * -algebras, we shall borrow the needed structures from representations.…”

Section: Quantization Of Observablesmentioning

confidence: 99%

“…Classically, the magnetic field changes the geometry of the phase-space and this is implemented by a modification of the standard symplectic form. Consequently, it also modifies the Poisson algebra structure of the smooth functions on phase-space, which are interpreted as classical observables, see [18,14,15] for details. At the quantum level, one introduces algebras of observables defined only in terms of the magnetic field [5,6,13,15,19].…”

Section: Introductionmentioning

confidence: 99%

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Coherent States and Pure State Quantization in the Presence of a Variable Magnetic Field

Măntoiu

Purice

Richard

2011

Int. J. Geom. Methods Mod. Phys.

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Starting with a previously constructed family of coherent states, we introduce the Berezin quantization for a particle in a variable magnetic field and we show that it constitutes a strict quantization of a natural Poisson algebra. The phase-space reinterpretation involves a magnetic version of the Bargmann space and leads naturally to Berezin-Toeplitz operators.2000 Mathematics Subject Classification: 81S10, 46L65.

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The Mathematical Formalism of a Particle in a Magnetic Field

Cited by 9 publications

References 21 publications

Magnetic Pseudodifferential Operators

Magnetic Pseudodifferential Operators

On the continuity of spectra for families of magnetic pseudodifferential operators

Coherent States and Pure State Quantization in the Presence of a Variable Magnetic Field

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