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.

In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on "the minimal coupling principle" at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product. from the Schwartz space S(R N ) to its dual. This is based on a study of the distribution kernel of this operator. These and some other fundamental facts are also considered in Section 3.Beyond these operators lies a symbolic calculus which is manifestly gauge invariant, being defined only in terms of the magnetic field. The composition is a magnetic correction of the Moyal product, while the involution is just the usual complex conjugation of functions. In Section 4 we study this symbolic calculus. Once again an important problem is to extend the formulae when obvious integrability conditions are not satisfied. This is a more complicated task for the product than it was for the quantization itself, especially if one aims at obtaining * -algebras. We postpone the application of the machinery of oscillatory integrals and classical symbol function spaces to a future article. For our present purposes the strategy of extension by duality methods (see [12], [13] for the non-magnetic case) is more fruitful. It will lead to a large, interesting * -algebra of distributions which will be called the magnetic Moyal algebra.Of course, for certain problems, a well-justified norm on (restricted) * -algebras of symbols could be very useful. It happens that this is easier to achieve after performing a partial Fourier transform. Surprisingly, one naturally encounters certain C * -algebras which were studied in pure mathematics, with little connection with physics. These are special types of twisted crossed products, associated to twisted actions of R N on suitable abelian C * -algebras of position observables. They were already related to quantum magnetic fields in [23]; see also [11], [5] and [6] for related works. In a future publication we shall extend their study and outline the connection with our pseudodifferential calculus.Concerning the difference between the expressions (0.2) (or (0.3)) and (0.4) some comments are necessary. In Subsection 3.4 we shall outline some situations when they give the same result. By admitting the convenient assumption that the components of A are smooth functions with tempered growth, both (0.2) and (0.4) can be extended to any tempered distribution f . Then we shall have Op A (f ) = Op A (f ) for all f if and only if A is linear (this is one of the most important cases appearing usually in the literature). Remark that this condition corresponds to a constant magnetic field, but it is not gauge invariant: for some other A ′ with dA ′ = B = const, Op A ′ and Op A ′ will be diffe...

The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudo-differential operators of suitable symbol classes. The approach is completely intrinsic; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudo-differential operator.Comment: Acknowledgements adde

Recently, we introduced a mathematical framework for the quantization of a particle in a variable magnetic field. It consists in a modified form of the Weyl pseudodifferential calculus and a C * -algebraic setting, these two points of view being isomorphic in a suitable sense. In the present paper we leave Planck's constant vary, showing that one gets a strict deformation quantization in the sense of Rieffel. In the limit → 0 one recovers a Poisson algebra induced by a symplectic form defined in terms of the magnetic field.

We analyze the Ginzburg-Landau equation for a superconductor in the case of a 2-dimensional model: a cylindrical conductor with a magnetic field parallel to the axis. This amounts to find the extrema of the free energywhere Ω is a bounded domain with smooth boundary in IR 2 , A = (A ί9 A 2 ) the vector potential, B A = d 1 A 2 -d 2 A ί the magnetic field, Φ Ά complex field. We describe the connected components of the maximal configuration space, i.e. of the set of all {A, Φ) with components in the Sobolev space H ί (Ω) and such that | Φ\ = 1 on the boundary, modulo the action of the gauge group. In the critical case K = 1 we give a complete description of the minimal configurations in each component.

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We consider anisotropic Schrödinger operators H = −∆ + V in L 2 (Ê n ).

We consider a periodic Schr\"odinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schr\"odinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.Comment: 55 pages, to appear in J. Funct. Ana