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.
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
We prove the coincidence of the two definitions of the integrated density of states (IDS) for Schrödinger operators with strongly singular magnetic fields and scalar potentials: the first one using the counting function of eigenvalues of the induced operator on a bounded open set with Dirichlet boundary conditions, the second one using the spectral projections of the whole space operator. Thus we generalize a result of [5], where the scalar potential was non-negative. Moreover, we prove the existence of IDS for the case of periodical magnetic field and scalar potential. §1. Introduction One considers the vector potential a = (a 1 , . . . , a n ) :(which is identified to the differential form 1≤j≤n a j dx j ) and the scalar potential V : R n → R satisfying the following hypotheses:loc (R n ) and V − := max(0, −V ) belongs to the Kato class K n , that is, one has:
We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudodifferential calculus for weak magnetic fields with no spatial decay conditions, when the nonmagnetic symbols have a certain spatial periodicity. We show in great generality that the symbol of the magnetic band Hamiltonian admits a convergent expansion. Moreover, if the non-magnetic band Hamiltonian admits a localized composite Wannier basis, we show that the magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix. In addition, if the magnetic field perturbation is slowly variable, then the spectrum of this matrix is close to the spectrum of a Weyl quantized, minimally coupled symbol.
We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schrödinger operators with magnetic fields and scalar potentials introduced in [21,22], the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes the results of [10,20] for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus developed in [21], as well as a Feynman-Kac-Itô formula for Lévy processes [19,22]. In addition, in case when both the magnetic field and the scalar potential are periodic, one also proves the existence of the IDS.
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