Marius Măntoiu

The magnetic Weyl calculus

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 formulas 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.

Magnetic Pseudodifferential Operators

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 .A mong 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.

Commutator Criteria for Magnetic Pseudodifferential Operators

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 pseudodifferential operators of suitable symbol classes; 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 pseudodifferential operator.

Strict deformation quantization for a particle in a magnetic field

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 ℏ→0 one recovers a Poisson algebra induced by a symplectic form defined in terms of the magnetic field.

On the continuity of spectra for families of magnetic pseudodifferential operators

For families of magnetic pseudodifierential operators deflned by symbols and magnetic flelds depending continuously on a real parameter †, we show that the corresponding family of spectra also varies continuously with †. 1

Spectral Analysis for Adjacency Operators on Graphs

Abstract.We put into evidence graphs with adjacency operator whose singular subspace is prescribed by the kernel of an auxiliary operator. In particular, for a family of graphs called admissible, the singular continuous spectrum is absent and there is at most an eigenvalue located at the origin. Among other examples, the one-dimensional XY model of solid-state physics is covered. The proofs rely on commutators methods.

A-Priori Decay for Eigenfunction of Perturbed Periodic Schrödinger Operators

Abstract. Recently Neumayr and Metzner [1] have shown that the connected N-point density-correlation functions of the two-dimensional and the one-dimensional Fermi gas at one-loop order generically (i.e.for nonexceptional energy-momentum configurations) vanish/are regular in the small momentum/small energy-momentum limits. Their result is based on an explicit analysis in the sequel of the results of Feldman et al. [2]. In this note we use Ward identities to give a proof of the same fact – in a considerably shortened and simplified way – for any dimension of space.

Weighted Estimations from a Conjugate Operator

In this Letter we develop a general procedure leading from a Mourre-type estimation for a given self-adjoint operator H to a Hardy-type weighted inequality. We use this method in order to prove exponential decay for eigenvectors of a large class of perturbations of operators of convolution with bounded analytic functions.

Toeplitz algebras and spectral results for the one-dimensional Heisenberg model

We determine the structure of the spectrum and obtain nonpropagation estimates for a class of Toeplitz operators acting on a subset of the lattice ZN. This class contains the Hamiltonian of the one-dimensional Heisenberg model.

On Fréchet–Hilbert algebras

We consider Hilbert algebras with a supplementary Fréchet topology and get various extensions of the algebraic structure by using duality techniques. In particular we obtain optimal multiplier-type involutive algebras which in applications are large enough to be of significant practical use. The setting covers many situations arising from quantization rules, as those involving square-integrable families of bounded operators