Radu Purice - Academia.edu (original) (raw)

Papers by Radu Purice

Reviews in Mathematical Physics, Mar 19, 2019

We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial d... more We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic [Formula: see text]-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by [Formula: see text] exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in [Formula: see text]. In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on [Formula: see text] and is [Formula: see text]-periodic; if [Formula: see text], the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

Journal of spectral theory, 2015

Using the procedures in [ ] and [ ] and the magnetic pseudodi erential calculus we have developpe... more Using the procedures in [ ] and [ ] and the magnetic pseudodi erential calculus we have developped in [ , , , ] we construct an e ective Hamiltonian that describes the spectrum in any compact subset of the real axis for a large class of periodic pseudodi erential Hamiltonians in a bounded smooth magnetic eld, in a completely gauge covariant setting, without any restrictions on the vector potential and without any adiabaticity hypothesis.

HAL (Le Centre pour la Communication Scientifique Directe), 2018

We consider a 2d magnetic Schrödinger operator perturbed by a weak magnetic field which slowly va... more We consider a 2d magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a 'Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

Journal of Mathematical Physics, 2007

Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small ... more Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer-Büttiker type formula.

arXiv (Cornell University), Dec 23, 2022

In this paper we use some ideas from and consider the description of Hörmander type pseudo-differ... more In this paper we use some ideas from and consider the description of Hörmander type pseudo-differential operators on R d (d ě 1), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.

Journal of Spectral Theory, 2019

The aim of these notes is to show how the magnetic calculus developed in [MP, IMP1, IMP2, MPR, LM... more The aim of these notes is to show how the magnetic calculus developed in [MP, IMP1, IMP2, MPR, LMR] permits to give a new information on the nature of the coefficients of the expansion of the trace of a function of the magnetic Schrödinger operator whose existence was established in [HR2].

arXiv (Cornell University), Jul 31, 2009

We show coincidence of the two definitions of the integrated density of states (IDS) for a class ... more 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.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

We introduce magnetic coherent states for a particle in a variable magnetic field. They provide a... more We introduce magnetic coherent states for a particle in a variable magnetic field. They provide a pure state quantization of the phase space R 2N endowed with a magnetic symplectic form.

arXiv (Cornell University), Mar 2, 2005

We study generalised magnetic Schrödinger operators of the form H h (A, V) = h(Π A) + V , where h... more We study generalised magnetic Schrödinger operators of the form H h (A, V) = h(Π A) + V , where h is an elliptic symbol, Π A = −i∇ − A, with A a vector potential defining a variable magnetic field B, and V is a scalar potential. We are mainly interested in anisotropic functions B and V. The first step is to show that these operators are affiliated to suitable C *-algebras of (magnetic) pseudodifferential operators. A study of the quotient of these C *-algebras by the ideal of compact operators leads to formulae for the essential spectrum of H h (A, V), expressed as a union of spectra of some asymptotic operators, supported by the quasi-orbits of a suitable dynamical system. The quotient of the same C *-algebras by other ideals give localization results on the functional calculus of the operators H h (A, V), which can be interpreted as non-propagation properties of their unitary groups. 1

arXiv (Cornell University), Mar 11, 2004

There is a connection between the Weyl pseudodifferential calculus and crossed product C *-algebr... more There is a connection between the Weyl pseudodifferential calculus and crossed product C *-algebras associated with certain dynamical systems. And in fact both topics are involved in the quantization of a non-relativistic particle moving in R N. Our paper studies the situation in which a variable magnetic field is also present. The Weyl calculus has to be modified, giving a functional calculus for a family of operators (positions and magnetic momenta) with highly non-trivial commutation relations. On the algebraic side, the dynamical system is twisted by a cocycle defined by the flux of the magnetic field, leading thus to twisted crossed products. Following mainly [MP1] and [MP2], we outline the interplay between the modified pseudodifferential setting and the C *-algebraic formalism at an abstract level as well as in connection with magnetic fields.

arXiv (Cornell University), Jul 30, 2009

We extend the Bargmann transform to the magnetic pseudodifferential calculus, using gauge-covaria... more We extend the Bargmann transform to the magnetic pseudodifferential calculus, using gauge-covariant families of coherent states. We also introduce modulation mappings, a first step towards adapting modulation spaces to the magnetic case. 1

In previous papers, a generalization of the Weyl calculus was introduced in connection with the q... more 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 selfadjoint operators of the form H = h(Q, Π A), where h is an elliptic symbol, Π A = D − A and A is the vector potential corresponding to a short-range magnetic field.

ABSTRACT We consider the Dirac evolution equation in presence of a time dependent electromagnetic... more ABSTRACT We consider the Dirac evolution equation in presence of a time dependent electromagnetic potential which is a solution of the homogeneous wave equation with regular and compactly supported initial data. We prove a propagation property for the free Dirac Hamiltonian using the explicit form of the free propagator that we use together with an energy estimation and the finite propagation speed of the Dirac evolution. We prove existence and unitarity of the wave operators associated to the couple of Dirac evolutions: the free one and the time dependent one.

ABSTRACT In this paper we develop a general procedure to prove Hardy type estimations for an oper... more ABSTRACT In this paper we develop a general procedure to prove Hardy type estimations for an operator that admits a conjugate operator, starting from the Mourre estimation. We use this method for operators of convolution with analytic functions, obtaining Hardy type estimations with exponential weights, for sufficiently small exponents. Present address: Departement de Physique Th'eorique; Universit'e de Gen`eve; 32, bd. d'Yvoy; CH-1211 Gen`eve 4; SUISSE e-mail: mantoiu@kalymnos.unige.ch 1 1 INTRODUCTION The main aim of this paper is to prove weighted estimations of the type: kw 1 uk C kw 2 ((Gammair) Gamma E)uk (1.1) where (Gammair) is the convolution operator with the Fourier transform of the function , the norm is the L 2 Gammanorm on the space R n for some n 1, E is a real number that may also belong to the spectrum of the operator (Gammair) in L 2 (R n ), w 1 and w 2 are weight functions that grow at infinity and u is a function in L 2 (R n ) with support far from t...

ABSTRACT We define a conjugate operator for Schrödinger Hamiltonians with magnetic field, which a... more ABSTRACT We define a conjugate operator for Schrödinger Hamiltonians with magnetic field, which allows a non-perturbative and gauge invariant analysis for these Hamiltonians. We prove the absence of singular continuous spectrum, the discreteness and finite degeneracy of the point spectrum and a limiting absorption principle for Schrödinger Hamiltonians with magnetic field decaying slightly faster than |x| -1 , in dimension n≥2.

arXiv (Cornell University), Feb 28, 2023