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Radu Purice

Publications and source records attributed to Radu Purice.

At least 19 recordsLinked to original sources
arXiv2026

A fresh look at the Peierls-Onsager substitution

We formulate a general version of the Peierls-Onsager substitution for a finite family of Bloch eigenvalues under a local spectral gap hypothesis, via strongly localized tight-frames and magnetic matrices. This extends the existing results to long-range magnetic fields without any slow-variation hypothesis and without any triviality assumption for the associated Bloch sub-bundle. Moreover, our results cover a large class of periodic, elliptic pseudo-differential operators. We also prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.

math-ph↗
arXiv2025

The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators

We study the structure of the fibre operators corresponding to periodic magnetic pseudo-differential operators having periodic magnetic potentials. We obtain explicit formulas for their distribution kernel, both when these fibres are seen as operators on the $d$-dimensional torus, and also when they are seen as infinite matrices acting on a discrete $\ell^2$ space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.

math.AP↗
arXiv2024

Spectral regularity with respect to dilations for a class of pseudodifferential operators

We continue the study of the perturbation problem discussed in \cite{CP3} and get rid of the 'slow variation' assumption by considering symbols of the form $a\big(x+\delta\,F(x),\xi\big)$ with $a$ a real H\"{o}rmander symbol of class $S^0_{0,0}(\mathbb{R}^d\times\mathbb{R}^d)$ and $F$ a smooth function with all its derivatives globally bounded, with $|\delta|\leq1$. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of $\delta=0$ is still of the order $\sqrt{|\delta|}$, the distance between their spectral edges behaves like $|\delta|^\nu$ with $\nu\in[1/2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.

math.AP↗
arXiv2024

A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields

We consider periodic (pseudo)differential {elliptic operators of Schr\"odinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.

math-ph↗
arXiv2023

Sharp spectral stability for a class of singularly perturbed pseudo-differential operators

Let $a(x,\xi)$ be a real H\"ormander symbol of the type $S_{0,0}^0(\mathbb{R}^{d}\times \mathbb{R}^d)$, let $F$ be a smooth function with all its derivatives globally bounded, and let $K_\delta$ be the self-adjoint Weyl quantization of the perturbed symbols $a(x+F(\delta\, x),\xi)$, where $|\delta|\leq 1$. First, we prove that the Hausdorff distance between the spectra of $K_\delta$ and $K_{0}$ is bounded by $\sqrt{|\delta|}$, and we give examples where spectral gaps of this magnitude can open when $\delta\neq 0$. Second, we show that the distance between the spectral edges of $K_\delta$ and $K_0$ (and also the edges of the inner spectral gaps, as long as they remain open at $\delta=0$) are of order $|\delta|$, and give a precise dependence on the width of the spectral gaps.

math-ph↗
arXiv2022

Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of H\"{o}rmander type pseudo-differential operators on $\mathbb{R}^d$ ($d\geq1$), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, 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 Calder{\'o}n-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.

math.AP↗
arXiv2020

Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field

This is the last paper in a series of three in which we have studied the Peierls substitution in the case of a weak magnetic field. Here we deal with two $2d$ Bloch eigenvalues which have a conical crossing. It turns out that in the presence of an almost constant weak magnetic field, the spectrum near the crossing develops gaps which remind of the Landau levels of an effective mass-less magnetic Dirac operator.

math-ph↗
arXiv2018

Quantum observables as magnetic pseudodifferential operators

In a series of papers we have argued that the 'basic' physical procedure of minimal coupling giving the quantum description of a Hamiltonian system interacting with a magnetic field, can be given a very satisfactory mathematical formulation as a twisted Weyl quantization \cite{MP2}. In this paper we shall present a review of some of these results with some modified proofs that allow a special focus on the dependence on the behavior of the magnetic field, having in view possible developments towards problems with unbounded magnetic fields. The main new result is contained in Theorem 2.9 and states that the the symbol of the evolution group of the self-adjoint operator defined by a real elliptic symbol of strictly positive order in a smooth bounded magnetic field is in the associated magnetic Moyal algebra, i.e. leaves invariant the space of Schwartz test functions and its dual

math-ph↗
arXiv2017

Peierls' substitution for low lying spectral energy windows

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.

math-ph↗
arXiv2017

Low lying spectral gaps induced by slowly varying magnetic fields

We consider a periodic Schrödinger 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 Schrödinger 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.

math.SP↗
arXiv2015

Peierls substitution and magnetic pseudo-differential calculus

We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudo-differential calculus for weak magnetic fields with no spatial decay conditions, when the non-magnetic 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.

math-ph↗
arXiv2014

Spectral edge regularity of magnetic Hamiltonians

We analyse the spectral edge regularity of a large class of magnetic Hamiltonians when the perturbation is generated by a globally bounded magnetic field. We can prove Lipschitz regularity of spectral edges if the magnetic field perturbation is either constant or slowly variable. We also recover an older result by G. Nenciu who proved Lipschitz regularity up to a logarithmic factor for general globally bounded magnetic field perturbations.

math.SP↗
arXiv2013

The Peierls-Onsager Effective Hamiltonian in a complete gauge covariant setting: Description of the spectrum

Using the procedures in \cite{Bu} and \cite{GMS} and the magnetic pseudodifferential calculus we have developped in \cite{MP1,MPR1,IMP1,IMP2} we construct an effective Hamitonian that describes the spectrum in any compact subset of the real axis for a large class of periodic pseudodifferential Hamiltonians in a bounded smooth magnetic field, in a completely gauge covariant setting, without any restrictions on the vector potential and without any adiabaticity hypothesis.

math-ph↗
arXiv2012

Abstract composition laws and their modulation spaces

On classes of functions defined on R^2n we introduce abstract composition laws modelled after the pseudodifferential product of symbols. We attach to these composition laws modulation mappings and spaces with useful algebraic and topological properties.

math.FA↗
arXiv2010

Magnetic Fourier Integral Operators

In some previous papers we have defined and studied a 'magnetic' pseudodifferential calculus as a gauge covariant generalization of the Weyl calculus when a magnetic field is present. In this paper we extend the standard Fourier Integral Operators Theory to the case with a magnetic field, proving composition theorems, continuity theorems in 'magnetic' Sobolev spaces and Egorov type theorems. The main application is the representation of the evolution group generated by a 1-st order 'magnetic' pseudodifferential operator (in particular the relativistic Schrödinger operator with magnetic field) as such a 'magnetic' Fourier Integral Operator. As a consequence of this representation we obtain some estimations for the distribution kernel of this evolution group and a result on the propagation of singularities.

math-ph↗