Searcharxiv⌕ Search

arXiv subjects

Jean Nourrigat

Publications and source records attributed to Jean Nourrigat.

At least 19 recordsLinked to original sources
arXiv2026

Quantization and process

This article is concerned with generalizations of pseudo-differential operators in $L^2(\mathbb{R}^n)$, $n\geq 1$. The definition of the new calculi depends only on bounded measures on the phase space $\mathbb{R}^{2n}$ and each measure gives rise to a specific calculus. Quantizations of anti-Wick, Weyl, classical and Born-Jordan are particular cases of the general calculi. Classes of symbols in this framework are then studied. The Gevrey class of parameter 1/2 is a class of symbol that is common to all the general calculi, that is, a class of symbols independent on the bounded measures parametrizing the quantizations. Precise additional hypotheses on the measures are necessary in the aim to consider the larger class of symbols $L^{\infty}(\mathbb{R}^{2n})$. This result can be applied for anti-Wick but not for Weyl quantization. Concerning Weyl pseudo-differential calculus, we recover the standard class of Sj\"ostrand and Gr\"ochening. Then, we prove that probability measures of L\'evy processes on the phase space $\mathbb{R}^{2n}$ with diffusion larger than 1/4 are natural examples of measures satisfying the latter additional hypotheses in order to consider $L^{\infty}(\mathbb{R}^{2n})$ symbols. This relation is derived using the L\'evy-Khintchine formula. Composition laws in that general context are next investigated. In that purpose, we give a formula for the composition of two symbols in some precise class of symbols valid for all general quantizations. General calculi are relying on Wick quantization which is therefore primarily examined for some precise classes of symbols. Additional results in that context are provided, such as Mizrahi series expansions and Banach algebra isomorphisms between operators and symbol classes.

math.AP↗
arXiv2023

Time evolution for the Pauli-Fierz operator (Markov approximation and Rabi cycle)

This article is concerned with a system of particles interacting with the quantized electromagnetic field (photons) in the non relativistic Quantum Electrodynamics (QED) framework and governed by the Pauli-Fierz Hamiltonian. We are interested not only in deriving approximations of several quantities when the coupling constant is small but also in obtaining different controls of the error terms. First, we investigate the time dynamics approximation in two situations, the Markovian (Theorem 1.4 completed by Theorem 1.16) and non Markovian (Theorem 1.6) cases. These two contexts differ in particular regarding the approximation leading terms, the error control and the initial states. Second, we examine two applications. The first application is the study of marginal transition probabilities related to those analyzed by Bethe and Salpeter in \cite{B-S}, such as proving the exponential decay in the Markovian case assuming the Fermi Golden Rule (FGR) hypothesis (Theorem 1.17 or Theorem 1.15) and obtaining a FGR type approximation in the non Markovian case (Theorem 1.5). The second application, in the non Markovian case, includes the derivation of Rabi cycles from QED (Theorem 1.7). All the results are established under the following assumptions at some steps of the proofs: an ultraviolet and an infrared regularization are imposed, the quadratic terms of the Pauli-Fierz Hamiltonian are dropped, and the dipole approximation is assumed but only to obtain optimal error controls.

math-ph↗
arXiv2021

Markovian approximation for Pauli Fierz operators

The purpose of this article is to derive a Markovian approximation of the reduced time dynamics of observables for the Pauli-Fierz Hamiltonian with a precise control of the error terms. In that aim, we define a Lindblad operator associated to the corresponding quantum master equation. In a particular case, this allows to study the transition probability matrix.

quant-ph↗
arXiv2020

Classical and quantum energies for interacting magnetic systems

The purpose of this article is to give different interpretations of the first non vanishing term (quadratic) of the ground state asymptotic expansion for a spin system in quantum electrodynamics, as the spin magnetic moments go to $0$. One of the interpretations makes a direct link with some classical physics laws. A central role is played by an operator $A_M$ acting only in the finite dimensional spin state space and making the connections with the different interpretations, and also being in close relation with the multiplicity of the ground state.

math-ph↗
arXiv2019

Lindblad approximation and spin relaxation in quantum electrodynamics

This article is concerned with the time evolution of a $\frac{1}{2}$-spin particle in a constant external magnetic field with the quantized electromagnetic field (photons). We derive a Lindblad (or GKLS) type approximation of the spin dynamics together with a precise control of the error coming from this approximation. The error term is bounded by $g^2$ where $g$ is the coupling constant of the spin-photon interaction. The point here is the uniformity in time $t>0$ of this error control.

math-ph↗
arXiv2019

On the anti-Wick symbol as a Gelfand-Shilov generalized function

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping $ {\cal S}(\R^n)$ into ${\cal S}'(\R^n)$, which is generally not a tempered distribution, can still be defined as a Gelfand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gelfand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.

math.AP↗
arXiv2019

Ground state photon number at large distance

The purpose of this article is to give a result of localization in space of the ground states photons, in some sense, of a Hamiltonian modelling nuclear magnetic resonance in quantum electrodynamics. The asymptotics at infinity obtained for the density of the number of photons around x is the power -5 of the distance of x to the particles. Moreover, the number of photons at large distance is smallest in the ground state total mean spin direction.

math-ph↗
arXiv2018

Integral formulas for the Weyl and anti-Wick symbols

The first purpose of this article is to provide conditions for a bounded operator in $L^2(\R^n)$ to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on $\R^{2n}$. Then, explicit formulas for the Weyl (resp. anti-Wick) symbol are proved. Secondly, other formulas for the Weyl and anti-Wick symbols involving a kind of Campbell Hausdorff formula are obtained. A point here is that these conditions and explicit formulas depend on the dimension $n$ only through a Gaussian measure on $\R^{2n}$ of variance $1/2$ in the Weyl case (resp. variance $1$ in the anti-Wick case) suggesting that the infinite dimension setting for these issues could be considered. Besides, these conditions are related to iterated commutators recovering in particular the Beals characterization Theorem.

math.AP↗
arXiv2018

Composition of states and observables in Fock spaces

This article is concerned with compositions in the context of three standard quantizations in the Fock space framework, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for time dynamics issues. Anti-Wick quantization and Segal Bargmann transforms are implied there for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick symbol of the composition of two operators, we obtain an absolutely converging series, and for the Weyl symbol, the remainder term of the asymptotic expansion is absolutely converging, still in the Fock spaces framework.

math-ph↗
arXiv2018

Infinite dimensional semiclassical analysis and applications to a model in NMR

We are interested in this paper with the connection between the dynamics of a model related to Nuclear Magnetic Resonance (NMR) in Quantum Field Theory (QFT) with its classical counterpart known as the Maxwell-Bloch equations. The model in QFT is a model of Quantum Electrodynamics (QED) considering fixed spins interacting with the quantized electromagnetic field in an external constant magnetic field. This model is close to the common spin-boson model. The classical model goes back to F. Bloch [15] in 1946. Our goal is not only to study the derivation of the Maxwell-Bloch equations but to also establish a semiclassical asymptotic expansion of arbitrary high orders with control of the error terms of this standard nonlinear classical motion equations. This provides therefore quantum corrections of any order in powers of the semiclassical parameter of the Bloch equations. Besides, the asymptotic expansion for the photon number is also analyzed and a law describing the photon number time evolution is written down involving the radiation field polarization. Since the quantum photon state Hilbert space (radiation field) is infinite dimensional we are thus concerned in this article with the issue of semiclassical calculus in an infinite dimensional setting. In this regard, we are studying standard notions as Wick and anti-Wick quantizations, heat operator, Beals characterization theorem and compositions of symbols in the infinite dimensional context which can have their own interest.

math.AP↗
arXiv2017

On the domain of a magnetic Schrödinger operator with complex electric potential

The aim of this paper is to review and compare the spectral properties of (the closed extension of) --$Δ$ + U (V $\ge$ 0) and --$Δ$ + iV in L 2 (R^d) for C $\infty$ real potentials U or V with polynomial behavior. The case with magnetic field will be also considered. More precisely, we would like to present the existing criteria for: $\bullet$ essential selfadjointness or maximal accretivity $\bullet$ Compactness of the resolvent. $\bullet$ Maximal inequalities, i.e. the existence of C > 0 such that, $\forall$u $\in$ C^$\infty$\_0 (R ^d), ||u||^2 \_{H^2 (R^d)} + ||U u||^2 \_{L^2 (R^d)}$\le$ C ||(--$Δ$ + U)u||^2\_{L^2 (R^d)} + ||u||^2\_{L^2 (R^d)}or similar estimates for $-Δ+ i V$.

math-ph↗
arXiv2017

Quantum radiative corrections for a model in NMR in quantum electrodynamics

In this article, we are interested in a spin model including the quantized electromagnetic field (photons). With this model of quantum electrodynamics (QED) related to nuclear magnetic resonance (NMR) we give explicit quantum radiative corrections of the time evolution for the spin observables, for the electric and magnetic fields observables and for the photon number observable. As a by-product, this underlines that Bloch equations are the semiclassical limit of the model in QED considered here. In addition, transition probabilities for the same model are investigated.

math-ph↗
arXiv2016

Weyl calculus in Wiener spaces and in QED

The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space $H$. If $(i, H, B)$ is a Wiener triplet associated to $H$, the quantum state space will be the space of $L^2$ functions on $B$ with respect to a Gaussian measure with $h/2$ variance, where $h$ is the semiclassical parameter. We prove the boundedness of our pseudodifferential operators (PDO) in the spirit of Calderón-Vaillancourt with an explicit bound, a Beals type characterization, and metaplectic covariance. An application to a model of quantum electrodynamics (QED) is added in the last section, for fixed spin $1/2$ particles interacting with the quantized electromagnetic field (photons). We prove that some observable time evolutions, the spin evolutions, the magnetic and electric evolutions when subtracting their free evolutions, are PDO in our class.

math.AP↗
arXiv2016

Semiclassical expansion of the ground state for a model of interacting spins in QED

In this article, we consider fixed spin 1/2 particles interacting through the quantized electromagnetic field in a constant magnetic field. We give some asymptotic expansions for the ground state and the ground state energy of the Hamiltonian operator $H(h)$ describing this system. The first terms of these expansions enable to recover elementary formulas for the energy and the magnetic field of the spins when considered as magnets. A first order radiative correction is computed for the energy.

math-ph↗
arXiv2014

On bounded pseudodifferential operators in Wiener spaces

We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing the phase space $ \mathbb{R}^{2n}$ by $B^2$, where $(i,H,B)$ is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on $B^2$ and belonging to a $L^1$ space for a gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of $L^2(B)$. For example, the symbol can be the stochastic extension on $B^2$, in the sense of L. Gross, of a function $F$ which is continuous and bounded on $H^2$. In the second approach, this function $F$ defined on $H^2$ satisfies differentiability conditions analogous to the finite dimensional ones. One needs to introduce hybrid operators acting as Weyl operators on the variables of finite dimensional subset of $H$ and as Anti-Wick operators on the rest of the variables. The final Weyl operator is then defined as a limit and it is continuous on a $L^2$ space. Under rather weak conditions, it is an extension of the operator defined by the first approach. We give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject.

math.AP↗
arXiv2014

On bounded pseudodifferential operators in a high-dimensional setting

This work is concerned with extending the results of Calder\' on and Vaillancourt proving the boundedness of Weyl pseudo differential operators Op_h^{weyl} (F) in L^2(\R^n). We state conditions under which the norm of such operators has an upper bound independent of n. To this aim, we apply a decomposition of the identity to the symbol F, thus obtaining a sum of operators of a hybrid type, each of them behaving as a Weyl operator with respect to some of the variables and as an anti-Wick operator with respect to the other ones. Then we establish upper bounds for these auxiliary operators, using suitably adapted classical methods like coherent states.

math.AP↗
arXiv2013

The Weyl symbol of Schrödinger semigroups

In this paper, we study the Weyl symbol of the Schrödinger semigroup $e^{-tH}$, $H=-Δ+V$, $t>0$, on $L^2(\mathbb{R}^n)$, with nonnegative potentials $V$ in $L^1_{\rm loc}$. Some general estimates like the $L^{\infty}$ norm concerning the symbol $u$ are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives $\partial_x^α\partial_ξ^βu$. In particular, this implies that the symbol of the Schrödinger semigroups belongs to the class of symbols introduced in [1] in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.

math.AP↗