arXiv subjects

Clotilde Fermanian Kammerer

Publications and source records attributed to Clotilde Fermanian Kammerer.

At least 19 recordsLinked to original sources

arXiv2026

Measuring a Quantum System Thanks to the Correspondence Principle

The measurement problem is one of the longest-standing open problems in quantum mechanics: noncommutative observables are incompatible with the Born and collapse rules, prescribing that the outcome of their observation is distributed according to the laws of classical probability. In this paper, we solve the measurement problem through the correspondence principle with a semiclassical approach. We prove that Born rule, the wavefunction collapse, and macroscopic observability all hold true, approximately, but with arbitrary precision, whenever a measuring device (in a semiclassical state) is coupled adiabatically to the observed system. Our measurement device is based on the propagation properties of semiclassical bosonic coherent states through an adiabatic two level system on scale times of the order of the Ehrenfest time.

math-ph↗

arXiv2026

Edge modes generated by intersection between 3 energy bands

In this contribution, we investigate 2-dimensional model problems of three coupled equations. We assume that the underlying Hamiltonian presents a symmetric intersection between three eigenvalues of Dirac's type with a mass function that vanish along a curve. We investigate the existence of edge modes for these models: such functions are solutions of the semiclassical associated evolution problem that are asymptotic to a coherent state with zero energy and break the correspondence principle by propagating inside the crossing set and not along a classical trajectory.

math.AP↗

arXiv2026

Quantization on filtered manifolds

In this article, we develop a pseudodifferential calculus on a general filtered manifold M . The symbols are fields of operators $σ$(x, $π$) parametrised by x $\in$ M and the unitary dual G x M of the osculating Lie group G x M . We define classes of symbols and a local quantization formula associated to a local frame adapted to the filtration. We prove that the collection of operators on M coinciding locally with the quantization of symbols enjoys the essential properties of a pseudodifferential calculus: composition, adjoint, parametrices, continuity on adapted Sobolev spaces. Moreover, we show that the polyhomogeneous subcalculus coincides with the calculus constructed by van Erp and Yuncken via groupoids.

math.FA↗

arXiv2025

Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings

This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schrödinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schrödinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter.

math.AP↗

arXiv2025

The superadiabatic projectors method applied to the spectral theory of magnetic operators

This article deals with a generalization of the superadiabatic projectors method. In a general framework, the well-known superadiabatic projectors are constructed and accurately described in the case of rank one, when a remarkable factorization occurs. We apply these ideas to spectral theory and we explain how our abstract results allow to recover or improve recent results about the semiclassical magnetic Laplacian.

math.SP↗

arXiv2024

Semi-classical analysis

We introduce three representative topics in semi-classical analysis. Starting from the correspondence between classical and quantum mechanics, basic semi-classical analysis tools and results are presented. The three topics are investigated in the light of the introduced techniques allowing one to emphasize different aspects of semi-classical analysis.

math.AP↗

arXiv2024

Quantization on Groups and Garding inequality

In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn-Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on representation theory via the group Fourier transform and the Plancherel theorem. As an application, we give a simple proof of Garding inequalities for three globally symbolic pseudo-differential calculi on groups.

math.FA↗

arXiv2023

Geometric invariance of the semi-classical calculus on nilpotent graded Lie groups

In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. We analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. We interpret the geometric meaning of this invariance in the setting of filtered manifolds.

math.FA↗

arXiv2022

Some remarks on semi-classical analysis on two-step Nilmanifolds

In this paper, we present recent results about the developement of a semiclassical approach in the setting of nilpotent Lie groups and nilmanifolds. We focus on two-step nilmanifolds and exhibit some properties of the weak limits of sequence of densities associated with eigenfunctions of a sub-Laplacian. We emphasize the influence of the geometry on these properties.

math.AP↗

arXiv2022

On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schr{ö}dinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac-Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations.

math.AP↗

arXiv2022

Dynamical approximations for composite quantum systems: Assessment of error estimates for a separable ansatz

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work [I. Burghardt, R. Carles, C. Fermanian Kammerer, B. Lasorne, C. Lasser, J. Phys. A. 54, 414002 (2021)]. Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The timedependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate.

physics.chem-ph↗

arXiv2022

Effective mass theorems with Bloch modes crossings

We study a Schr{ö}dinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wavelength comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

math.AP↗

arXiv2021

Propagation of Coherent States through Conical Intersections

In this paper, we analyze the propagation of a wave packet through a conical intersection. This question has been addressed for Gaussian wave packets in the 90s by George Hagedorn and we consider here a more general setting. We focus on the case of Schr{ö}dinger equation but our methods are general enough to be adapted to systems presenting codimension 2 crossings and to codimension 3 ones with specific geometric conditions. Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase. Its proof is based on a normal form approach combined with the use of superadiabatic projectors and the analysis of their degeneracy close to the crossing.

math.AP↗

arXiv2021

Separation of scales: Dynamical approximations for composite quantum systems

We consider composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system-bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation.Specifically, we investigate a coupling r{é}gime that is partially flat, i.e., slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum-classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations.

math.AP↗

arXiv2021

Observability and Controllability for the Schr{ö}dinger Equation on Quotients of Groups of Heisenberg Type

We give necessary and sufficient conditions for the controllability of a Schr\''odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\''odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\''odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

math.AP↗

arXiv2021

Edge state dynamics along curved interfaces

We study the propagation of wavepackets along weakly curved interfaces between topologically distinct media. Our Hamiltonian is an adiabatic modulation of Dirac operators omnipresent in the topological insulators literature. Using explicit formulas for straight edges, we construct a family of solutions that propagates, for long times, unidirectionally and dispersion-free along the curved edge. We illustrate our results through various numerical simulations.

math-ph↗

arXiv2020

Coherent electronic transport in periodic crystals

We consider independent electrons in a periodic crystal in their ground state, and turn on a uniform electric field at some prescribed time. We rigorously define the current per unit volume and study its properties using both linear response and adiabatic theory. Our results provide a unified framework for various phenomena such as the quantification of Hall conductivity of insulators with broken time-reversibility, the ballistic regime of electrons in metals, Bloch oscillations in the long-time response of metals, and the static conductivity of graphene. We identify explicitly the regime in which each holds.

math-ph↗

arXiv2020

A Nonlinear Quantum Adiabatic Approximation

This paper is devoted to a generalisation of the quantum adiabatic theorem to a nonlinear setting. We consider a Hamiltonian operator which depends on the time variable and on a finite number of parameters and acts on a separable Hilbert space of which we select a fixed basis. We study an evolution equation in which this Hamiltonian acts on the unknown vector, while depending on coordinates of the unknown vector in the selected basis, thus making the equation nonlinear. We prove existence of solutions to this equation and consider their asymptotics in the adiabatic regime, i.e. when the Hamiltonian is slowly varying in time. Under natural spectral hypotheses, we prove the existence of normalised time dependent vectors depending on the Hamiltonian only, called instantaneous nonlinear eigenvectors, and show the existence of solutions which remain close to these vectors, up to a rapidly oscillating phase, in the adiabatic regime. We first investigate the case of bounded operators and then exhibit a set of spectral assumptions under which the result extends to unbounded Hamiltonians.

math-ph↗