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Philippe Briet

Publications and source records attributed to Philippe Briet.

At least 19 recordsLinked to original sources
arXiv2025

Geometric spectral properties of electromagnetic waveguides

Consider a reference homogeneous and isotropic electromagnetic waveguide with a simply connected cross-section embedded in a perfect conductor. In this setting, when the waveguide is straight, the spectrum of the associated self-adjoint Maxwell operator with a constant twist (which may be zero) lies on the real line and is symmetric with respect to zero and exhibits a spectral gap around the origin. Moreover, the spectrum is purely essential, and contains 0 which is an eigenvalue of infinite multiplicity. In this work, we present new results on the effects of geometric deformations, specifically bending and twisting, on the spectrum of the Maxwell operator. More precisely, we provide, on the one hand, sufficient conditions on the asymptotic behavior of curvature and twist that ensure the preservation of the essential spectrum of the reference waveguide. Our approach relies on a Birman-Schwinger-type principle, which may be of independent interest and applicable in other contexts. On the other hand, we give sufficient conditions (involving in particular the geometrical shape of the cross-section of the waveguide) so that the geometrical deformation creates discrete spectrum (namely isolated eigenvalues of finite multiplicity) within the gap of the essential spectrum. In addition, we give some results on the localization of these discrete eigenvalues. The sufficient condition involving the cross-section is then studied both analytically and numerically. Finally, we examine its stability under shape deformations of the cross-section, focusing in particular on the case of a waveguide with a rectangular cross-section.

math.AP↗
arXiv2021

Spectral optimisation of Dirac rectangles

We are concerned with the dependence of the lowest positive eigenvalue of the Dirac operator on the geometry of rectangles, subject to infinite-mass boundary conditions. We conjecture that the square is a global minimiser both under the area or perimeter constraints. Contrary to well-known non-relativistic analogues, we show that the present spectral problem does not admit explicit solutions. We prove partial optimisation results based on a variational reformulation and newly established lower and upper bounds to the Dirac eigenvalue. We also propose an alternative approach based on symmetries of rectangles and a non-convex minimisation problem; this implies a sufficient condition formulated in terms of a symmetry of the minimiser which guarantees the conjectured results.

math.SP↗
arXiv2021

Spectral properties of relativistic quantum waveguides

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist.

math.SP↗
arXiv2018

Molecular dynamics at an energy-level crossing

This paper is a continuation of a previous work about the study of the survival probability modelizing the molecular predissociation in the Born-Oppenheimer framework. Here we consider the critical case where the reference energy corresponds to the value of a crossing of two electronic levels, one of these two levels being confining while the second dissociates. We show that the survival probability associated to a certain initial state is a sum of the usual time-dependent exponential contribution, and a reminder term that is jointly polynomially small with respect to the time and the semiclassical parameter. We also compute explicitly the main contribution of the remainder.

math-ph↗
arXiv2018

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

math.SP↗
arXiv2018

Spectral analysis of sheared nanoribbons

We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip subject to a new deformation of "shearing": the strip is built by translating a segment oriented in a constant direction along an unbounded curve in the plane. We locate the essential spectrum under the hypothesis that the projection of the tangent vector of the curve to the direction of the segment admits a (possibly unbounded) limit at infinity and state sufficient conditions which guarantee the existence of discrete eigenvalues. We justify the optimality of these conditions by establishing a spectral stability in opposite regimes. In particular, Hardy-type inequalities are derived in the regime of repulsive shearing.

math-ph↗
arXiv2017

Twisted waveguide with a Neumann window

This paper is concerned with the study of theexistence/non-existence of the discrete spectrum of the Laplaceoperator on a domain of $\mathbb R ^3$ which consists in atwisted tube. This operator is defined by means of mixed boundaryconditions. Here we impose Neumann Boundary conditions on abounded open subset of the boundary of the domain (the Neumannwindow) and Dirichlet boundary conditions elsewhere.

math.SP↗
arXiv2016

Stark resonances in 2-dimensional curved quantum waveguides

In this paper we study the influence of an electric field on a two dimen-sional waveguide. We show that bound states that occur under a geometrical deformation of the guide turn into resonances when we apply an electric field of small intensity having a nonzero component on the longitudinal direction of the system. MSC-2010 number: 35B34,35P25, 81Q10, 82D77.

math.SP↗
arXiv2016

Stark resonances in a quantum waveguide with analytic curvature

We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide ${\bf Ω}$ i.e. bound states of the corresponding Laplace operator $-Δ\_{\bf Ω}$. Here the curvature of the guide is supposed to satisfy some assumptions of analyticity, and decays as $O(|s|^{-\varepsilon}), \varepsilon > 3$ at infinity. We show that under conditions on the electric field $ \bf F$, ${\bf H}(F):= -Δ\_{\bf Ω} + {\bf F}. {\bf x} $ has resonances near the discrete eigenvalues of $-Δ\_{\bf Ω}$.

math.SP↗
arXiv2015

Estimates on the molecular dynamics for the predissociation process

We study the survival probability associated with a semi-classical matrix Shrödinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is exponentially small (in the semiclassical parameter) with arbitrarily large rate of decay. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.

math.SP↗
arXiv2014

Hardy inequalities in globally twisted waveguides

We establish various Hardy-type inequalities for the Dirichlet Laplacian in perturbed periodically twisted tubes of non-circular cross-sections. We also state conjectures about the existence of such inequalities in more general regimes, which we support by heuristic and numerical arguments.

math.SP↗
arXiv2013

Scattering in twisted waveguides

We consider a twisted quantum waveguide i.e. a domain of the form Ω_θ : = r_θω\times R, where ω\subset R^2 is a bounded domain, and r_θ= r_θ(x_3) is a rotation by the angle θ(x_3) depending on the longitudinal variable x_3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H_θ, self-adjoint in L^2 (Ω_θ), and consider related scattering problems. First, we show that if the derivative of the difference θ_1 - θ_2 decays fast enough as |x_3| goes to infinity, then the wave operators for the operator pair (H_{θ_1}, H_{θ_2}) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ' = β- ε, with constant β\in R, and εwhich decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to εis an analytically fibered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of H_θ, is empty.

math.SP↗
arXiv2013

Exponential decay and resonances in a driven system

We study the resonance phenomena for time periodic perturbations of a Hamiltonian $H$ on the Hilbert space $L^2(\mathbb R ^d)$. Here, resonances are characterized in terms of time behavior of the survival probability. Our approach uses the Floquet-Howland formalism combined with the results of L. Cattaneo, J.M. Graf and W. Hunziker on resonances for time independent perturbations.

math.SP↗
arXiv2012

A rigorous approach to the magnetic response in disordered systems

This paper is a part of an ongoing study on the diamagnetic behavior of a 3-dimensional quantum gas of non-interacting charged particles subjected to an external uniform magnetic field together with a random electric potential. We prove the existence of an almost-sure non-random thermodynamic limit for the grand-canonical pressure, magnetization and zero- field orbital magnetic susceptibility. We also give an explicit formulation of these thermodynamic limits. Our results cover a wide class of physically relevant random potentials which model not only crystalline disordered solids, but also amorphous solids.

math-ph↗
arXiv2011

A rigorous proof of the Landau-Peierls formula and much more

We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of a non-interacting Bloch electron gas, at fixed temperature and density, for both metals and semiconductors/insulators. In particular, we obtain the Landau-Peierls formula in the low temperature and density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.

math-ph↗
arXiv2010

Diamagnetism of quantum gases with singular potentials

We consider a gas of quasi-free quantum particles confined to a finite box, subjected to singular magnetic and electric fields. We prove in great generality that the finite volume grand-canonical pressure is jointly analytic in the chemical potential ant the intensity of the external magnetic field. We also discuss the thermodynamic limit.

math-ph↗