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1. Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces

Author

Tebbani Hossni, Achache Mahdi, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, [MATH]Mathematics [math]
Abstract

We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations $$\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\, t \text {-a.e.}, \, u(0)=u_0. \end{aligned}$$ u ′ ( t ) + A ( t ) u ( t ) = F ( t , u ) , t -a.e. , u ( 0 ) = u 0 . Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $$\mathcal {H}.$$ H . We prove the maximal regularity result in temporally weighted $$L^2$$ L 2 -spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value $$u_0$$ u 0 and the inhomogeneous term F. Our results are motivated by boundary value problems.

Published
2022
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2. Lieb–Thirring inequalities for an effective Hamiltonian of bilayer graphene

Author

Philippe Briet, Jean-Claude Cuenin, Stanislas Kupin, Leonid Golinskii, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Loughborough University, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0035,REPKA,Noyaux reproduisants en Analyse et au-delà(2018), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Condensed Matter - Mesoscale and Nanoscale Physics, FOS: Physical sciences, 35P15, 30C35, 47A75, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Mathematics::Spectral Theory, Discrete spectrum, Mathematics - Spectral Theory, Quantum mechanics, Mesoscale and Nanoscale Physics (cond-mat.mes-hall), FOS: Mathematics, Geometry and Topology, Bilayer graphene, Spectral Theory (math.SP), Mathematical Physics, Hamiltonian (control theory), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

Combining the methods of Cuenin [2019] and Borichev-Golinskii-Kupin [2009, 2018], we obtain the so-called Lieb-Thirring inequalities for non-selfadjoint perturbations of an effective Hamiltonian for bilayer graphene., 22 pages; few typos corrected

Published
2021
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3. Stable recovery of noncompactly supported electromagnetic potentials in unbounded domain

Author

Yavar Kian, Yosra Soussi, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université de Tunis El Manar (UTM), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
General Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, 01 natural sciences, Domain (software engineering), 010101 applied mathematics, Mathematics - Analysis of PDEs, 35R30, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Electromagnetic potential, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Analysis of PDEs (math.AP), Mathematics
Abstract

We consider the inverse problem of determining an electromagnetic potential appearing in an infinite cylindrical domain from boundary measurements. More precisely, we prove the stable recovery of some general class of magnetic field and electric potential from boundary measurements. Assuming some knowledge of the unknown coefficients close to the boundary, we obtain also some results of stable recovery with measurements restricted to some portion of the boundary. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

Published
2021
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4. Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Author

Masahiro Yamamoto, Yavar Kian, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Spacetime, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), 35R11, 35B30, 35R05, 01 natural sciences, 010101 applied mathematics, Sobolev space, Strong solutions, Mathematics - Analysis of PDEs, Compatibility (mechanics), FOS: Mathematics, Fractional diffusion, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Analysis, Well posedness, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; Abstract We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

Published
2021
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5. Markovian Repeated Interaction Quantum Systems

Author

Jean-François Bougron, Alain Joye, Claude-Alain Pillet, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Université de Cergy Pontoise (UCP), Université Paris-Seine, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), CPT - E5 Physique statistique et systèmes complexes, and ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017)

Subjects
fluctuation relations, Quantum Physics, Statistical Mechanics (cond-mat.stat-mech), fluctuation, Markov chain, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), entropy production, dynamical system, Open quantum systems, linear response, quantum, thermodynamics, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], adiabatic, spectral, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [PHYS.COND]Physics [physics]/Condensed Matter [cond-mat], entropy, Quantum Physics (quant-ph), Mathematical Physics, Condensed Matter - Statistical Mechanics, nonequilibrium statistical mechanics
Abstract

International audience; We study a class of dynamical semigroups $(\mathbb{L}^n)_{n\in\mathbb{N}}$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system $(\mathcal{L}_{\omega_n}\circ\cdots\circ\mathcal{L}_{\omega_1}(\rho_{\omega_0}))_{n\in\mathbb{N}}$ driven by a Markov chain $(\omega_n)_{n\in\mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $\mathbb{L}$. As a physical application, we consider the case where the $\mathcal{L}_\omega$'s are the reduced dynamical maps describing the repeated interactions of a system $\mathcal{S}$ with thermal probes $\mathcal{C}_\omega$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.

Published
2022
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6. Multidimensional Borg-Levinson inverse spectral problems

Author

Eric Soccorsi, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Inverse, Applied mathematics, Mathematics::Spectral Theory, MSC: 35R30, Mathematics
Abstract

International audience; This text deals with multidimensional Borg-Levinson inverse theory. Its main purpose is to establish that the Dirichlet eigenvalues and Neumann boundary data of the operator −∆+q, acting in a bounded domain of R d with d 2, uniquely determine the realvalued bounded potential q. We first address the case of incomplete spectral data, where finitely many boundary spectral eigen-pairs remain unknown. Under suitable summability condition on the Neumann data, we also consider the case where only the asymptotic behavior of the eigenvalues is known. Finally, we use the multidimensional Borg-Levinson theory for solving parabolic inverse coefficient problems.

Published
2020
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7. Molecular dynamics at an energy-level crossing

Author

André Martinez, Philippe Briet, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Alma Mater Studiorum Università di Bologna [Bologna] (UNIBO), Université de Bologne, Briet P., and Martinez A.

Subjects
Work (thermodynamics), FOS: Physical sciences, Semiclassical physics, Resonance, 01 natural sciences, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum evolution, Statistical physics, 0101 mathematics, Remainder, Born-Oppenheimer approximation, Mathematical Physics, Mathematics, 35P15, 35C20, 35S99, 47A75, Applied Mathematics, 010102 general mathematics, Eigenvalue crossing, Mathematical Physics (math-ph), State (functional analysis), Level crossing, Term (time), Exponential function, 010101 applied mathematics, Survival probability, Analysis
Abstract

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., 37 pages, 1 figure

Published
2019
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8. Carleman estimate for the Schrödinger equation and application to magnetic inverse problems

Author

Xinchi Huang, Masahiro Yamamoto, Yavar Kian, Eric Soccorsi, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, The University of Tokyo (UTokyo), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Boundary (topology), Mathematics::Spectral Theory, Inverse problem, Lipschitz continuity, 01 natural sciences, Magnetic field, Schrödinger equation, 010101 applied mathematics, symbols.namesake, Mathematics - Analysis of PDEs, 35R30, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Symmetrization, Electric potential, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Analysis, Mathematics
Abstract

In this paper, we prove Lipschitz stable determination of the complex-valued electric potential and the direction of the magnetic field appearing in the dynamic Schrodinger equation with static coefficients, by finitely many partial boundary measurements of the solution. This is by means of the Bukhgeim–Klibanov method, based on an appropriate Carleman estimate. Since the time symmetrization of the static magnetic Schrodinger equation around t = 0 is not possible, we preliminarily establish a Carleman inequality specifically designed for this problem.

Published
2019
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9. Recovery of the Order of Derivation for Fractional Diffusion Equations in an Unknown Medium

Author

Bangti Jin, Yavar Kian, Department of Computer science [University College of London] (UCL-CS), University College of London [London] (UCL), CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, Applied Mathematics, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH]Mathematics [math], Analysis of PDEs (math.AP)
Abstract

In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and time-fractional diffusion equations, from the flux measurement observed at one point on the boundary. The mathematical model for time-fractional diffusion equations involves a Djrbashian-Caputo fractional derivative in time. We prove a uniqueness result in an unknown medium (e.g., diffusion coefficients, obstacle, initial condition and source), i.e., the recovery of the order of derivation in a diffusion process having several pieces of unknown information. The proof relies on the analyticity of the solution at large time, asymptotic decay behavior, strong maximum principle of the elliptic problem and suitable application of the Hopf lemma. Further we provide an easy-to-implement reconstruction algorithm based on a nonlinear least-squares formulation, and several numerical experiments are presented to complement the theoretical analysis., Comment: 22 pages, 3 figures, to appear at SIAM Journal on Applied Mathematics

Published
2021
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10. Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide

Author

Diomba Sambou, Yavar Kian, Eric Soccorsi, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Facultad de Matemáticas [Santiago de Chile], Pontificia Universidad Católica de Chile (UC), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Logarithm, partial boundary data, Boundary (topology), Carleman estimate, 01 natural sciences, Mathematics - Analysis of PDEs, 35R30, 35K05, FOS: Mathematics, Free boundary problem, time-dependent source term, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Waveguide (acoustics), 0101 mathematics, Mathematics, heat equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Inverse problem, Parabolic partial differential equation, 010101 applied mathematics, Inverse scattering problem, stability inequality, Heat equation, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.

Published
2018
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11. Local decay for weak interactions with massless particles

Author

Jean-Marie Barbaroux, Jérémy Faupin, Jean-Claude Guillot, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), and École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Standard Model, Spectral theory, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Weak Interactions, FOS: Physical sciences, 01 natural sciences, Fock space, Mathematics - Spectral Theory, symbols.namesake, Quantum mechanics, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Spectral Theory (math.SP), Mathematical Physics, Boson, Physics, 010102 general mathematics, Statistical and Nonlinear Physics, Observable, Mourre Theory, Mathematical Physics (math-ph), Massless particle, Spectral Theory, symbols, High Energy Physics::Experiment, Local Decay, 010307 mathematical physics, Geometry and Topology, Neutrino, Hamiltonian (quantum mechanics), Ground state, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We consider a mathematical model for the weak decay of the intermediate boson $Z^0$ into neutrinos and antineutrinos. We prove that the total Hamiltonian has a unique ground state in Fock space and we establish a limiting absorption principle, local decay and a property of relaxation to the ground state for initial states and observables suitably localized in energy and position. Our proofs rest, in particular, on Mourre's theory and a low-energy decomposition., Comment: 42 pages

Published
2018
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12. Spectral optimization of Dirac rectangles

Author

David Krejcirik, Philippe BRIET, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Czech Technical University in Prague (CTU)

Subjects
010308 nuclear & particles physics, 010102 general mathematics, FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), 01 natural sciences, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics - Optimization and Control, Spectral Theory (math.SP), Mathematical Physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP], Analysis of PDEs (math.AP)
Abstract

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., Comment: 11 pages; due to a gap in the proof in our previous version (see Remark 1), we obtain just partial results, by an alternative approach; version accepted for publication in Journal of Mathematical Physics

Published
2022
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13. Global uniqueness in an inverse problem for time fractional diffusion equations

Author

Eric Soccorsi, Yavar Kian, Lauri Oksanen, Masahiro Yamamoto, Université de Toulon (UTLN), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), University College of London [London] (UCL), and The University of Tokyo (UTokyo)

Subjects
Inverse problems, fractional diffusion equation, Pure mathematics, Applied Mathematics, 010102 general mathematics, Boundary (topology), Inverse problem, Riemannian manifold, partial data, Isometry (Riemannian geometry), 01 natural sciences, Manifold, 010101 applied mathematics, Mathematics - Analysis of PDEs, 35R30, 35R11, 58J99, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Order (group theory), Boundary value problem, Uniqueness, 0101 mathematics, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

Given ( M , g ) , a compact connected Riemannian manifold of dimension d ⩾ 2 , with boundary ∂M, we consider an initial boundary value problem for a fractional diffusion equation on ( 0 , T ) × M , T > 0 , with time-fractional Caputo derivative of order α ∈ ( 0 , 1 ) ∪ ( 1 , 2 ) . We prove uniqueness in the inverse problem of determining the smooth manifold ( M , g ) (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ∂M at fixed time. In the “flat” case where M is a compact subset of R d , two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation ρ ∂ t α u − div ( a ∇ u ) + q u = 0 on ( 0 , T ) × M are recovered simultaneously.

Published
2018
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14. Simultaneous determination of coefficients, internal sources and an obstacle of a diffusion equation from a single measurement

Author

Kian, Yavar, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Inverse problems, inverse obstacle problem, 35R11, Mathematics - Analysis of PDEs, diffusion equation, inverse source prob- lem, FOS: Mathematics, uniqueness, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], inverse coefficient problem, partial data Mathematics subject classification 2010 : 35R30, 35R30, 35R11, Analysis of PDEs (math.AP)
Abstract

International audience; This article is devoted to the simultaneous resolution of three inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single boundary measurement. Namely, we consider the simultaneous unique determination of several class of coefficients, some internal sources (a source term and an initial condition) and an obstacle appearing in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium), an obstacle and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection-diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation.

Published
2020
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15. Reflection Probabilities of One-Dimensional Schrödinger Operators and Scattering Theory

Author

Jane Panangaden, Benjamin Landon, Justine Zwicker, Annalisa Panati, Department of Mathematics (Harvard University), Harvard University [Cambridge], Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Harvard University

Subjects
Nuclear and High Energy Physics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 010102 general mathematics, Statistical and Nonlinear Physics, Statistical mechanics, 01 natural sciences, symbols.namesake, Operator (computer programming), Quantum mechanics, 0103 physical sciences, symbols, 010307 mathematical physics, Scattering theory, 0101 mathematics, Mathematical Physics, Schrödinger's cat, Mathematics, Mathematical physics
Abstract

The dynamic reflection probability of Davies and Simon (Commun Math Phys 63(3):277–301, 1978) and the spectral reflection probability of Gesztesy et al. (Diff Integral Eqs 10(3):521–546, 1997) and Gesztesy and Simon (Helv Phys Acta 70:66–71, 1997) for a one-dimensional Schrodinger operator \(H = - \Delta + V\) are characterized in terms of the scattering theory of the pair \((H, H_\infty )\) where \(H_\infty \) is the operator obtained by decoupling the left and right half-lines \(\mathbb {R}_{\le 0}\) and \(\mathbb {R}_{\ge 0}\). An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer et al. (Commun Math Phys 295(2):531–550, 2010). This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black-box model and follows a strategy parallel to Jaksic (Commun Math Phys 332:827–838, 2014).

Published
2017
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16. On the Well-posedness of the Magnetic Schrödinger-Poisson System in ℝ3

Author

Vitali Vougalter, Jean-Marie Barbaroux, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [University of Toronto], and University of Toronto

Subjects
density matrices, magnetic fields, 01 natural sciences, Schrödinger-Poisson system, Strong solutions, symbols.namesake, functional spaces, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Coulomb, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, global existence and uniqueness, Magnetic field, Classical mechanics, Modeling and Simulation, symbols, 010307 mathematical physics, Poisson system, Schrödinger's cat, Well posedness, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

International audience; We prove global existence and uniqueness of strong solutions for the Schrödinger-Poisson system in the repulsive Coulomb case in R 3 in the presence of a smooth magnetic field.

Published
2017
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17. Some trace monotonicity properties and applications

Author

Jean-Michel Combes, Peter D. Hislop, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), University of Kentucky, Department of Mathematics, University of Kentucky (UK)-University of Kentucky (UK), and University of Kentucky-University of Kentucky

Subjects
Pure mathematics, Control and Optimization, Trace (linear algebra), Relation (database), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Loewner’s Theorem, FOS: Physical sciences, Monotonic function, 47B80, 47A60, operator monotone functions, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], random Schrodinger operators, symbols.namesake, FOS: Mathematics, Mathematical Physics, Mathematics, Algebra and Number Theory, Continuum (topology), operator trace inequalities, Wegner estimate, Mathematical Physics (math-ph), Mathematics::Spectral Theory, Functional Analysis (math.FA), Mathematics - Functional Analysis, 47B10, 47B10, 81Q10, 47B80, symbols, Analysis, Schrödinger's cat
Abstract

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. The relation between these results and Löwner’s Theorem is discussed. We also apply these results to complete a proof of the Wegner estimate for continuum models of random Schrödinger operators as given in a 1994 paper by Combes and Hislop.

Published
2016
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18. An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation

Author

Mourad Bellassoued, Yavar Kian, Eric Soccorsi, Fédération de recherche Denis Poisson (FDP), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Université de Carthage - University of Carthage, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), and Université d'Orléans (UO)-Université de Tours-Centre National de la Recherche Scientifique (CNRS)

Subjects
Infinite cylindrical waveguide, Logarithm, Inverse Problems, Boundary (topology), Inverse, Schrödinger equation, Scalar potential, Carleman estimate, 01 natural sciences, Domain (mathematical analysis), 35R30, 35Q41, symbols.namesake, Stability estimate, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Waveguide (acoustics), 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Inverse problem, 16. Peace & justice, 010101 applied mathematics, symbols, Analysis
Abstract

International audience; In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.

Published
2016
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19. Dynamical localization of Dirac particles in electromagnetic fields with dominating magnetic potentials

Author

Josef Mehringer, Jean-Marie Barbaroux, Edgardo Stockmeyer, Amal Taarabt, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Pontificia Universidad Católica de Chile (UC)

Subjects
Electromagnetic field, Dirac (software), FOS: Physical sciences, Dirac operator, 01 natural sciences, law.invention, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], law, Quantum mechanics, 0103 physical sciences, Point (geometry), [MATH]Mathematics [math], 0101 mathematics, 010306 general physics, Mathematical Physics, Physics, Graphene, Applied Mathematics, 010102 general mathematics, Spectrum (functional analysis), Mathematical Physics (math-ph), Massless particle, Quantum electrodynamics, symbols, Analysis
Abstract

We consider two-dimensional massless Dirac operators in a radially symmetric electromagnetic field. In this case the fields may be described by one-dimensional electric and magnetic potentials $V$ and $A$. We show dynamical localization in the regime when $\displaystyle\lim_{r\to\infty}|V|/|A

Published
2016
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20. Recovery of non-smooth coefficients appearing in anisotropic wave equations

Author

Yavar Kian, Ali Feizmohammadi, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Spectral Theory, Inverse problem, Lambda, Wave equation, 01 natural sciences, Dirichlet distribution, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Mathematics - Analysis of PDEs, symbols, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Magnetic potential, 0101 mathematics, Anisotropy, Scalar field, Hyperbolic partial differential equation, Analysis, Mathematics, Analysis of PDEs (math.AP)
Abstract

We study the problem of unique recovery of a non-smooth one-form $\mathcal A$ and a scalar function $q$ from the Dirichlet to Neumann map, $\Lambda_{\mathcal A,q}$, of a hyperbolic equation on a Riemannian manifold $(M,g)$. We prove uniqueness of the one-form $\mathcal A$ up to the natural gauge, under weak regularity conditions on $\mathcal A,q$ and under the assumption that $(M,g)$ is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function $q$. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier Integral Operator to non-smooth parameters and functions.

Published
2019
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21. Large deviations and entropy production in viscous fluid flows

Author

Armen Shirikyan, Vahagn Nersesyan, Vojkan Jaksic, Claude-Alain Pillet, McGill University = Université McGill [Montréal, Canada], Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université de Cergy Pontoise (UCP), Université Paris-Seine, CNRS PICS Fluctuation theorems in stochastic systems, Initiative d'excellence Paris-Seine, NSERC, ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), and ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011)

Subjects
regular densities, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Markov process, FOS: Physical sciences, 01 natural sciences, entropy production, Navier-Stokes system, symbols.namesake, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics - Optimization and Control, Mathematical Physics, Probability measure, Mathematics, Lebesgue measure, Entropy production, Mechanical Engineering, Lagrangian trajectories, 010102 general mathematics, Mathematical analysis, Probability (math.PR), Mathematical Physics (math-ph), 16. Peace & justice, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Large deviations, 35Q30, 35R60, 60B12, 60F10, 76D05, 93B05, Optimization and Control (math.OC), Bounded function, symbols, Large deviations theory, Vector field, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Rate function, Mathematics - Probability, Analysis, Analysis of PDEs (math.AP)
Abstract

We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier-Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time average is a bounded function of the trajectory. The proofs are based on a new criterion for the validity of the level-3 LDP for Markov processes and an application of a general result on the image of probability measures under smooth maps to the laws associated with the motion of the particle., Comment: 52 pages

Published
2019
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22. Identification of time-varying source term in time-fractional diffusion equations

Author

Yavar Kian, Éric Soccorsi, Qi Xue, Masahiro Yamamoto, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Institut des Sciences de la Terre (ISTerre), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Institut national des sciences de l'Univers (INSU - CNRS)-Institut de recherche pour le développement [IRD] : UR219-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Tokyo University of Science [Tokyo], ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017), and Institut national des sciences de l'Univers (INSU - CNRS)-Institut de recherche pour le développement [IRD] : UR219-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-Université Grenoble Alpes (UGA)

Subjects
Inverse source problems, fractional diffusion equation, diffusion equation, Applied Mathematics, General Mathematics, numerical reconstruction, 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, FOS: Mathematics, uniqueness result, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Tikhonov regularization method, 0101 mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in (0, 2). We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.

Published
2019
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23. The inverse problem of two-state quantum systems with non-adiabatic static linear coupling

Author

Andrii Khrabustovskyi, Eric Soccorsi, Imen Rassas, Technische Universität Graz (TU Graz), University of Tunis El Manar, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
General Mathematics, Mathematics::Analysis of PDEs, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics::Metric Geometry, 0101 mathematics, Adiabatic process, Nonlinear Sciences::Pattern Formation and Solitons, Quantum, Mathematics, Coupling, Applied Mathematics, 010102 general mathematics, Mathematical analysis, State (functional analysis), Mathematics::Spectral Theory, Inverse problem, Lipschitz continuity, 010101 applied mathematics, 35R30, Zeroth law of thermodynamics, symbols, Schrödinger's cat, Analysis of PDEs (math.AP)
Abstract

International audience; We consider the inverse problem of determining the coupling coefficients in a two-state Schrödinger system. We prove a Lipschitz stability inequality for the zeroth- and first-order coupling terms by finitely many partial lateral measurements of the solution to the coupled Schrödinger equations.

Published
2020
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24. Uniqueness and stability results for an inverse spectral problem in a periodic waveguide

Author

Yavar Kian, Otared Kavian, Eric Soccorsi, Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, periodic potential 2010 AMS Subject Classification: Primary: 35R30, 35J10, Inverse, Boundary (topology), 01 natural sciences, Primary: 35R30, 35J10, secondary: 35P99, symbols.namesake, Mathematics - Analysis of PDEs, periodic waveg-uide, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Eigenvalues and eigenvectors, Mathematical physics, Mathematics, Schrödinger operator, Applied Mathematics, 010102 general mathematics, secondary: 35P99, uniqueness, Inverse spectral problem, stability, 010101 applied mathematics, Fourier transform, Dirichlet boundary condition, Bounded function, Domain (ring theory), symbols, Analysis of PDEs (math.AP)
Abstract

Let Ω : = ω × R where ω ⊂ R 2 is a bounded domain, and let V : Ω → R be a bounded potential which is 2π-periodic in the variable x 3 ∈ R . We study the inverse problem consisting in the determination of V, through the boundary spectral data of the operator u ↦ A u : = − Δ u + V u , acting on L 2 ( ω × ( 0 , 2 π ) ) , with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for j = 1 , 2 two potentials V j are given so that ‖ V j ‖ ∞ ≤ R , and if we denote by ( λ j , k ) k the eigenvalues of the operators A j (that is the operator A with V : = V j ), then for a constant c > 0 , depending on ω and R > 0 , we have ‖ F ( ( V 1 − V 2 ) 1 ω × ( 0 , 2 π ) ) ‖ ∞ ≤ c lim sup k → ∞ | λ 1 , k − λ 2 , k | , provided that ∑ k ≥ 1 ‖ ψ 1 , k − ψ 2 , k ‖ L 2 ( ∂ ω × [ 0 , 2 π ] ) 2 ∞ , where ψ j , k : = ∂ ϕ j , k / ∂ n (here F denotes the Fourier transform). The arguments developed here may be applied to other spectral inverse problems, and similar results can be obtained.

Published
2015
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25. Heat trace asymptotics and boundedness in the second order Sobolev space of isospectral potentials for the Dirichlet Laplacian

Author

Yavar Kian, Laurent Kayser, Mourad Choulli, Eric Soccorsi, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Degree (graph theory), General Mathematics, Mathematical analysis, Order (ring theory), Sobolev space, symbols.namesake, Isospectral, Dirichlet boundary condition, Domain (ring theory), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Nabla symbol, Asymptotic expansion, Mathematics
Abstract

International audience; Let $\Omega$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overline{\Omega};\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(\Omega)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (\Omega)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $\Omega$, that is when ${\bf a}$ is the identity of $\R^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^\infty$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s

Published
2015
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26. Asymptotics of Green function for the linear waves equations in a domain with a non-uniform bottom

Author

A. Yu. Anikin, Michel Rouleux, S. Yu. Dobrokhotov, Vladimir E. Nazaikinskii, Moscow Institute of Physics and Technology [Moscow] (MIPT), institute for Problems in Mechanics [Moscow], Russian Academy of Sciences [Moscow] (RAS), CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Operator (physics), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 010102 general mathematics, Mathematical analysis, FOS: Physical sciences, Harmonic (mathematics), Mathematical Physics (math-ph), 01 natural sciences, Electronic mail, 010101 applied mathematics, Quadratic equation, Free surface, Boundary value problem, 0101 mathematics, Constant (mathematics), Mathematical Physics, Convergent series
Abstract

International audience; We consider the linear problem for water-waves created by sources on the bottom and the free surface in a 3-D basin having slowly varying profile $z=-D(x)$. The fluid verifies Euler-Poisson equations. These (non-linear) equations have been given a Hamiltonian form by Zakharov, involving canonical variables $(\xi(x,t),\eta(x,t))$ describing the dynamics of the free surface; variables $(\xi,\eta)$ are related by the free surface Dirichlet-to-Neumann (DtN) operator. For a single variable $x\in{\bf R}$ and constant depth, DtN operator was explicitely computed in terms of a convergent series. Here we neglect quadratic terms in Zakharov equations, and consider the linear response to a disturbance of $D(x)$ harmonic in time when the wave-lenght is small compared to the depth of the basin. We solve the Green function problem for a matrix-valued DtN operator, at the bottom and the free-surface.

Published
2017
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27. Inverse moving source problems in electrodynamics

Author

Yavar Kian, Guanghui Hu, Peijun Li, Yue Zhao, Institut des Sciences de la Terre (ISTerre), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Institut national des sciences de l'Univers (INSU - CNRS)-Institut de recherche pour le développement [IRD] : UR219-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Peking University [Beijing], Concordia University [Montreal], and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Source function, Distribution (number theory), Applied Mathematics, Mathematical analysis, Dirac (software), Inverse, 010103 numerical & computational mathematics, Function (mathematics), 01 natural sciences, Computer Science Applications, Theoretical Computer Science, 010101 applied mathematics, symbols.namesake, Mathematics - Analysis of PDEs, Maxwell's equations, Signal Processing, FOS: Mathematics, symbols, Orbit (dynamics), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Mathematical Physics, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; This paper is concerned with the uniqueness on two inverse moving source problems in electrody-namics with partial boundary data. We show that (1) if the temporal source function is compactly supported, then the spatial source profile function or the orbit function can be uniquely determined by the tangential trace of the electric field measured on part of a sphere; (2) if the temporal function is given by a Dirac distribution, then the impulsive time point and the source location can be uniquely determined at four receivers on a sphere.

Published
2019
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28. On time-fractional diffusion equations with space-dependent variable order

Author

Masahiro Yamamoto, Yavar Kian, Eric Soccorsi, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and The University of Tokyo (UTokyo)

Subjects
Nuclear and High Energy Physics, Variables, media_common.quotation_subject, 010102 general mathematics, Statistical and Nonlinear Physics, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, 35R30, Fractional diffusion, FOS: Mathematics, Order (group theory), Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Diffusion (business), Mathematical Physics, Mathematics, Variable (mathematics), media_common, Analysis of PDEs (math.AP)
Abstract

We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.

Published
2017
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29. Semi-classical quantization rules for a periodic orbit of hyperbolic type

Author

Hanen Louati, Michel Rouleux, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diffraction, Physics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 010102 general mathematics, FOS: Physical sciences, Mathematical Physics (math-ph), 01 natural sciences, Electronic mail, Hamiltonian system, Quantization (physics), 0103 physical sciences, Periodic orbits, 010307 mathematical physics, 0101 mathematics, Mathematical Physics, Eigenvalues and eigenvectors, Poincaré map, Mathematical physics
Abstract

Determination of periodic orbits for a Hamiltonian system together with their semi-classical quantization has been a long standing problem. We consider here resonances for a $h$-Pseudo-Differential Operator $H(y,hD_y;h)$ induced by a periodic orbit of hyperbolic type at energy $E_0$. We generalize the framework of [G\'eSj], in the sense that we allow for both hyperbolic and elliptic eigenvalues of Poincar\'e map, and show that all resonances in $W=[E_0-\varepsilon_0,E_0+\varepsilon_0]-i]0,h^\delta]$, $0, Comment: arXiv admin note: substantial text overlap with arXiv:1606.06180

Published
2016
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30. Quantitative Estimates on the Binding Energy for Hydrogen in Non-Relativistic QED

Author

Thomas Chen, Semjon Vugalter, Jean-Marie Barbaroux, Vitali Vougalter, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), University of Texas at Austin [Austin], Department of Mathematics [University of Toronto], University of Toronto, and Karlsruhe Institute of Technology (KIT)

Subjects
Physics, Nuclear and High Energy Physics, Logarithm, 010102 general mathematics, Binding energy, Order (ring theory), Statistical and Nonlinear Physics, Term (logic), 01 natural sciences, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, 010307 mathematical physics, [MATH]Mathematics [math], 0101 mathematics, Remainder, Ground state, Constant (mathematics), ComputingMilieux_MISCELLANEOUS, Mathematical Physics, Mathematical physics, Analytic function
Abstract

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order $O(\alpha^5\log\alpha^{-1})$, where $\alpha$ denotes the finestructure constant, and prove rigorous bounds on the remainder term of the order $o(\alpha^5\log\alpha^{-1})$. As a consequence, we prove that the binding energy is not a real analytic function of $\alpha$, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of $\alpha$, as conjectured in the recent literature.

Published
2010
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31. 100 years of improvements of bounding properties of Padé approximants to the Stieltjes functions: One-point, two-point and N-point Padé approximants

Author

Jacek Gilewicz, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Pure mathematics, Applied Mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 010102 general mathematics, Mathematical analysis, Riemann–Stieltjes integral, 010103 numerical & computational mathematics, 01 natural sciences, Computational Mathematics, Bounding overwatch, Convergence (routing), Padé approximant, Order (group theory), Point (geometry), 0101 mathematics, Mathematics
Abstract

The inequalities between the convergents of the Stieltjes continued fractions (CF) allowing to prove their convergence to the Stieltjes functions were first introduced in the famous work of Stieltjes (1894) [26]. Of course, the name of Stieltjes was attributed to these objects more later, such as the identification of the convergents of CF with Pade approximants (PA). The present review relates the evolution of PA errors inequalities for the Stieltjes functions up to now. Alphonse Magnus and the author (Gilewicz and Magnus (1979) [13]) remarked that the original Stieltjes inequalities are not optimal, not order equilibrated. From this time up to now we have established the optimal inequalities for the errors of PA for the Stieltjes functions in the cases of the classical one-point PA, of two-point (0 and ~) PA and of N-point (N>2) PA. The last case is presented for the first time in the present review.

Published
2010
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32. Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

Author

Mourad Choulli, Yavar Kian, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Carleman inequality, Logarithm, General Mathematics, Mathematics::Analysis of PDEs, Boundary (topology), 01 natural sciences, Stability (probability), Dirichlet distribution, symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Parabolic equation, Zero order, 35R30, 35K20, 35K58, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Inverse problem, Term (time), 010101 applied mathematics, Nonlinear system, logarithmic stability, symbols, semilinear parabolic equation, partial Dirichlet-to-Neumann map, Analysis of PDEs (math.AP)
Abstract

We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding “linearized” Dirichlet-to-Neumann map. This is the first result for this kind of inverse problems. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann [11] . This parabolic Carleman inequality enters in an essential way in the construction of complex geometric optic (abbreviated to CGO in this text) solutions that vanish at a part of the lateral boundary and initial or final time.

Published
2016
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33. What is absolutely continuous spectrum?

Author

Bruneau, L., Jaksic, V., Last, Y., Pillet, C. -A., Pillet, Claude-Alain, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Einstein Institute of Mathematics, The Hebrew University of Jerusalem (HUJ), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)

Subjects
Quantum Physics, scattering theory, Statistical Mechanics (cond-mat.stat-mech), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, spectral theory, Mathematical Physics (math-ph), [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], [PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Quantum Physics (quant-ph), Mathematical Physics, Condensed Matter - Statistical Mechanics, quantum transport, nonequilibrium statistical mechanics
Abstract

This note is an expanded version of the author's contribution to the Proceedings of the ICMP Santiago, 2015, and is based on a talk given by the second author at the same Congress. It concerns a research program devoted to the characterization of the absolutely continuous spectrum of a self-adjoint operator H in terms of the transport properties of a suitable class of open quantum systems canonically associated to H.

Published
2016
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34. Full statistics of energy conservation in two-time measurement protocols

Author

Annalisa Panati, Yan Pautrat, Vojkan Jaksic, Tristan Benoist, Claude-Alain Pillet, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), CPT - E5 Physique statistique et systèmes complexes, Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), NSERC, ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), ANR-12-JS01-0008,SQFT,Théories spectrales et de la diffusion pour des modèles de théorie quantique des champs(2012), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Quantum Physics, Current (mathematics), Statistical Mechanics (cond-mat.stat-mech), FOS: Physical sciences, Mathematical Physics (math-ph), Time limit, Energy conservation, Distribution (mathematics), Probability theory, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Statistics, Statistical physics, Quantum Physics (quant-ph), Quantum statistical mechanics, Anderson impurity model, Mathematical Physics, Condensed Matter - Statistical Mechanics, First law of thermodynamics
Abstract

The first law of thermodynamics states that the average total energy current between different reservoirs vanishes at large times. In this note we examine this fact at the level of the full statistics of two times measurement protocols also known as the Full Counting Statistics. Under very general conditions, we establish a tight form of the first law asserting that the fluctuations of the total energy current computed from the energy variation distribution are exponentially suppressed in the large time limit. We illustrate this general result using two examples: the Anderson impurity model and a 2D spin lattice model., 5 pages, 1 figure. Accepted for publication in Phys. Rev. E

Published
2015
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35. Universal Lower Bounds for Quantum Diffusion

Author

Barbaroux, Jean-Marie, Barbaroux, J.M., Tcheremchantsev, S., Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
dynamical localization, Pure mathematics, double-space method, spectral measure, 01 natural sciences, Upper and lower bounds, Combinatorics, symbols.namesake, moment of order p, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Order (group theory), 0101 mathematics, Schrödinger operators, ComputingMilieux_MISCELLANEOUS, Mathematics, 010102 general mathematics, Spectrum (functional analysis), correlation dimensions, Function (mathematics), Absolute continuity, Moment (mathematics), Fourier transform, Hausdorff dimension, symbols, 010307 mathematical physics, Analysis
Abstract

We study the connections between dynamical properties of Schrodinger operators H on separable Hilbert space H and the properties of corresponding spectral measures. Our main result establishes a relation for the moment of order p of the form 〈〈|X| p 〉 ψ ( t ) 〉(T)≡T −1 ∫ T 0 ‖|X| p /2 e − itH ψ‖ 2 H dt⩾L ψ , p / d (T). (1) Here L ψ , p / d ( T ) is a function connected to the behavior of the Fourier transform of measures in the subclass of measures absolutely continuous with respect to the spectral measure μ ψ . Beyond the intrinsic interest of the general formulation (1), this result allows us to derive necessary conditions for dynamical localization in the presence of a pure point spectrum. On the other hand, if we focus on subsequences of time T k ↗+∞, we can exhibit lower bounds which are, in certain cases, strictly larger than the well-known power-law lower bound for all T expressed in terms of the Hausdorff dimension of spectral measures.

Published
1999
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36. Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field

Author

Bach, Volker, Barbaroux, Jean-Marie, Helffer, Bernard, Siedentop, Heinz, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
generalized Hartree-Fock states, QED, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Dirac operator, General Mathematics, Nuclear Theory, stability of matter
Abstract

International audience; We investigate stability of matter of the Hartree-Fock functional of the relativistic electron-positron field -- in suitable second quantization -- interacting via a second quantized Coulomb field and a classical magnetic field. We are able to show that stability holds for a range of nuclear charges $Z_1,..,Z_K\leq Z$ and fine structure constants $\alpha$ that include the physical value of $\alpha$ and elements up to holmium ($Z=67$).

Published
1998
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37. Hardy inequalities in globally twisted waveguides

Author

Hiba Hammedi, Philippe Briet, David Krejčiřík, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Czech Academy of Sciences [Prague] (CAS)

Subjects
Inequality, Heuristic, media_common.quotation_subject, Mathematical analysis, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Complex system, FOS: Physical sciences, Statistical and Nonlinear Physics, State (functional analysis), Mathematical Physics (math-ph), Mathematics::Spectral Theory, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Dirichlet laplacian, FOS: Mathematics, Applied mathematics, Spectral Theory (math.SP), Mathematical Physics, Mathematics, media_common, Analysis of PDEs (math.AP)
Abstract

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., Comment: 18 pages, 1 figure

Published
2014
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38. Remarks on the Relation between Quantum Dynamics and Fractal Spectra

Author

Barbaroux, Jean-Marie, Barbaroux, J.M, Combes, J.M, Montcho, R, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Quantum dynamics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Hausdorff space, 01 natural sciences, Stability (probability), Spectral line, Schrödinger equation, Global dimension, symbols.namesake, Fractal, Position (vector), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, symbols, 010307 mathematical physics, Statistical physics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Analysis, Mathematics
Abstract

We investigate the large time behaviour of various solutions of the Schrodinger equation in terms of some local and global dimensions of the spectral measure. We emphasize in particular the role of the Hausdorff and correlation dimensions for the growth exponents of position moments. We also discuss the stability of such exponents under local perturbations.

Published
1997
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39. A note on reflectionless Jacobi matrices

Author

Benjamin Landon, Annalisa Panati, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Department of Mathematics (Harvard University), Harvard University [Cambridge], CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Harvard University

Subjects
Pure mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Characterization (mathematics), 01 natural sciences, Mathematics - Spectral Theory, symbols.namesake, 0103 physical sciences, Diagonal matrix, FOS: Mathematics, 0101 mathematics, Spectral Theory (math.SP), Mathematical Physics, Resolvent, Mathematics, 010102 general mathematics, Statistical and Nonlinear Physics, Statistical mechanics, Mathematical Physics (math-ph), 16. Peace & justice, Connection (mathematics), Reflection (mathematics), Jacobian matrix and determinant, symbols, 010307 mathematical physics, Scattering theory, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of [Breuer-Ryckman-Simon]. The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection. To appear in Commun. Math. Phys., Comment: 10 pages

Published
2013
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40. Quantitative estimates on the binding energy for hydrogen in non-relativistic QED, II: The spin case

Author

Semjon Vugalter, Jean-Marie Barbaroux, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Mathematik, Universitaet Muenchen, and ANR-12-JS01-0008,SQFT,Théories spectrales et de la diffusion pour des modèles de théorie quantique des champs(2012)

Subjects
Spectral theory, Hydrogen, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Binding energy, FOS: Physical sciences, chemistry.chemical_element, Pauli-Fierz hamiltonian, 01 natural sciences, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Physics::Atomic Physics, 0101 mathematics, Mathematical Physics, Spin-½, Physics, Zeeman effect, 010308 nuclear & particles physics, 010102 general mathematics, 81Q10, 35P15, 46N50, 81T10, Statistical and Nonlinear Physics, spectral theory, Mathematical Physics (math-ph), chemistry, symbols, Atomic physics, Constant (mathematics), Ground state
Abstract

The hydrogen binding energy in the Pauli–Fierz model with the spin Zeeman term is determined up to the order α3, where α denotes the fine-structure constant.

Published
2014
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