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12 results on '"Magnetic Laplacian"'

1. Resolvent kernel for the Kohn Laplacian on Heisenberg groups

Author

Neur Eddine Askour and Zouhair Mouayn

Subjects
Kohn Laplacian, Heisenberg group, magnetic Laplacian, resolvent kernel, Green kernel, spectral density., Mathematics, QA1-939
Abstract

We present a formula that relates the Kohn Laplacian on Heisenberg groups and the magnetic Laplacian. Then we obtain the resolvent kernel for the Kohn Laplacian and find its spectral density. We conclude by obtaining the Green kernel for fractional powers of the Kohn Laplacian.

Published
2002

2. Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

Author

Alessandro Savo and Bruno Colbois

Subjects
Mathematics - Differential Geometry, Upper bounds, Mathematical analysis, Zero (complex analysis), Planar domains, Ground state energy, Magnetic Laplacian, Zero magnetic field, Computer Science::Digital Libraries, Upper and lower bounds, 58J50, 35P15, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), Domain (ring theory), Simply connected space, Neumann boundary condition, FOS: Mathematics, Geometry and Topology, Ground state, Constant (mathematics), Laplace operator, Analysis, Mathematics, Analysis of PDEs (math.AP)
Abstract

We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain $$\Omega $$ Ω , with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $$\epsilon $$ ϵ -net. In the last part, we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.

Published
2020

3. Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

Author

Bruno Colbois and Alessandro Savo

Subjects
Mathematics - Differential Geometry, Boundary (topology), Lowest eigenvalue, Magnetic Laplacian, Planar domains, Spectrum, 01 natural sciences, Upper and lower bounds, Domain (mathematical analysis), Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 0103 physical sciences, Simply connected space, FOS: Mathematics, Neumann boundary condition, 0101 mathematics, Spectral Theory (math.SP), Eigenvalues and eigenvectors, Mathematics, 010102 general mathematics, Mathematical analysis, Spectrum (functional analysis), Differential Geometry (math.DG), 010307 mathematical physics, Laplace operator, Analysis, Analysis of PDEs (math.AP)
Abstract

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain $\Omega$, with magnetic Neumann boundary conditions. If $\Omega$ is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of $\Omega$. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles., Comment: 32 pages, 11 figures

Published
2021

4. Lower bounds for the first eigenvalue of the magnetic Laplacian

Author

Bruno Colbois and Alessandro Savo

Subjects
Mathematics - Differential Geometry, 01 natural sciences, Upper and lower bounds, Domain (mathematical analysis), Zero magnetic field, 58J50, 35P15, Mathematics - Analysis of PDEs, Eigenvalues, Magnetic Laplacian, Analysis, 0103 physical sciences, FOS: Mathematics, Neumann boundary condition, 0101 mathematics, Eigenvalues and eigenvectors, Mathematics, 010102 general mathematics, Mathematical analysis, Differential Geometry (math.DG), Bounded function, Product (mathematics), Metric (mathematics), 010307 mathematical physics, Laplace operator, Analysis of PDEs (math.AP)
Abstract

We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate., Replaces in part arXiv:1611.01930

Published
2017

5. Interpolation inequalities and spectral estimates for magnetic operators

Author

Ari Laptev, Jean Dolbeault, Maria J. Esteban, Michael Loss, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Mathematics [Imperial College London], Imperial College London, School of Mathematics - Georgia Institute of Technology, Georgia Institute of Technology [Atlanta], ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013), and ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017)

Subjects
Gagliardo-Nirenberg inequalities, Nuclear and High Energy Physics, SYMMETRY, Physics, Multidisciplinary, POSITIVE SOLUTIONS, 01 natural sciences, Upper and lower bounds, Physics, Particles & Fields, 0202 Atomic, Molecular, Nuclear, Particle And Plasma Physics, Mathematics - Analysis of PDEs, 35P30, 26D10, 46E35, 35J61, 35J10, 35Q40, 35Q55, 35B40, 46N50, 47N50, 47N50, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, spectral gap, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, Science & Technology, 0105 Mathematical Physics, logarithmic Sobolev inequalities, Physics, Numerical analysis, 010102 general mathematics, Principal (computer security), magnetic Schrödinger operator, Statistical and Nonlinear Physics, Magnetic Laplacian, optimal constants, Mathematics::Spectral Theory, FIELDS, interpolation, Keller-Lieb-Thirring estimates, Physics, Mathematical, UNIQUENESS, Physical Sciences, SCHRODINGER-EQUATION, 010307 mathematical physics, Analysis of PDEs (math.AP), Interpolation
Abstract

International audience; We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring estimates for the principal eigenvalue of magnetic Schrödinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate.

Published
2017

6. From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit

Author

Nicolas Raymond, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Spectral theory, 35J10, 35B25, 35P15, 35J10, Semiclassical physics, magnetic field, 01 natural sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], normal form, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Eigenvalues and eigenvectors, Mathematics, Numerical Analysis, Applied Mathematics, 010102 general mathematics, Mathematical analysis, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Magnetic Laplacian, spectral theory, Mathematics::Spectral Theory, Vector Laplacian, 010101 applied mathematics, Agmon estimates, Bounded function, Laplacian matrix, Laplace operator, Analysis, 35P15, semiclassical analysis
Abstract

43 pages; International audience; We consider a twisted magnetic Laplacian with Neumann condition on a smooth and bounded domain of $\R^2$ in the semiclassical limit $h\to 0$. Under generic assumptions, we prove that the eigenvalues admit complete asymptotic expansions in powers of $h^{1/4}$.

Published
2013

7. Magnetic WKB Constructions

Author

Virginie Bonnaillie-Noël, Nicolas Raymond, Frédéric Hérau, Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray ( LMJL ), Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ), Labex Centre Henri Lebesgue n° ANR-11-LABX-0020-01, ANR-11-BS01-0019,NOSEVOL,Opérateurs non-autoadjoints, analyse semiclassique et problèmes d'évolution ( 2011 ), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), ANR-11-BS01-0019,NOSEVOL,Opérateurs non-autoadjoints, analyse semiclassique et problèmes d'évolution(2011), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), École normale supérieure - Paris (ENS-PSL), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
FOS: Physical sciences, Semiclassical physics, coherent states, 01 natural sciences, Upper and lower bounds, WKB approximation, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], magnetic laplacian, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Born-Oppenheimer approximation, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, Mathematical physics, WKB expansion, Eikonal equation, Mechanical Engineering, 010102 general mathematics, Mathematical analysis, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Mathematical Physics (math-ph), Eigenfunction, tunnel effect, 010101 applied mathematics, Agmon estimates, 67 pages, Coherent states, Laplace operator, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman-Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions. We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal estimates of Agmon and upper bound of the tunelling effect (in symmetric cases). We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades.

Published
2016

8. Peak power in the 3D magnetic Schrödinger equation

Author

Virginie Bonnaillie-Noël, Nicolas Raymond, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Aperture, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Magnetic Laplacian, Mathematics::Spectral Theory, 01 natural sciences, Schrödinger equation, Magnetic field, 010101 applied mathematics, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Cone (topology), symbols, singular 3D domain, 35P15, 35J10, 81Q10, 81Q15, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Constant (mathematics), Laplace operator, Analysis, Eigenvalues and eigenvectors, Mathematics, spectral asymptotics
Abstract

28 pages; International audience; This paper is devoted to the spectral analysis of the magnetic Neumann Laplacian on an infinite cone of aperture $\alpha$. When the magnetic field is constant and parallel to the revolution axis and when the aperture goes to zero, we prove that the first $n$ eigenvalues exist and admit asymptotic expansions in powers of $\alpha^2$.

Published
2013

9. Magnetic Neumann Laplacian on a sharp cone

Author

Nicolas Raymond, Virginie Bonnaillie-Noël, Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS-PSL), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Essential spectrum, FOS: Physical sciences, 01 natural sciences, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Neumann boundary condition, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], singular 3D domain, 0101 mathematics, Eigenvalues and eigenvectors, Mathematical Physics, Mathematics, spectral asymptotics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Magnetic Laplacian, Mathematical Physics (math-ph), Eigenfunction, Mathematics::Spectral Theory, Magnetic field, 35P15, 15J10, 81Q10, 81Q15, 010101 applied mathematics, Orientation (vector space), Asymptotic expansion, Laplace operator, Analysis, Analysis of PDEs (math.AP)
Abstract

This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture $\alpha$ and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit $\alpha\to 0$ and establish a full asymptotic expansion for the $n$-th eigenvalue and the $n$-th eigenfunction., Comment: 22 pages

Published
2013
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10. When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit

Author

Nicolas Raymond, Nicolas Popoff, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Spectral theory, Semiclassical analysis, Normal form, Semiclassical physics, Boundary (topology), Edge (geometry), 01 natural sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 35P15, 35Q40, 81Q20, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Singular domain, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Order (ring theory), Magnetic Laplacian, Computational Mathematics, 010307 mathematical physics, Reduction (mathematics), Laplace operator, Analysis
Abstract

International audience; We study the magnetic Laplacian in the case when the Neumann boundary contains an edge. We provide complete asymptotic expansions in powers of $h^{1/4}$ of the low lying eigenpairs in the semiclassical limit $h\to 0$. In order to get our main result we establish a general method based on a normal form procedure, microlocal arguments, the Feshbach-Grushin reduction and the Born-Oppenheimer approximation.

Published
2013

11. Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model

Author

Nicolas Raymond, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Spectral theory, Semiclassical physics, 01 natural sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Eigenvalues and eigenvectors, Mathematics, Toy model, Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], 35P15, 35J15, 81Q20, Magnetic Laplacian, spectral theory, Eigenfunction, Mathematics::Spectral Theory, 010101 applied mathematics, Agmon estimates, Asymptotic expansion, Laplace operator, Analysis, semiclassical analysis
Abstract

25 pages; International audience; In this paper we investigate the semiclassical behavior of the lowest eigenvalues of a model Schrödinger operator with variable magnetic field. This work aims at proving an accurate asymptotic expansion for these eigenvalues, the corresponding upper bound being already proved in the general case. The present work also aims at establishing localization estimates for the attached eigenfunctions.

Published
2012

12. Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

Author

Abderemane Morame, Françoise Truc, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Institut Fourier (IF ), Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)

Subjects
Betti number, General Mathematics, Star (game theory), magnetic Laplacian, Boundary (topology), FOS: Physical sciences, 01 natural sciences, Combinatorics, cuspidal manifold, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 35P20, 0103 physical sciences, Asymptotic formula, 0101 mathematics, Remainder, Mathematical Physics, Mathematics, spectral asymptotics, Cusp (singularity), 010102 general mathematics, Mathematical analysis, Mathematical Physics (math-ph), Mathematics::Spectral Theory, Manifold, 010307 mathematical physics, Laplace operator, embedded eigenvalues
Abstract

International audience; We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to ${\bf{X}}\times ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2\delta}.$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of the Laplace-Beltrami operator $-\Delta =-\Delta_0 .$

Published
2011

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