Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Virginie Bonnaillie-Noël; Frédéric Hérau; Nicolas Raymond
Curvature induced magnetic bound states: towards the tunneling effect for the ellipse
Journées équations aux dérivées partielles (2016), Exp. No. 3, 14 p., doi: 10.5802/jedp.644
Article PDF

Résumé - Abstract

This article is devoted to the semiclassical analysis of the magnetic Laplacian on a smooth domain of the plane carrying Neumann boundary conditions. We provide WKB expansions of the eigenfunctions when Neumann boundary traps the lowest eigenfunctions near the points of maximal curvature. We also explain and illustrate a conjecture of magnetic tunneling when the domain is an ellipse.

Bibliographie

[1] A. Bernoff & P. Sternberg, “Onset of superconductivity in decreasing fields for general domains”, J. Math. Phys. 39 (1998) no. 3, p. 1272-1284 Article
[2] V. Bonnaillie-Noël, “Harmonic oscillators with Neumann condition of the half-line”, Commun. Pure Appl. Anal. 11 (2012) no. 6, p. 2221-2237 Article
[3] V. Bonnaillie-Noël, F. Hérau & N. Raymond, “Magnetic WKB constructions”, Arch. Ration. Mech. Anal. 221 (2016) no. 2, p. 817-891 Article
[4] V. Bonnaillie-Noël, F. Hérau & N. Raymond, “Semiclassical tunneling and magnetic flux effects on the circle”, J. Spectr. Theory (2017)
[5] M. Dauge & B. Helffer, “Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators”, J. Differential Equations 104 (1993) no. 2, p. 243-262 Article
[6] M. Dimassi & J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, Cambridge, 1999 Article
[7] N. Dombrowski & N. Raymond, “Semiclassical analysis with vanishing magnetic fields”, J. Spectr. Theory 3 (2013) no. 3, p. 423-464
[8] S. Fournais & B. Helffer, “Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian”, Ann. Inst. Fourier (Grenoble) 56 (2006) no. 1, p. 1-67
[9] S. Fournais & B. Helffer, Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, 77, Birkhäuser Boston Inc., Boston, MA, 2010
[10] B. Helffer, A. Kachmar & N. Raymond, “Tunneling for the Robin Laplacian in smooth planar domains”, To appear in Commun. Contempt. Math. (arXiv:1509.03986) (2016)
[11] B. Helffer & Y. A. Kordyukov, Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells, Spectral theory and geometric analysis, Contemp. Math. 535, Amer. Math. Soc., 2011, p. 55–78 Article
[12] B. Helffer & Y. A. Kordyukov, “Accurate semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator”, Ann. Henri Poincaré 16 (2015) no. 7, p. 1651-1688 Article
[13] B. Helffer & A. Morame, “Magnetic bottles in connection with superconductivity”, J. Funct. Anal. 185 (2001) no. 2, p. 604-680 Article
[14] B. Helffer & J. Sjöstrand, “Multiple wells in the semiclassical limit. I”, Comm. Partial Differential Equations 9 (1984) no. 4, p. 337-408 Article
[15] D. Martin, “Mélina, bibliothèque de calculs éléments finis.”, http://anum-maths.univ-rennes1.fr/melina (2010)
[16] A. Outassourt, “Comportement semi-classique pour l’opérateur de Schrödinger à potentiel périodique”, J. Funct. Anal. 72 (1987) no. 1, p. 65-93 Article
[17] N. Raymond, “From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit”, Anal. PDE 6 (2013) no. 6, p. 1289-1326 Article
[18] N. Raymond, Bound states of the Magnetic Schrödinger Operator, EMS Tracts in Mathematics 27, European Mathematical Society, 2017
[19] N. Raymond & S. Vũ Ngọc, “Geometry and spectrum in 2D magnetic wells”, Ann. Inst. Fourier (Grenoble) 65 (2015) no. 1, p. 137-169
Copyright Cellule MathDoc 2018 | Crédit | Plan du site