Curvature induced magnetic bound states: towards the tunneling effect for the ellipse
Journées équations aux dérivées partielles (2016), Talk no. 3, 14 p.

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.

Published online:
DOI: 10.5802/jedp.644
Bonnaillie-Noël, Virginie 1; Hérau, Frédéric 2; Raymond, Nicolas 3

1 DMA - UMR CNRS 8553 PSL Research University CNRS, ENS Paris 45 rue d’Ulm F-75230 Paris cedex 05, France
2 LMJL - UMR CNRS 6629 Université de Nantes, CNRS 2 rue de la Houssinière BP 92208 F-44322 Nantes cedex 3, France
3 IRMAR - UMR CNRS 8625 Université Rennes 1, CNRS Campus de Beaulieu F-35042 Rennes cedex, France
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Bonnaillie-Noël, Virginie; Hérau, Frédéric; Raymond, Nicolas. Curvature induced magnetic bound states:  towards the tunneling effect for the ellipse. Journées équations aux dérivées partielles (2016), Talk no. 3, 14 p. doi : 10.5802/jedp.644. http://archive.numdam.org/articles/10.5802/jedp.644/

[1] Bernoff, A.; Sternberg, P. Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., Volume 39 (1998) no. 3, pp. 1272-1284 | DOI | MR

[2] Bonnaillie-Noël, V. Harmonic oscillators with Neumann condition of the half-line, Commun. Pure Appl. Anal., Volume 11 (2012) no. 6, pp. 2221-2237 | DOI | MR

[3] Bonnaillie-Noël, V.; Hérau, F.; Raymond, N. Magnetic WKB constructions, Arch. Ration. Mech. Anal., Volume 221 (2016) no. 2, pp. 817-891 | DOI | MR

[4] Bonnaillie-Noël, V.; Hérau, F.; Raymond, N. Semiclassical tunneling and magnetic flux effects on the circle, J. Spectr. Theory (2017), to appear pages

[5] Dauge, M.; Helffer, B. Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators, J. Differential Equations, Volume 104 (1993) no. 2, pp. 243-262 | DOI | MR

[6] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999, xii+227 pages | DOI | MR

[7] Dombrowski, N.; Raymond, N. Semiclassical analysis with vanishing magnetic fields, J. Spectr. Theory, Volume 3 (2013) no. 3, pp. 423-464 | MR

[8] Fournais, S.; Helffer, B. Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 1, pp. 1-67 http://aif.cedram.org/item?id=AIF_2006__56_1_1_0 | MR

[9] Fournais, S.; Helffer, B. Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, 77, Birkhäuser Boston Inc., Boston, MA, 2010, xx+324 pages | MR

[10] Helffer, B.; Kachmar, A.; Raymond, N. Tunneling for the Robin Laplacian in smooth planar domains, To appear in Commun. Contempt. Math. (arXiv:1509.03986) (2016)

[11] Helffer, B.; Kordyukov, Y. A. Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells, Spectral theory and geometric analysis (Contemp. Math.), Volume 535, Amer. Math. Soc., Providence, RI, 2011, pp. 55-78 | DOI | MR

[12] Helffer, B.; Kordyukov, Y. A. Accurate semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator, Ann. Henri Poincaré, Volume 16 (2015) no. 7, pp. 1651-1688 | DOI | MR

[13] Helffer, B.; Morame, A. Magnetic bottles in connection with superconductivity, J. Funct. Anal., Volume 185 (2001) no. 2, pp. 604-680 | DOI | MR

[14] Helffer, B.; Sjöstrand, J. Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations, Volume 9 (1984) no. 4, pp. 337-408 | DOI | MR

[15] Martin, D. Mélina, bibliothèque de calculs éléments finis., http://anum-maths.univ-rennes1.fr/melina (2010)

[16] Outassourt, A. Comportement semi-classique pour l’opérateur de Schrödinger à potentiel périodique, J. Funct. Anal., Volume 72 (1987) no. 1, pp. 65-93 | DOI | MR

[17] Raymond, N. From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit, Anal. PDE, Volume 6 (2013) no. 6, pp. 1289-1326 | DOI | MR

[18] Raymond, N. Bound states of the Magnetic Schrödinger Operator, EMS Tracts in Mathematics, 27, European Mathematical Society, 2017

[19] Raymond, N.; Vũ Ngọc, S. Geometry and spectrum in 2D magnetic wells, Ann. Inst. Fourier (Grenoble), Volume 65 (2015) no. 1, pp. 137-169 http://aif.cedram.org/item?id=AIF_2015__65_1_137_0 | MR

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