Geometry and Spectrum  in 2D Magnetic Wells

Raymond, Nicolas; Vũ Ngọc, San

doi:10.5802/aif.2927

Geometry and Spectrum in 2D Magnetic Wells

Raymond, Nicolas; Vũ Ngọc, San

Annales de l'Institut Fourier, Volume 65 (2015) no. 1, p. 137-169

Abstract
Résumé

This paper is devoted to the classical mechanics and spectral analysis of a pure magnetic Hamiltonian in $ℝ^{2}$ . It is established that both the dynamics and the semiclassical spectral theory can be treated through a Birkhoff normal form and reduced to the study of a family of one dimensional Hamiltonians. As a corollary, recent results by Helffer-Kordyukov are extended to higher energies.

Cet article est consacré à la mécanique classique et l’analyse spectrale d’un hamiltonien purement magnétique dans $ℝ^{2}$ . On démontre que la dynamique et la théorie spectrale semi-classique peuvent être traitées par une forme normale de Birkhoff, et ainsi réduites à l’étude d’une famille d’hamiltoniens à un degré de liberté. Corollairement, on obtient une extension de résultats récents de Helffer et Kordyukov à de plus hautes énergies.

Zbl 1327.81207

DOI : https://doi.org/10.5802/aif.2927
Classification: 81Q20, 35Pxx, 35S05, 70Hxx, 37Jxx
Keywords: magnetic field, normal form, spectral theory, semiclassical limit, Hamiltonian flow, microlocal analysis

@article{AIF_2015__65_1_137_0,
     author = {Raymond, Nicolas and V\~u Ng\d oc, San},
     title = {Geometry and Spectrum  in 2D Magnetic Wells},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     pages = {137-169},
     doi = {10.5802/aif.2927},
     zbl = {1327.81207},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_1_137_0}
}

Raymond, Nicolas; Vũ Ngọc, San. Geometry and Spectrum  in 2D Magnetic Wells. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 137-169. doi : 10.5802/aif.2927. http://www.numdam.org/item/AIF_2015__65_1_137_0/

References

[1] Agmon, Shmuel Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$ -body Schrödinger operators, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 29 (1982), pp. 118 | MR 745286 | Zbl 0503.35001

[2] Agmon, Shmuel Bounds on exponential decay of eigenfunctions of Schrödinger operators, Schrödinger operators (Como, 1984), Springer, Berlin (Lecture Notes in Math.) Tome 1159 (1985), pp. 1-38 | Article | MR 824986 | Zbl 0583.35027

[3] ArnolʼD, V. I. Remarks on the Morse theory of a divergence-free vector field, the averaging method, and the motion of a charged particle in a magnetic field, Tr. Mat. Inst. Steklova, Tome 216 (1997), pp. 9-19 (Din. Sist. i Smezhnye Vopr.) | MR 1632109 | Zbl 0923.58010

[4] Charles, Laurent; Vũ Ngọc, San Spectral asymptotics via the semiclassical Birkhoff normal form, Duke Math. J., Tome 143 (2008) no. 3, pp. 463-511 | Article | MR 2423760 | Zbl 1154.58015

[5] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B. Schrödinger operators with application to quantum mechanics and global geometry, Springer-Verlag, Berlin, Texts and Monographs in Physics (1987), pp. x+319 | MR 883643 | Zbl 0619.47005

[6] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 268 (1999), pp. xii+227 | Article | MR 1735654 | Zbl 0926.35002

[7] Dombrowski, Nicolas; Raymond, Nicolas Semiclassical analysis with vanishing magnetic fields, J. Spectr. Theory, Tome 3 (2013) no. 3, pp. 423-464 | Article | MR 3073418

[8] Fournais, Søren; Helffer, Bernard Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 1, pp. 1-67 http://aif.cedram.org/item?id=AIF_2006__56_1_1_0 | Article | Numdam | MR 2228679 | Zbl 1097.47020

[9] Fournais, Søren; Helffer, Bernard Spectral methods in surface superconductivity, Birkhäuser Boston Inc., Boston, MA, Progress in Nonlinear Differential Equations and their Applications, 77 (2010), pp. xx+324 | MR 2662319 | Zbl 1256.35001

[10] Fournais, Søren; Persson, Mikael Strong diamagnetism for the ball in three dimensions, Asymptot. Anal., Tome 72 (2011) no. 1-2, pp. 77-123 | MR 2919872 | Zbl 1222.35194

[11] Helffer, Bernard Semi-classical analysis for the Schrödinger operator and applications, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1336 (1988), pp. vi+107 | MR 960278 | Zbl 0647.35002

[12] Helffer, Bernard; Kordyukov, Yuri A. Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom, J. Funct. Anal., Tome 257 (2009) no. 10, pp. 3043-3081 | Article | MR 2568685 | Zbl 1184.35233

[13] Helffer, Bernard; Kordyukov, Yuri A. Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells, Spectral theory and geometric analysis, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 535 (2011), pp. 55-78 | Article | MR 2560751 | Zbl 1218.58017

[14] Helffer, Bernard; Kordyukov, Yuri A. Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator, Asymptot. Anal., Tome 82 (2013), pp. 65-89 | MR 3088341 | Zbl 1325.35123

[15] Helffer, Bernard; Mohamed, Abderemane Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., Tome 138 (1996) no. 1, pp. 40-81 | Article | MR 1391630 | Zbl 0851.58046

[16] Helffer, Bernard; Morame, Abderemane Magnetic bottles in connection with superconductivity, J. Funct. Anal., Tome 185 (2001) no. 2, pp. 604-680 | Article | MR 1856278 | Zbl 1078.81023

[17] Helffer, Bernard; Morame, Abderemane Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case), Ann. Sci. École Norm. Sup. (4), Tome 37 (2004) no. 1, pp. 105-170 | Article | Numdam | MR 2050207 | Zbl 1057.35061

[18] Helffer, Bernard; Sjöstrand, J. Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mém. Soc. Math. France (N.S.), Tome 39 (1989), pp. 1-124 | Numdam | MR 1041490 | Zbl 0725.34099

[19] Ivrii, Victor Microlocal analysis and precise spectral asymptotics, Springer-Verlag, Berlin, Springer Monographs in Mathematics (1998), pp. xvi+731 | MR 1631419 | Zbl 0906.35003

[20] Kato, Tosio Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Tome 13 (1972), p. 135-148 (1973) | MR 333833 | Zbl 0246.35025

[21] Lieb, E.H.; Thirring, W. Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics (1976), pp. 269-303 | Zbl 0342.35044

[22] Littlejohn, Robert G. A guiding center Hamiltonian: a new approach, J. Math. Phys., Tome 20 (1979) no. 12, pp. 2445-2458 | Article | MR 553507 | Zbl 0444.70020

[23] Martinez, André An introduction to semiclassical and microlocal analysis, Springer-Verlag, New York, Universitext (2002), pp. viii+190 | MR 1872698 | Zbl 0994.35003

[24] Mcduff, Dusa; Salamon, Dietmar Introduction to symplectic topology, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1998), pp. x+486 | MR 1698616 | Zbl 1066.53137 | Zbl 0844.58029

[25] Montgomery, Richard Hearing the zero locus of a magnetic field, Comm. Math. Phys., Tome 168 (1995) no. 3, pp. 651-675 http://projecteuclid.org/getRecord?id=euclid.cmp/1104272494 | Article | MR 1328258 | Zbl 0827.58076

[26] Popoff, Nicolas; Raymond, Nicolas When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit, SIAM J. Math. Anal., Tome 45 (2013) no. 4, pp. 2354-2395 | Article | MR 3085117 | Zbl 1345.35110

[27] Raymond, Nicolas Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2, Ann. Henri Poincaré, Tome 10 (2009) no. 1, pp. 95-122 | Article | MR 2496304 | Zbl 1210.81034

[28] Raymond, Nicolas Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model, Comm. Partial Differential Equations, Tome 37 (2012) no. 9, pp. 1528-1552 | Article | MR 2969489 | Zbl 1254.35167

[29] Raymond, Nicolas From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit, Anal. PDE, Tome 6 (2013) no. 6, pp. 1289-1326 | Article | MR 3148056 | Zbl 1288.82060

[30] Robert, Didier Autour de l’approximation semi-classique, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 68 (1987), pp. x+329 | MR 897108 | Zbl 0621.35001

[31] Simon, Barry Kato’s inequality and the comparison of semigroups, J. Funct. Anal., Tome 32 (1979) no. 1, pp. 97-101 | Article | MR 533221 | Zbl 0413.47037

[32] Truc, Françoise Semi-classical asymptotics for magnetic bottles, Asymptot. Anal., Tome 15 (1997) no. 3-4, pp. 385-395 | MR 1487718 | Zbl 0902.35079

[33] Colin De Verdière, Yves L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Tome 105 (1986) no. 2, pp. 327-335 http://projecteuclid.org/getRecord?id=euclid.cmp/1104115337 | Article | MR 849211 | Zbl 0612.35102

[34] Vũ Ngọc, San Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math., Tome 53 (2000) no. 2, pp. 143-217 | Article | MR 1721373 | Zbl 1027.81012

[35] Vũ Ngọc, San Systèmes intégrables semi-classiques: du local au global, Société Mathématique de France, Paris, Panoramas et Synthèses, Tome 22 (2006), pp. vi+156 | MR 2331010 | Zbl 1118.37001

[36] Vũ Ngọc, San Quantum Birkhoff normal forms and semiclassical analysis, Noncommutativity and singularities, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 55 (2009), pp. 99-116 | MR 2463493 | Zbl 1185.58012

[37] Weinstein, Alan Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., Tome 6 (1971), p. 329-346 (1971) | Article | MR 286137 | Zbl 0213.48203

[38] Zworski, Maciej Semiclassical analysis, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 138 (2012), pp. xii+431 | MR 2952218 | Zbl 1252.58001