Cet article est consacré à la mécanique classique et l’analyse spectrale d’un hamiltonien purement magnétique dans . 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.
This paper is devoted to the classical mechanics and spectral analysis of a pure magnetic Hamiltonian in . 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.
Classification : 81Q20, 35Pxx, 35S05, 70Hxx, 37Jxx
Mots clés : champ magnétique, forme normale, théorie spectrale, limite semi-classique, flot hamiltonien, analyse microlocale
@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}, pages = {137--169}, publisher = {Association des Annales de l'institut Fourier}, volume = {65}, number = {1}, year = {2015}, doi = {10.5802/aif.2927}, zbl = {1327.81207}, language = {en}, url = {archive.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, Tome 65 (2015) no. 1, pp. 137-169. doi : 10.5802/aif.2927. http://archive.numdam.org/item/AIF_2015__65_1_137_0/
[1] Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of -body Schrödinger operators, Mathematical Notes, Volume 29, Princeton University Press, Princeton, NJ, 1982, pp. 118 | MR 745286 | Zbl 0503.35001
[2] Bounds on exponential decay of eigenfunctions of Schrödinger operators, Schrödinger operators (Como, 1984) (Lecture Notes in Math.) Volume 1159, Springer, Berlin, 1985, pp. 1-38 | Article | MR 824986 | Zbl 0583.35027
[3] 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, Volume 216 (1997), pp. 9-19 (Din. Sist. i Smezhnye Vopr.) | MR 1632109 | Zbl 0923.58010
[4] Spectral asymptotics via the semiclassical Birkhoff normal form, Duke Math. J., Volume 143 (2008) no. 3, pp. 463-511 | Article | MR 2423760 | Zbl 1154.58015
[5] Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987, pp. x+319 | MR 883643 | Zbl 0619.47005
[6] Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, Volume 268, Cambridge University Press, Cambridge, 1999, pp. xii+227 | Article | MR 1735654 | Zbl 0926.35002
[7] Semiclassical analysis with vanishing magnetic fields, J. Spectr. Theory, Volume 3 (2013) no. 3, pp. 423-464 | Article | MR 3073418
[8] 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 | Article | Numdam | MR 2228679 | Zbl 1097.47020
[9] Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, 77, Birkhäuser Boston Inc., Boston, MA, 2010, pp. xx+324 | MR 2662319 | Zbl 1256.35001
[10] Strong diamagnetism for the ball in three dimensions, Asymptot. Anal., Volume 72 (2011) no. 1-2, pp. 77-123 | MR 2919872 | Zbl 1222.35194
[11] Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, Volume 1336, Springer-Verlag, Berlin, 1988, pp. vi+107 | MR 960278 | Zbl 0647.35002
[12] Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom, J. Funct. Anal., Volume 257 (2009) no. 10, pp. 3043-3081 | Article | MR 2568685 | Zbl 1184.35233
[13] 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 | Article | MR 2560751 | Zbl 1218.58017
[14] Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator, Asymptot. Anal., Volume 82 (2013), pp. 65-89 | MR 3088341 | Zbl 1325.35123
[15] Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., Volume 138 (1996) no. 1, pp. 40-81 | Article | MR 1391630 | Zbl 0851.58046
[16] Magnetic bottles in connection with superconductivity, J. Funct. Anal., Volume 185 (2001) no. 2, pp. 604-680 | Article | MR 1856278 | Zbl 1078.81023
[17] Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case), Ann. Sci. École Norm. Sup. (4), Volume 37 (2004) no. 1, pp. 105-170 | Article | EuDML 82625 | Numdam | MR 2050207 | Zbl 1057.35061
[18] Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mém. Soc. Math. France (N.S.), Volume 39 (1989), pp. 1-124 | EuDML 94884 | Numdam | MR 1041490 | Zbl 0725.34099
[19] Microlocal analysis and precise spectral asymptotics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998, pp. xvi+731 | MR 1631419 | Zbl 0906.35003
[20] Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Volume 13 (1972), p. 135-148 (1973) | MR 333833 | Zbl 0246.35025
[21] 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] A guiding center Hamiltonian: a new approach, J. Math. Phys., Volume 20 (1979) no. 12, pp. 2445-2458 | Article | MR 553507 | Zbl 0444.70020
[23] An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, New York, 2002, pp. viii+190 | MR 1872698 | Zbl 0994.35003
[24] Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998, pp. x+486 | MR 1698616 | Zbl 0844.58029
[25] Hearing the zero locus of a magnetic field, Comm. Math. Phys., Volume 168 (1995) no. 3, pp. 651-675 http://projecteuclid.org/getRecord?id=euclid.cmp/1104272494 | Article | MR 1328258 | Zbl 0827.58076
[26] When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit, SIAM J. Math. Anal., Volume 45 (2013) no. 4, pp. 2354-2395 | Article | MR 3085117 | Zbl 1345.35110
[27] Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2, Ann. Henri Poincaré, Volume 10 (2009) no. 1, pp. 95-122 | Article | MR 2496304 | Zbl 1210.81034
[28] Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model, Comm. Partial Differential Equations, Volume 37 (2012) no. 9, pp. 1528-1552 | Article | MR 2969489 | Zbl 1254.35167
[29] 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 | Article | MR 3148056 | Zbl 1288.82060
[30] Autour de l’approximation semi-classique, Progress in Mathematics, Volume 68, Birkhäuser Boston Inc., Boston, MA, 1987, pp. x+329 | MR 897108 | Zbl 0621.35001
[31] Kato’s inequality and the comparison of semigroups, J. Funct. Anal., Volume 32 (1979) no. 1, pp. 97-101 | Article | MR 533221 | Zbl 0413.47037
[32] Semi-classical asymptotics for magnetic bottles, Asymptot. Anal., Volume 15 (1997) no. 3-4, pp. 385-395 | MR 1487718 | Zbl 0902.35079
[33] L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Volume 105 (1986) no. 2, pp. 327-335 http://projecteuclid.org/getRecord?id=euclid.cmp/1104115337 | Article | MR 849211 | Zbl 0612.35102
[34] Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math., Volume 53 (2000) no. 2, pp. 143-217 | Article | MR 1721373 | Zbl 1027.81012
[35] Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses, Volume 22, Société Mathématique de France, Paris, 2006, pp. vi+156 | MR 2331010 | Zbl 1118.37001
[36] Quantum Birkhoff normal forms and semiclassical analysis, Noncommutativity and singularities (Adv. Stud. Pure Math.) Volume 55, Math. Soc. Japan, Tokyo, 2009, pp. 99-116 | MR 2463493 | Zbl 1185.58012
[37] Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., Volume 6 (1971), p. 329-346 (1971) | Article | MR 286137 | Zbl 0213.48203
[38] Semiclassical analysis, Graduate Studies in Mathematics, Volume 138, American Mathematical Society, Providence, RI, 2012, pp. xii+431 | MR 2952218 | Zbl 1252.58001