Geometry and Spectrum in 2D Magnetic Wells

Raymond, Nicolas; Vũ Ngọc, San

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

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.

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.

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

[2] Agmon, Shmuel 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 | DOI | MR | Zbl

[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, Volume 216 (1997), pp. 9-19 (Din. Sist. i Smezhnye Vopr.) | MR | Zbl

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

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

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

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

[8] Fournais, Søren; Helffer, Bernard Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 1, pp. 1-67 | DOI | Numdam | MR | Zbl

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

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

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

[12] Helffer, Bernard; Kordyukov, Yuri A. 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 | DOI | MR | Zbl

[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 (Contemp. Math.), Volume 535, Amer. Math. Soc., Providence, RI, 2011, pp. 55-78 | DOI | MR | Zbl

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

[15] Helffer, Bernard; Mohamed, Abderemane 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 | DOI | MR | Zbl

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

[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), Volume 37 (2004) no. 1, pp. 105-170 | DOI | EuDML | Numdam | MR | Zbl

[18] Helffer, Bernard; Sjöstrand, J. 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 | Numdam | MR | Zbl

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

[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), Volume 13 (1972), p. 135-148 (1973) | MR | Zbl

[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

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

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

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

[25] Montgomery, Richard 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 | DOI | MR | Zbl

[26] Popoff, Nicolas; Raymond, Nicolas 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 | DOI | MR | Zbl

[27] Raymond, Nicolas 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 | DOI | MR | Zbl

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

[29] Raymond, Nicolas 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 | Zbl

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

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

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

[33] Colin de Verdière, Yves 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 | DOI | MR | Zbl

[34] Vũ Ngọc, San 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 | DOI | MR | Zbl

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

[36] Vũ Ngọc, San 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 | Zbl

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

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