Geodesic
Microlocal Normal Forms for the Magnetic Laplacian
Vũ Ngọc, San1
1 IRMAR (UMR CNRS 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex, France
Journées équations aux dérivées partielles (2014), article no. 12, 12 p.

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We explore symplectic techniques to obtain long time estimates for a purely magnetic confinement in two degrees of freedom. Using pseudo-differential calculus, the same techniques lead to microlocal normal forms for the magnetic Laplacian. In the case of a strong magnetic field, we prove a reduction to a 1D semiclassical pseudo-differential operator. This can be used to derive precise asymptotic expansions for the eigenvalues at any order.

DOI : 10.5802/jedp.115

Vũ Ngọc, San 1

1 IRMAR (UMR CNRS 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex, France 
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Vũ Ngọc, San. Microlocal Normal Forms for the Magnetic Laplacian. Journées équations aux dérivées partielles (2014), article  no. 12, 12 p. doi : 10.5802/jedp.115. http://geodesic.mathdoc.fr/articles/10.5802/jedp.115/

[1] Arnol’d, 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) no. Din. Sist. i Smezhnye Vopr., pp. 9-19 | Zbl | MR

[2] Boutet de Monvel, L.; Trèves, F. On a class of pseudodifferential operators with double characteristics, Invent. Math., Volume 24 (1974), pp. 1-34 | Zbl | MR

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

[4] Cheverry, C. Can one hear whistler waves ? (2014) (preprint hal-00956458) | HAL

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

[6] 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 | Zbl | MR

[7] Hörmander, L. A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann., Volume 217 (1975) no. 2, pp. 165-188 | Zbl | MR

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

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

[10] Raymond, N.; Vũ Ngọc, S Geometry and spectrum in 2D magnetic wells, Ann. Inst. Fourier (Grenoble) (2014) (to appear)

[11] Sjöstrand, J. Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat., Volume 12 (1974), pp. 85-130 | Zbl | MR

[12] Sjöstrand, J. Semi-excited states in nondegenerate potential wells, Asymptotic Analysis, Volume 6 (1992), pp. 29-43 | Zbl | MR

[13] Weinstein, A. Symplectic manifolds and their lagrangian submanifolds, Adv. in Math., Volume 6 (1971), pp. 329-346 | Zbl | MR

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