Microlocal Normal Forms for the Magnetic Laplacian
Vũ Ngọc, San
Journées équations aux dérivées partielles, (2014), p. 1-12 / Harvested from Numdam
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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.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/jedp.115 
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     author = {V\~u Ng\d oc, San},
     title = {Microlocal Normal Forms for the Magnetic Laplacian},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2014},
     pages = {1-12},
     doi = {10.5802/jedp.115},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2014____A12_0}
}

Vũ Ngọc, San. Microlocal Normal Forms for the Magnetic Laplacian. Journées équations aux dérivées partielles,  (2014), pp. 1-12. doi : 10.5802/jedp.115. http://gdmltest.u-ga.fr/item/JEDP_2014____A12_0/

[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, Tome 216 (1997) no. Din. Sist. i Smezhnye Vopr., pp. 9-19 | MR 1632109 | Zbl 0923.58010

[2] Boutet De Monvel, L.; Trèves, F. On a class of pseudodifferential operators with double characteristics, Invent. Math., Tome 24 (1974), pp. 1-34 | MR 353064 | Zbl 0281.35083

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

[4] Cheverry, C. Can one hear whistler waves ? (2014) https://hal.archives-ouvertes.fr/hal-00956458 (preprint hal-00956458)

[5] Fournais, S.; Helffer, B. 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

[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, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 535 (2011), pp. 55-78 | Article | MR 2560751 | Zbl 1218.58017

[7] Hörmander, L. A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann., Tome 217 (1975) no. 2, pp. 165-188 | MR 377603 | Zbl 0306.35032

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

[9] Littlejohn, R. 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

[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., Tome 12 (1974), pp. 85-130 | MR 352749 | Zbl 0317.35076

[12] Sjöstrand, J. Semi-excited states in nondegenerate potential wells, Asymptotic Analysis, Tome 6 (1992), pp. 29-43 | MR 1188076 | Zbl 0782.35050

[13] Weinstein, A. Symplectic manifolds and their lagrangian submanifolds, Adv. in Math., Tome 6 (1971), pp. 329-346 | MR 286137 | Zbl 0213.48203