Microlocal Normal Forms for the Magnetic Laplacian
San Vũ Ngọc1
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.

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
San Vũ Ngọc 1

1 IRMAR (UMR CNRS 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex, France 
@incollection{JEDP_2014____A12_0,
     author = {San V\~{u} Ngọc},
     title = {Microlocal {Normal} {Forms} for the {Magnetic} {Laplacian}},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {12},
     pages = {1--12},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2014},
     doi = {10.5802/jedp.115},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.115/}
}
TY  - JOUR
AU  - San Vũ Ngọc
TI  - Microlocal Normal Forms for the Magnetic Laplacian
JO  - Journées équations aux dérivées partielles
PY  - 2014
SP  - 1
EP  - 12
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.115/
DO  - 10.5802/jedp.115
LA  - en
ID  - JEDP_2014____A12_0
ER  - 
%0 Journal Article
%A San Vũ Ngọc
%T Microlocal Normal Forms for the Magnetic Laplacian
%J Journées équations aux dérivées partielles
%D 2014
%P 1-12
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.115/
%R 10.5802/jedp.115
%G en
%F JEDP_2014____A12_0
San Vũ Ngọc. Microlocal Normal Forms for the Magnetic Laplacian. Journées équations aux dérivées partielles (2014), article  no. 12, 12 p. doi : 10.5802/jedp.115. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.115/

[1] V. I. Arnol’d 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 | MR | Zbl

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

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

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

[5] S. Fournais; B. Helffer 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

[6] B. Helffer; Y. A. Kordyukov 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

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

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

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

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

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

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

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

Cited by Sources: