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Microlocal Normal Forms for the Magnetic Laplacian

San Vũ Ngọc[1]

  • [1] IRMAR (UMR CNRS 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex, France

Journées Équations aux dérivées partielles (2014)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Vũ Ngọc, San. "Microlocal Normal Forms for the Magnetic Laplacian." Journées Équations aux dérivées partielles (2014): 1-12. <http://eudml.org/doc/275560>.

@article{VũNgọc2014,
abstract = {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.},
affiliation = {IRMAR (UMR CNRS 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex, France},
author = {Vũ Ngọc, San},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
pages = {1-12},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Microlocal Normal Forms for the Magnetic Laplacian},
url = {http://eudml.org/doc/275560},
year = {2014},
}

TY - JOUR
AU - Vũ Ngọc, San
TI - Microlocal Normal Forms for the Magnetic Laplacian
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 12
AB - 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.
LA - eng
UR - http://eudml.org/doc/275560
ER -

References

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  7. L. Hörmander, A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann. 217 (1975), 165-188 Zbl0306.35032MR377603
  8. V. Ivrii, Microlocal analysis and precise spectral asymptotics, (1998), Springer-Verlag, Berlin Zbl0906.35003MR1631419
  9. R. G. Littlejohn, A guiding center Hamiltonian: a new approach, J. Math. Phys. 20 (1979), 2445-2458 Zbl0444.70020MR553507
  10. N. Raymond, S Vũ Ngọc, Geometry and spectrum in 2D magnetic wells, Ann. Inst. Fourier (Grenoble) (2014) Zbl1327.81207
  11. J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat. 12 (1974), 85-130 Zbl0317.35076MR352749
  12. J. Sjöstrand, Semi-excited states in nondegenerate potential wells, Asymptotic Analysis 6 (1992), 29-43 Zbl0782.35050MR1188076
  13. A. Weinstein, Symplectic manifolds and their lagrangian submanifolds, Adv. in Math. 6 (1971), 329-346 Zbl0213.48203MR286137

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