no. 5
The level crossing problem in semi-classical analysis. II. The hermitian case
[Le problème des croisements des valeurs propres en analyse semi-classique. II : le cas hermitien]
Colin de Verdière, Yves1
1 Institut Fourier, 100 rue des Maths, 38402 Saint-Martin d'Hères (France)
Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1423-1441.

Cet article est la seconde partie de l'article «The level crossing problem in semi-classical analysis I. The symmetric case» (Annales de l'Institut Fourier, volume en l'honneur de Frédéric Pham). Nous considérons ici le cas où la matrice de dispersion est hermitienne.

This paper is the second part of the paper “The level crossing problem in semi-classical analysis I. The symmetric case”(Annales de l'Institut Fourier in honor of Frédéric Pham). We consider here the case where the dispersion matrix is complex hermitian.

DOI : 10.5802/aif.2054
Classification : 35C20, 35Q40, 35S30, 53D05

Colin de Verdière, Yves 1

1 Institut Fourier, 100 rue des Maths, 38402 Saint-Martin d'Hères (France) 
@article{AIF_2004__54_5_1423_0,
     author = {Colin de Verdi\`ere, Yves},
     title = {The level crossing problem in semi-classical analysis. {II.} {The} hermitian case},
     journal = {Annales de l'Institut Fourier},
     pages = {1423--1441},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     doi = {10.5802/aif.2054},
     zbl = {1067.35162},
     mrnumber = {2127853},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2054/}
}
TY  - JOUR
AU  - Colin de Verdière, Yves
TI  - The level crossing problem in semi-classical analysis. II. The hermitian case
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 1423
EP  - 1441
VL  - 54
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2054/
DO  - 10.5802/aif.2054
LA  - en
ID  - AIF_2004__54_5_1423_0
ER  - 
%0 Journal Article
%A Colin de Verdière, Yves
%T The level crossing problem in semi-classical analysis. II. The hermitian case
%J Annales de l'Institut Fourier
%D 2004
%P 1423-1441
%V 54
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2054/
%R 10.5802/aif.2054
%G en
%F AIF_2004__54_5_1423_0
Colin de Verdière, Yves. The level crossing problem in semi-classical analysis. II. The hermitian case. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1423-1441. doi : 10.5802/aif.2054. https://aif.centre-mersenne.org/articles/10.5802/aif.2054/

[1] V. Arnold, Mathematical Methods of Classical Mechanics, 2d Edition, Springer, 1988 | Zbl

[2] P. Braam & H. Duistermaat, Normal forms of real symmetric systems with multiplicity, Indag. Math., N.S. (4) (1993) no.4 p. 407-421 | MR | Zbl

[3] Y. Colin De Verdière, The level crossing problem in semi-classical analysis I. The symmetric case, Ann. Inst. Fourier 53 (2003) p. 1023-1054 | Numdam | MR | Zbl

[4] Y. Colin De Verdière, Bohr-Sommerfeld phases for avoided crossings, preprint, 2004

[5] Y. Colin De Verdière, M. Lombardi & J. Pollet, The microlocal Landau-Zener formula, Annales de l'IHP (Physique théorique) 71 (1999) p. 95-127 | Numdam | MR | Zbl

[6] Y. Colin De Verdière & J. Vey, Le lemme de Morse isochore, Topology 18 (1979) p. 283-293 | MR | Zbl

[7] F. Faure & B. Zhilinskii, Topological Chern Indices in Molecular Spectra, Phys. Rev. Letters 85 (2000) p. 960-963

[8] F. Faure & B. Zhilinskii, Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum, Lett. Math. Phys 55 (2001) p. 219-239 | MR | Zbl

[9] C. Fermanian-Kammerer, A non-commutative Landau-Zener formula, Math. Nacht 271 (2004) p. 22-50 | MR | Zbl

[10] C. Fermanian-Kammerer, Wigner measures and molecular propagation through generic energy level crossings, Rev. Math. Phys 15 (2003) p. 1285-1317 | MR | Zbl

[11] C. Fermanian-Kammerer, Semi-classical analysis of generic codimension 3 crossings, International Math. Research Notices 45 (2004) p. 2391-2435 | MR | Zbl

[12] C. Fermanian-Kammerer & P. Gérard, A Landau-Zener formula for non-degenerated involutive codimension 3 crossings, Ann. Henri Poincaré 4 (2003) p. 513-552 | MR | Zbl

[13] W. Flynn & R. Littlejohn, Semi-classical theory of spin-orbit coupling, Phys. Rev. A 45 (1992) p. 7697-7717 | MR

[14] G. Hagedorn, Molecular Propagation through Electron Energy Level Crossings, Memoirs of the AMS 536 (1994) | MR | Zbl

[15] A. Joye, Proof of the Landau-Zener Formula, Asymptotic Analysis 9 (1994) p. 209-258 | MR | Zbl

[16] A. Kaufman & E. Tracy, Ray Helicity: a Geometric Invariant for Multidimensional Resonant Wave Conversion, Phys. Rev. Lett 91 (2003)

[17] J. Moser, On the generalization of a theorem of Liapounoff, Comm. Pure Appl. Math 11 (1958) p. 257-271 | MR | Zbl

[18] E. Nelson, Topics in dynamics, I: Flows, Princeton Univ. Press, 1969 | MR | Zbl

[19] V. Rousse, Landau-Zener Transitions for Eigenvalue Avoided Crossings in the Adiabatic and Born-Oppenheimer Approximations, Asymptotic Analysis 37 (2004) p. 293-328 | MR | Zbl

[20] J. Williamson, On an algebraic problem concerning the normal forms of linear dynamical systems, American J. Maths 58 (1936) p. 141-163 | JFM | MR | Zbl

Cité par Sources :