We describe a microlocal normal form for a symmetric system of pseudo-differential equations whose principal symbol is a real symmetric matrix with a generic crossing of eigenvalues. We use it in order to give a precise description of the microlocal solutions.
Nous décrivons une forme normale microlocale pour un système symétrique d'équations pseudo-différentielles dont le symbole principal est à valeurs matrices symétriques réelles ayant un croisement générique de valeurs propres. Nous utilisons cette forme normale pour décrire de façon précise les solutions mucrolocales.
Classification: 35C20, 35Q40, 35S30
Keywords: mode conversion, polarization, Born-Oppenheimer approximation, Maxwell equations, eigenvalue crossing, pseudo-differential systems, semi-classical analysis, lagrangian manifold, propagation of singularities, coherent states, symplectic spinors
@article{AIF_2003__53_4_1023_0, author = {Colin de Verdi\`ere, Yves}, title = {The level crossing problem in semi-classical analysis {I.} {The} symmetric case}, journal = {Annales de l'Institut Fourier}, pages = {1023--1054}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {4}, year = {2003}, doi = {10.5802/aif.1973}, mrnumber = {2033509}, zbl = {02014671}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1973/} }
TY - JOUR TI - The level crossing problem in semi-classical analysis I. The symmetric case JO - Annales de l'Institut Fourier PY - 2003 DA - 2003/// SP - 1023 EP - 1054 VL - 53 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1973/ UR - https://www.ams.org/mathscinet-getitem?mr=2033509 UR - https://zbmath.org/?q=an%3A02014671 UR - https://doi.org/10.5802/aif.1973 DO - 10.5802/aif.1973 LA - en ID - AIF_2003__53_4_1023_0 ER -
Colin de Verdière, Yves. The level crossing problem in semi-classical analysis I. The symmetric case. Annales de l'Institut Fourier, Volume 53 (2003) no. 4, pp. 1023-1054. doi : 10.5802/aif.1973. https://aif.centre-mersenne.org/articles/10.5802/aif.1973/
[1] Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publication Inc., New York, 1970 | MR: 1225604
[2] Singularities of Caustics and Wave Fronts, Kluwer, 1990 | MR: 1151185 | Zbl: 0734.53001
[3] On the interior scattering of waves, defined by hyperbolic variational principles, J. Geom. Phys., Tome 5 (1988), pp. 305-315 | Article | MR: 1048504 | Zbl: 0699.35199
[4] Born-Oppenheimer wave function near level crossing, Phys. Rev. A, Tome 62-06254 (2000)
[5] Zür Quantentheorie der Molekeln, Annal. Phys., Tome 84 (1927), pp. 457-484 | Article | JFM: 53.0845.04
[6] The Spectral Theory of Toeplitz Operators, Princeton Univ. Press, 1981 | MR: 620794 | Zbl: 0469.47021
[7] Normal forms of real symmetric systems with multiplicity, Indag. Math., N.S., Tome 4 (1993) no. 4, pp. 407-421 | Article | MR: 1252985 | Zbl: 0802.35176
[8] Singular Lagrangian manifolds and semi-classical analysis, Duke Math. J., Tome 116 (2003), pp. 263-298 | Article | MR: 1953293 | Zbl: 01911548
[9] Spectres de Graphes, Soc. Math. France, 1998 | MR: 1652692 | Zbl: 0913.05071
[10] The level crossing problem in semi-classical analysis. II. The Hermitian case (2003) (Prépublication Institut Fourier no 603)
[11] The microlocal Landau-Zener formula, Ann. IHP (Phys. théorique), Tome 71 (1999), pp. 95-127 | Numdam | MR: 1704655 | Zbl: 0986.81027
[12] Équilibre instable en régime semi-classique. I. Concentration microlocale, Commun. in PDE, Tome 19 (1994), pp. 1535-1563 | Article | MR: 1294470 | Zbl: 0819.35116
[13] Le lemme de Morse isochore, Topology, Tome 18 (1979), pp. 283-293 | Article | MR: 551010 | Zbl: 0441.58003
[14] On the Born-Oppenheimer approximation, International Symposium on Mathematical Problems in Theoretical Physics, Kyoto Univ., Kyoto (Lecture Notes in Phys.) Tome 39 (1975), pp. 467-471 | Zbl: 0372.47026
[15] Spectral properties of atomic and molecular systems, Quantum dynamics of molecules (Proc. NATO Adv. Study Inst.) (1979), pp. 435-482
[16] Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptotic Anal., Tome 14 (1997), pp. 377-404 | MR: 1461126 | Zbl: 0894.35026
[17] Avoided Crossings in Mesoscopic Systems: Electron Propagation on a Non-uniform Magnetic Cylinder, J. Math. Phys., Tome 42 (2001), pp. 4707-4738 | Article | MR: 1855092 | Zbl: 1017.81050
[18] Geometry of the transport Equation in Multicomponent WKB Approximations, Commun. Math. Phys., Tome 176 (1996), pp. 701-711 | Article | MR: 1376438 | Zbl: 0848.34039
[19] Topological Chern Indices in Molecular Spectra, Phys. Rev. Letters, Tome 85 (2000), pp. 960-963 | Article
[20] Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum, Lett. Math. Phys., Tome 55 (2001), pp. 219-239 | Article | MR: 1843445 | Zbl: 0981.81028
[21] A non-commutative Landau-Zener formula (2002) (Prépublication Université de Cergy-Pontoise, 01/02, 1-34) | Zbl: 1050.35093
[22] Une formule de Landau-Zener pour un croisement de codimension 2, Séminaire Équations aux dérivées partielles, École Polytechnique, Tome exposé XXI, 9/04 (2002)
[23] Wigner measures and codimension two crossings (2002) (Preprint mp-arc, 02-186) | MR: 1952198 | Zbl: 1061.81078
[24] Normal forms for linear mode conversion and Landau-Zener transitions in one dimension, Ann. Physics, Tome 234 (1994) no. 2, pp. 334-403 | Article | MR: 1289934
[25] Semi-classical theory of spin-orbit coupling, Phys. Rev. A, Tome 45 (1992), pp. 7697-7717 | Article | MR: 1167156
[26] Harmonic Analysis on Phase Space, Princeton Univ. Press, 1989 | MR: 983366 | Zbl: 0682.43001
[27] Mesures semi-classiques et croisement de modes, Bull. Soc. Math. France, Tome 130 (2002), pp. 123-168 | Numdam | MR: 1906196 | Zbl: 0996.35004
[28] A Landau-Zener formula for non-degenerated involutive codimension 3 crossings (sept. 2002) (Preprint) | MR: 2007256 | Zbl: 1049.81029
[29] Symplectic spinors and PDE, Géométrie Symplectique et Physique mathématique (Colloque CNRS Aix-en-Provence) (1974)
[30] Molecular Propagation through Electron Energy Level Crossings, Memoirs of the AMS, Tome 536 (1994) | MR: 1234882 | Zbl: 0833.92025
[31] Higher Order Corrections to the Time-Dependent Born-Oppenheimer Approximation. I: Smooth Potentials, Ann. Math., Tome 124 (1986), pp. 571-590 | Article | MR: 866709 | Zbl: 0619.35094
[32] Raising and lowering operators for semiclassical wave packets, Ann. Phys., Tome 269 (1998), pp. 77-104 | Article | MR: 1650826 | Zbl: 0929.34067
[33] Landau-Zener Transitions through small Electronic Eigenvalues Gaps in the Born-Oppenheimer Approximation, Ann. IHP (Phys. théorique), Tome 68 (1998), pp. 85-134 | Numdam | MR: 1618922 | Zbl: 0915.35090
[34] Molecular Propapagation through small avoided Crossings of Electron Energy Levels, Rev. Math. Phys., Tome 11 (1999), pp. 41-101 | Article | MR: 1668071 | Zbl: 0965.81138
[35] Localized coupling between surface- and bottom-intensified flow over topography, J. Phys. Oceanogr., Tome 27 (1997), pp. 977-999 | Article
[36] Generic elastic Media, Physica Scripta, Tome 44 (1992), pp. 122-127 | Article
[37] Proof of the Landau-Zener Formula, Asymptotic Analysis, Tome 9 (1994), pp. 209-258 | MR: 1295294 | Zbl: 0814.35109
[38] Forme normale d'un hamiltonien à deux niveaux près d'un point de branchement (limite semi-classique), C. R. Acad. Sci. Paris, Sér. I, Tome 317 (1993) no. 4, pp. 359-364 | MR: 1235449 | Zbl: 0797.58083
[39] Integral representations over isotropic submanifolds and equations of zero curvature, Adv. Math., Tome 135 (1998), pp. 220-286 | Article | MR: 1620834 | Zbl: 0926.53030
[40] On the Born-Oppenheimer Approximation of Wave Operators in Molecular Scattering Theory, Commun. Math. Phys., Tome 143 (1992), pp. 607-639 | MR: 1145603 | Zbl: 0754.35099
[41] Electromagnetic theory and geometrical optics., Interscience Publishers, 1965 | MR: 180094 | Zbl: 0123.23602
[42] Zur Theorie der Energieübertragung. II., Z. Phys. Sowjet., Tome 2 (1932), pp. 46-51 | Zbl: 0005.27801
[42] Collected papers, Pergamon Press, 1965
[43] Mécanique quantique (théorie non relativiste), Mir, Moscou, 1967 | Zbl: 0148.43806
[44] Fourier integral operators with complex valued phase functions, Lecture Notes in Math. (1975) no. 459 | Article | MR: 431289 | Zbl: 0306.42007
[45] Microlocal structure of involutive conical refraction, Duke Math. J., Tome 46 (1979), pp. 571-582 | Article | MR: 544247 | Zbl: 0422.58026
[46] On the generalization of a theorem of Liapounoff, Comm. Pure Appl. Math., Tome 11 (1958), pp. 257-271 | Article | MR: 96021 | Zbl: 0082.08003
[47] Topics in dynamics. I: Flows, Princeton Univ. Press, 1969 | MR: 282379 | Zbl: 0197.10702
[48] WKB expansions for systems of Schrödinger operators with crossing eigenvalues, Asymptotic Anal., Tome 14 (1997), pp. 1-48 | MR: 1437193 | Zbl: 0885.35104
[49] Mode Conversion for Rossby Waves over Topography, J. Phys. Oceanogr., Tome 31 (2001), pp. 1922-1925 | Article
[50] Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Phys. Zeit., Tome 30 (1929), pp. 467-470 | JFM: 55.0520.05
[51] Symplectic manifolds and their Lagrangian submanifolds., Adv. Math., Tome 6 (1971), pp. 329-346 | Article | MR: 286137 | Zbl: 0213.48203
[52] Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond., Tome 137 (1932), pp. 696-702 | Article | Zbl: 0005.18605
Cited by Sources: