@article{AIHPA_1990__52_3_175_0,
author = {Bellissard, Jean and Vittot, Michel},
title = {Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics},
journal = {Annales de l'I.H.P. Physique th\'eorique},
pages = {175--235},
publisher = {Gauthier-Villars},
volume = {52},
number = {3},
year = {1990},
mrnumber = {1057445},
zbl = {0705.46037},
language = {en},
url = {http://www.numdam.org/item/AIHPA_1990__52_3_175_0/}
}
TY - JOUR AU - Bellissard, Jean AU - Vittot, Michel TI - Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics JO - Annales de l'I.H.P. Physique théorique PY - 1990 SP - 175 EP - 235 VL - 52 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/item/AIHPA_1990__52_3_175_0/ LA - en ID - AIHPA_1990__52_3_175_0 ER -
%0 Journal Article %A Bellissard, Jean %A Vittot, Michel %T Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics %J Annales de l'I.H.P. Physique théorique %D 1990 %P 175-235 %V 52 %N 3 %I Gauthier-Villars %U http://www.numdam.org/item/AIHPA_1990__52_3_175_0/ %G en %F AIHPA_1990__52_3_175_0
Bellissard, Jean; Vittot, Michel. Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics. Annales de l'I.H.P. Physique théorique, Tome 52 (1990) no. 3, pp. 175-235. http://www.numdam.org/item/AIHPA_1990__52_3_175_0/
[1] and , Solid state physics, Saunders, Philadelphia, Tokyo, 1976.
[2] and , Quantization of the Hall Conductance for General Multiparticule Schrödinger Hamiltonians, Phys. Rev. Lett., Vol. 54, 1985, pp. 259-262. | MR
[3] and , Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: I-Three-Dimensional Problem with Smooth Boundary Surface, Ann. Phys., Vol. 60, 1970, pp. 401-447. | MR | Zbl
[4] and , Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: II-Electromagnetic Field, Riemannian Spaces, Ann. Phys., Vol. 64, 1971, pp. 271-307. | MR | Zbl
[5] and , Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: III-Eigenfrequency Density Oscillations, Ann. Phys., Vol. 69, 1972, pp. 76-160. | MR | Zbl
[6] and , Solution of the Schrödinger Equation in Terms of Classical Paths, Ann. Phys., Vol. 85, 1974, pp. 514-545. | MR | Zbl
[7] and , Multiphotonic Ionization of Highly Excited Hydrogen Atoms, Phys. Rev., Vol. 33, 1974, p. 258.
[8] , Experiment and Theory for the Classically Chaotic Motion of the Driven Bound Electron, in Non linear evolution and chaotic phenomena, G. GALLAVOTTI and P. F. ZWEIFEL Eds., Plenum, New York, 1988. | MR
[9] and , Excited Atoms in Strong Microwaves: Classical Resonances and Localization in Experimental Final States Distributions, Phys. Rev. Lett., Vol. 61, 1988, p. 2007.
[10] , , and , Localization of Classically Chaotic Diffusion for Hydrogen Atoms in Microwave Fields, submitted to Phys. Rev. Lett.
[11] , Stability and Instability in Quantum Mechanics, in Trends and developments in the eighties, S. ALBEVERIO and P. BLANCHARD Eds., World Scientific, Singapore, 1985. | MR | Zbl
[12] , K-Theory of C*-Algebras in Solid State Physics, in Statistical mechanics and field theory, mathematical aspects, T. C. DORLAS, M. N. HUGENHOLTZ and M. WINNINK Eds., Lect. Notes Phys., Vol. 257, Springer, Berlin, 1986. | MR | Zbl
[13] , Ordinary Quantum Hall Effect and Non Commutative Cohomology, in Bad Schandau conference on localization, W. WELLER and P. ZIESCHE Eds., Teubner, Leipzig, 1988. | MR
[14] , C*-Algebras in Solid State Physics: 2D Electrons in a Uniform Magnetic Field, in Operators Algebras and Applications, Vol. II, E. V. EVANS and M. TAKESAKI Eds., Cambridge University Press, Cambridge, 1988. | MR | Zbl
[15] , Almost Periodicity in Solid State Physics and C*-Algebras, in The Harald Bohr Centennary, C. BERG and F. FLUGEDE Eds., Royal Danish Acad. Science, Copenhagen, 1989. | MR | Zbl
[16] and , Level Clustering in the Regular Spectrum, Proc. R. Soc. London, Vol. A356, 1977, pp. 375-394. | Zbl
[17] , Semiclassical Mechanics of Regular and Irregular Motion, in Chaotic Behavior of Deterministic Systems, G. Iooss, R. H. G. HELLEMAN and R. STORA Eds., North-Holland, Amsterdam, 1983. | MR | Zbl
[18] , Quantal Phase Factors Accompanying Adiabatic Changes, Proc. R. Soc. London, Vol. A392, 1984, pp. 45-57. [19] , Semiclassical Theory of Spectral Rigidity, Proc. R. Soc. London, Vol. A400, 1985, pp. 229-251. | MR | Zbl
[20] and , Statistics of Energy Levels Without Time-Reversal Symmetry, Aharonov-Bohm Chaotic Billards, J. Phys., Vol. A19, 1986, pp. 649-668. | MR
[21] , Semiclassical Formula for the Number Variance of the Riemann Zeros, Nonlinearity, Vol. 1, 1988, pp. 399-407. | MR | Zbl
[22] , Dynamical systems, A.M.S. Coll. Pub., Vol. 9, AMS, Providence, Rhode Island, 1927. | JFM
[23] , and , Fluctuation Properties of Nuclear Energy Levels and widths: Comparison of Theory with Experiment, in Nuclear data for science and technology, K. H. BÖCKHOFF Ed., ECSC, EEC, EAEC, Brussels and Luxembourg, 1983.
[24] and , Chaotic Motion and Random Matrix Theories, in Mathematical and computational methods in nuclear physics, J. S. DEHESA, J. M. G. GOMEZ and A. POLLS Eds., Lect. Notes Phys., Vol. 209, Springer, Berlin, 1984. | MR
[25] , and , Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett., Vol. 52, 1984, pp. 1-4. | MR | Zbl
[26] , and , Spectral Fluctuations of Classically Chaotic Quantum Systems, in Quantum chaos and statistical nuclear physics, T. H. SELIGMAN and H. NISHIOKA Eds., Lect. Notes Phys., Vol. 263, Springer, Berlin, 1986. | MR
[27] and , Operator Algebras and Quantum Statistical Mechanics, I, II, Springer, Berlin, 1979-1981. | Zbl
[28] , and , On the location of resonances for Schrödinger operators in the semiclassical limit: II, Comm. P.D.E., Vol. 12, 1987, pp. 201-222. | MR | Zbl
[29] , and , Spectral Stability Under Tunneling, Comm. Math. Phys., 1989 (to appear). | MR | Zbl
[30] , J. Phys. Radium, Vol. 7, 1926, pp. 353-368.
[31] , , , , and , Random-Matrix Physics: Spectrum and Strenght Fluctuations, Rev. Mod. Phys., Vol. 53, 1981, pp. 385-479. | MR
[32] , , and , Stochastic Behavior of a Quantum Pendulum Under a Periodic Perturbation, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, G. CASATI and J. FORD Eds., Lect. Notes Phys., Vol. 93, Springer, Berlin, 1979. | MR | Zbl
[33] , , and , Dynamical Stability of Quantum Chaotic Motion in a Hydrogen Atom, Phys. Rev. Lett., Vol. 56, 1986, p. 2437.
[34] , , and , New Photoelectric Ionization Peak in the Hydrogen Atom, Phys. Rev. Lett., Vol. 57, 1986, p. 823.
[35] , , and , Localization of Diffusive Excitation in the Two-Dimensional Hydrogen Atom in a Monochromatic Field, Phys. Rev. Lett., Vol. 59, 1987, p. 2927.
[36] , Résumé des leçons données à l'École Royale Polytechnique sur le calcul infinitésimal, Paris (1823) (repr. in Œuvres complètes IV, Gauthier-Villars, Paris, 1899).
[37] , Spectre d'un hamiltonien quantique et mécanique classique, Comm. P.D.E., Vol. 5, 1980, pp. 595-644. | MR | Zbl
[38] , A Universal Instability of Many Dimensional Oscillator Systems, Phys. Rep., Vol. 52, 1979, pp. 263. | MR
[39] , Compos. Math., Vol. 27, 1973, p. 83, Vol. 27, p. 159. | Numdam | MR | Zbl
[40] , Quasi-modes sur les variétés Riemanniennes, Inv. Math., 1977, Vol. 43, pp. 15-52. | EuDML | MR | Zbl
[41] , Non Commutative Differential Geometry, Pub. I.H.E.S., 1986, Vol. 62, pp. 43-144. | EuDML | Numdam | Zbl
[42] and , Proc. Acad. Sci. (Amsterdam), Vol. 36, 1933, p. 262.
[43] , and , High Rydberg States of an Atom in a Strong Magnetic Field, Phys. Rev. Lett., Vol. 50, 1983, pp. 579-583.
[44] , Les C*-algèbres et leurs représentations, Gauthiers-Villars, Paris, 1969. | MR | Zbl
[45] , Zum Quantensatz von Sommerfeld und Epstein, Verhandl. Deutsch. Phys. Ges., Vol. 19, 1917, p. 82-92.
[46] , Gaps in the Spectrum of an Almost Periodic Schrödinger Operator, C.R. Acad. Sci., Royal Soc. of Canada, Vol. IV, 1982, p. 255-259. | MR | Zbl
[47] , and , Chaos, Quantum Recurrence and Anderson Localization, Phys. Rev. Lett., Vol. 49, 1982, pp. 509-512. | MR
[48] , and , Quantum Dynamics of a Non Integrable System, Phys. Rev., Vol. A29, 1984, pp. 1639-1647.
[49] , Quasi-Integrable Mechanical Systems, in Critical Phenomena, Random Systems, Gauge Theories, K. OSTERWALDER and R. STORA Eds., North Holland, 1986. | MR | Zbl
[50] , and , Kolmogorov-Arnol'd-Moser Barriers in the Quantum Dynamics of Chaotic Systems, Phys. Rev. Lett., Vol. 57, 1986, pp. 2883- 2886.
[51] and , The Schrödinger Equation and Canonical Perturbation Theory, Comm. Math. Phys., Vol. 108, 1987, pp. 25-40. | MR | Zbl
[52] , A Method for Determining a Stochastic Transition, J. Math. Phys., Vol. 20, 1979, pp. 1183-1201.
[53] and , The Metaplectic Representation, Weyl Operators and Spectral Theory, in Differential Geometric Methods in Mathematical Physics, P. L. GARCIA, A. PÉREZ-RENDÓN and J. M. SOURIAU Eds., Lect. Notes Math., Vol. 836, Springer, Berlin, 1979. | MR | Zbl
[54] , Energy spectrum According to Classical Mechanics, J. Math. Phys., Vol. 11, 1970, pp. 1791-1806.
[55] , Periodic Orbits and Classical Quantization Conditions, J. Math. Phys., Vol. 12, 1971, pp. 343-358.
[56] and , Analyse semi-classique pour l'équation de Harper (avec application à l'étude de l'équation de Schrödinger avec champ magnétique) I, II, III, Preprint Univ. Orsay, Bull. Soc. Math. France, 1988 (submitted). | Numdam | Zbl
[57] , Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits, Phys. Rev. Lett., Vol. 53, 1984, pp. 1515-1518. | MR
[58] and , Quantum Ergodicity and Intensity Fluctuations, in Chaotic behavior in quantum systems, G. CASATI, Ed., NATO ASI, Vol. B120, Plenum, New York, 1985.
[59] and , Uniform Semiclassical Quantization of Regular and Chaotic Classical Dynamics on the Hénon-Heiles Surface, J. Chem. Phys., Vol. 77, 1982, pp. 5191-5203. | MR
[60] , An introduction to harmonic analysis, Wiley, New York, 1968. | MR | Zbl
[61] , Corrected Bohr-Sommerfeld Quantum Conditions for Nonseparable Systems, Ann. Phys., Vol. 4, 1958, pp. 180-188. | MR | Zbl
[62] , Z. Physik, Vol. 39, 1926, pp. 828-840. | JFM
[63] , The Quantum Hall Effect for Electron in a Random Potential, Comm. Math. Phys., Vol. 112, 1987, pp. 121-145. | MR | Zbl
[64] , Aspects of Intramolecular Dynamics in Chemistry, in Chaotic behavior in quantum systems, G. CASATI Ed., NATO ASI, Vol. B120, Plenum, New York, 1985.
[65] , Selberg's Trace Formula as Applied to a Compact Riemann Surface, Comm. Pure Appl. Math., Vol. 25, 1972, pp. 225-246. | MR
[66] , The Behavior of Hamiltonian Systems that Are Close to Integrable Ones, Funct. Anal. Appl., Vol. 5, 1971, pp. 338-339. | MR | Zbl
[67] , Exponential Estimates of the Time of Stability for Nearly Integrable Hamiltonians, Russ. Math. Surveys, Vol. 32, 1977, pp. 1-63. | Zbl
[68] , Interpretation of the Haas-van Alphen Effect, Phil. Mag., Vol. 43, 1952, pp. 1006-1008.
[69] , C*-Algebras and their Automorphisms Groups, Academic, New York, 1979. | MR | Zbl
[70] , Regular and Irregular Spectra, J. Phys., Vol. B6L, 1973, pp. 229-232.
[71] , Regular and Irregular Spectra in Molecules, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, G. CASATI and J. FORD Eds., Lect. Notes Phys., Vol. 93, Springer, Berlin, 1979. | MR
[72] , Numerical Identification of Regular and Irregular Spectra, J. Phys., Vol. B7, 1974, pp. 1909-1915.
[73] , Landau Level Spectrum of Bloch Electron in a Honeycomb Lattice, J. Phys. France, Vol. 46, 1985, pp. 1345-1354.
[73b] , PANNETIER and , Quasi Classical Approximations for Almost-Mathieu Equation, J. Phys. France, Vol. 48, 1987, pp. 2067-2079.
[74] , A Groupoid Approach to C*-Algebras; Lect. Notes Math., Vol. 793, Springer, Berlin, 1980. | MR | Zbl
[75] and , False Time-Reversal Violation and Energy Level Statistics : the Role of Anti-Unitary Symmetry, J. Phys., Vol. A19, 1986, pp. 669-682. | MR
[76] , Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series, J. Indian. Math. Soc., Vol. 20, 1956, pp. 47-87. | MR | Zbl
[77] and , Approximate Constants of Motion for Classically Chaotic Vibrational Dynamics: Vague Tori, Semiclassical Quantization, and Classical Intramolecular Energy Flow, J. Chem. Phys., Vol. 77, 1982, pp. 5204-5217. | MR
[78] , Unpublished (1968) (quoted in [52]).
[79] , , and , Quantized Hall Conductance in a Two Dimensional Periodic Potential, Phys. Rev. Lett., Vol. 49, 1982, pp. 405-408.
[80] , A Simple and Compact Presentation of Birkhoff Series, in Non Linear Evolution and Chaotic Phenomena, G. GALLAVOTTI and P. F. ZWEIFEL Eds., Plenum, New York, 1987. | MR | Zbl
[81] , Asymptotic h-Expansions of Stationnary Quantum States, Ann. I.H.P., Vol. A26, 1977, pp. 343-403. | EuDML | Numdam | MR
[82] , Critical Properties of Electrons Eigenstates in Incommensurate Systems, Proc. R. Soc. London, Vol. A391, 1984, pp. 305-350. | MR
[83] , Von Neumann Lattices of Wannier Functions for Bloch Electrons in a Magnetic Field, Proc. R. Soc. London, Vol. A403, 1986, pp. 135-166. | MR
[84] , An Example of Phase Holonomy and WKB Theory, J. Phys., Vol. A17, 1984, pp. 3459-3476. | MR
