Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics
Bellissard, Jean ; Vittot, Michel
Annales de l'I.H.P. Physique théorique, Tome 52 (1990), p. 175-235 / Harvested from Numdam
@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},
     volume = {52},
     year = {1990},
     pages = {175-235},
     mrnumber = {1057445},
     zbl = {0705.46037},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 175-235. http://gdmltest.u-ga.fr/item/AIHPA_1990__52_3_175_0/

[1] N. Ashcroft and D. Mermin, Solid state physics, Saunders, Philadelphia, Tokyo, 1976.

[2] J. Avron and R. Seiler, Quantization of the Hall Conductance for General Multiparticule Schrödinger Hamiltonians, Phys. Rev. Lett., Vol. 54, 1985, pp. 259-262. | MR 773033

[3] R. Balian and C. Bloch, 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 270008 | Zbl 0207.40202

[4] R. Balian and C. Bloch, 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 284729 | Zbl 0218.35071

[5] R. Balian and C. Bloch, Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: III-Eigenfrequency Density Oscillations, Ann. Phys., Vol. 69, 1972, pp. 76-160. | MR 289962 | Zbl 0226.35070

[6] R. Balian and C. Bloch, Solution of the Schrödinger Equation in Terms of Classical Paths, Ann. Phys., Vol. 85, 1974, pp. 514-545. | MR 438937 | Zbl 0281.35029

[7] J. Bayfield and P. Koch, Multiphotonic Ionization of Highly Excited Hydrogen Atoms, Phys. Rev., Vol. 33, 1974, p. 258.

[8] J. Bayfield, 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 1109986

[9] J. Bayfield and D.W. Sokol, Excited Atoms in Strong Microwaves: Classical Resonances and Localization in Experimental Final States Distributions, Phys. Rev. Lett., Vol. 61, 1988, p. 2007.

[10] J. Bayfield, G. Casati, I. Guarneri and D.W. Sokol, Localization of Classically Chaotic Diffusion for Hydrogen Atoms in Microwave Fields, submitted to Phys. Rev. Lett.

[11] J. Bellissard, Stability and Instability in Quantum Mechanics, in Trends and developments in the eighties, S. ALBEVERIO and P. BLANCHARD Eds., World Scientific, Singapore, 1985. | MR 853743 | Zbl 0584.35024

[12] J. Bellissard, 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 862832 | Zbl 0612.46066

[13] J. Bellissard, Ordinary Quantum Hall Effect and Non Commutative Cohomology, in Bad Schandau conference on localization, W. WELLER and P. ZIESCHE Eds., Teubner, Leipzig, 1988. | MR 965981

[14] J. Bellissard, 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 996451 | Zbl 0677.46055

[15] J. Bellissard, 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 1031737 | Zbl 0678.42007

[16] M.V. Berry and M. Tabor, Level Clustering in the Regular Spectrum, Proc. R. Soc. London, Vol. A356, 1977, pp. 375-394. | Zbl 1119.81395

[17] M.V. Berry, 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 724465 | Zbl 0571.70018

[18] M.V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proc. R. Soc. London, Vol. A392, 1984, pp. 45-57. [19] M.V. Berry, Semiclassical Theory of Spectral Rigidity, Proc. R. Soc. London, Vol. A400, 1985, pp. 229-251. | MR 738926 | Zbl 1113.81306 | Zbl 0875.35061

[20] M.V. Berry and M. Robnik, Statistics of Energy Levels Without Time-Reversal Symmetry, Aharonov-Bohm Chaotic Billards, J. Phys., Vol. A19, 1986, pp. 649-668. | MR 838450

[21] M.V. Berry, Semiclassical Formula for the Number Variance of the Riemann Zeros, Nonlinearity, Vol. 1, 1988, pp. 399-407. | MR 955621 | Zbl 0664.10022

[22] G.D. Birkhoff, Dynamical systems, A.M.S. Coll. Pub., Vol. 9, AMS, Providence, Rhode Island, 1927. | JFM 53.0732.01

[23] O. Bohigas, R.U. Haq and A. Pandey, 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] O. Bohigas and M.J. Giannoni, 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 769113

[25] O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett., Vol. 52, 1984, pp. 1-4. | MR 730191 | Zbl 1119.81326

[26] O. Bohigas, M.J. Giannoni and C. Schmit, 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 870168

[27] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, I, II, Springer, Berlin, 1979-1981. | Zbl 0421.46048

[28] P. Briet, J.M. Combes and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit: II, Comm. P.D.E., Vol. 12, 1987, pp. 201-222. | MR 876987 | Zbl 0622.47047

[29] P. Briet, J.M. Combes and P. Duclos, Spectral Stability Under Tunneling, Comm. Math. Phys., 1989 (to appear). | MR 1027916 | Zbl 0702.35189

[30] L. Brillouin, J. Phys. Radium, Vol. 7, 1926, pp. 353-368.

[31] T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey and S.S.M. Wong, Random-Matrix Physics: Spectrum and Strenght Fluctuations, Rev. Mod. Phys., Vol. 53, 1981, pp. 385-479. | MR 619406

[32] G. Casati, B.V. Chirikov, J. Ford and F.M. Izraelev, 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 550908 | Zbl 0498.60100

[33] G. Casati, B.V. Chirikov, I. Guarneri and D.L. Shepelyansky, Dynamical Stability of Quantum Chaotic Motion in a Hydrogen Atom, Phys. Rev. Lett., Vol. 56, 1986, p. 2437.

[34] G. Casati, B.V. Chirikov, I. Guarneri and D.L. Shepelyansky, New Photoelectric Ionization Peak in the Hydrogen Atom, Phys. Rev. Lett., Vol. 57, 1986, p. 823.

[35] G. Casati, B.V. Chirikov, I. Guarneri and D.L. Shepelyansky, Localization of Diffusive Excitation in the Two-Dimensional Hydrogen Atom in a Monochromatic Field, Phys. Rev. Lett., Vol. 59, 1987, p. 2927.

[36] A.L. Cauchy, 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] J. Chazarain, Spectre d'un hamiltonien quantique et mécanique classique, Comm. P.D.E., Vol. 5, 1980, pp. 595-644. | MR 578047 | Zbl 0437.70014

[38] B.V. Chirikov, A Universal Instability of Many Dimensional Oscillator Systems, Phys. Rep., Vol. 52, 1979, pp. 263. | MR 536429

[39] Y. Colin De Verdiere, Compos. Math., Vol. 27, 1973, p. 83, Vol. 27, p. 159. | Numdam | MR 348798 | Zbl 0281.53036 | Zbl 0272.53034

[40] Y. Colin De Verdiere, Quasi-modes sur les variétés Riemanniennes, Inv. Math., 1977, Vol. 43, pp. 15-52. | MR 501196 | Zbl 0449.53040

[41] A. Connes, Non Commutative Differential Geometry, Pub. I.H.E.S., 1986, Vol. 62, pp. 43-144. | Numdam | Zbl 0592.46056

[42] W.J. De Haas and P.M. Van Alphen, Proc. Acad. Sci. (Amsterdam), Vol. 36, 1933, p. 262.

[43] J.B. Delos, S.K. Knudson and D.W. Noid, High Rydberg States of an Atom in a Strong Magnetic Field, Phys. Rev. Lett., Vol. 50, 1983, pp. 579-583.

[44] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthiers-Villars, Paris, 1969. | MR 246136 | Zbl 0174.18601

[45] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verhandl. Deutsch. Phys. Ges., Vol. 19, 1917, p. 82-92.

[46] G.A. Elliot, 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 675127 | Zbl 0516.46048

[47] S. Fishman, D.R. Grempel and R.E. Prange, Chaos, Quantum Recurrence and Anderson Localization, Phys. Rev. Lett., Vol. 49, 1982, pp. 509-512. | MR 669169

[48] S. Fishman, D.R. Grempel and R.E. Prange, Quantum Dynamics of a Non Integrable System, Phys. Rev., Vol. A29, 1984, pp. 1639-1647.

[49] G. Gallavotti, Quasi-Integrable Mechanical Systems, in Critical Phenomena, Random Systems, Gauge Theories, K. OSTERWALDER and R. STORA Eds., North Holland, 1986. | MR 880535 | Zbl 0662.70022

[50] T. Geisel, G. Radons and J. Rubner, Kolmogorov-Arnol'd-Moser Barriers in the Quantum Dynamics of Chaotic Systems, Phys. Rev. Lett., Vol. 57, 1986, pp. 2883- 2886.

[51] S. Graffi and T. Paul, The Schrödinger Equation and Canonical Perturbation Theory, Comm. Math. Phys., Vol. 108, 1987, pp. 25-40. | MR 872139 | Zbl 0622.35071

[52] J.M. Greene, A Method for Determining a Stochastic Transition, J. Math. Phys., Vol. 20, 1979, pp. 1183-1201.

[53] V. Guillemin and S. Sternberg, 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 607713 | Zbl 0457.58018

[54] M.C. Gutzwiller, Energy spectrum According to Classical Mechanics, J. Math. Phys., Vol. 11, 1970, pp. 1791-1806.

[55] M.C. Gutzwiller, Periodic Orbits and Classical Quantization Conditions, J. Math. Phys., Vol. 12, 1971, pp. 343-358.

[56] B. Helffer and J. Sjöstrand, 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). | Zbl 0714.34130

[57] E.J. Heller, Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits, Phys. Rev. Lett., Vol. 53, 1984, pp. 1515-1518. | MR 762412

[58] E.J. Heller and R.L. Sundberg, Quantum Ergodicity and Intensity Fluctuations, in Chaotic behavior in quantum systems, G. CASATI, Ed., NATO ASI, Vol. B120, Plenum, New York, 1985.

[59] C. Jaffe and W.P. Reinhardt, Uniform Semiclassical Quantization of Regular and Chaotic Classical Dynamics on the Hénon-Heiles Surface, J. Chem. Phys., Vol. 77, 1982, pp. 5191-5203. | MR 681223

[60] Y. Katznelson, An introduction to harmonic analysis, Wiley, New York, 1968. | MR 248482 | Zbl 0352.43001 | Zbl 0169.17902

[61] J.B. Keller, Corrected Bohr-Sommerfeld Quantum Conditions for Nonseparable Systems, Ann. Phys., Vol. 4, 1958, pp. 180-188. | MR 99207 | Zbl 0085.43103

[62] H.A. Kramers, Z. Physik, Vol. 39, 1926, pp. 828-840. | JFM 52.0969.04

[63] H. Kunz, The Quantum Hall Effect for Electron in a Random Potential, Comm. Math. Phys., Vol. 112, 1987, pp. 121-145. | MR 904141 | Zbl 1108.81314

[64] R.A. Marcus, Aspects of Intramolecular Dynamics in Chemistry, in Chaotic behavior in quantum systems, G. CASATI Ed., NATO ASI, Vol. B120, Plenum, New York, 1985.

[65] H.P. Mckean, Selberg's Trace Formula as Applied to a Compact Riemann Surface, Comm. Pure Appl. Math., Vol. 25, 1972, pp. 225-246. | MR 473166

[66] N.N. Nekhoroshev, The Behavior of Hamiltonian Systems that Are Close to Integrable Ones, Funct. Anal. Appl., Vol. 5, 1971, pp. 338-339. | MR 294813 | Zbl 0254.70015

[67] N.N. Nekhoroshev, Exponential Estimates of the Time of Stability for Nearly Integrable Hamiltonians, Russ. Math. Surveys, Vol. 32, 1977, pp. 1-63. | Zbl 0389.70028

[68] L. Onsager, Interpretation of the Haas-van Alphen Effect, Phil. Mag., Vol. 43, 1952, pp. 1006-1008.

[69] G. Pedersen, C*-Algebras and their Automorphisms Groups, Academic, New York, 1979. | MR 548006 | Zbl 0416.46043

[70] I.C. Percival, Regular and Irregular Spectra, J. Phys., Vol. B6L, 1973, pp. 229-232.

[71] I.C. Percival, 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 550901

[72] N. Pomphrey, Numerical Identification of Regular and Irregular Spectra, J. Phys., Vol. B7, 1974, pp. 1909-1915.

[73] R. Rammal, Landau Level Spectrum of Bloch Electron in a Honeycomb Lattice, J. Phys. France, Vol. 46, 1985, pp. 1345-1354.

[73b] Y.Y. Wang, PANNETIER and R. Rammal, Quasi Classical Approximations for Almost-Mathieu Equation, J. Phys. France, Vol. 48, 1987, pp. 2067-2079.

[74] J. Renault, A Groupoid Approach to C*-Algebras; Lect. Notes Math., Vol. 793, Springer, Berlin, 1980. | MR 584266 | Zbl 0433.46049

[75] M. Robnik and M.V. Berry, False Time-Reversal Violation and Energy Level Statistics : the Role of Anti-Unitary Symmetry, J. Phys., Vol. A19, 1986, pp. 669-682. | MR 838451

[76] A. Selberg, 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 88511 | Zbl 0072.08201

[77] R.B. Shirts and W.P. Reinhardt, 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 681224

[78] J.B. Taylor, Unpublished (1968) (quoted in [52]).

[79] D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. Den Nijs, Quantized Hall Conductance in a Two Dimensional Periodic Potential, Phys. Rev. Lett., Vol. 49, 1982, pp. 405-408.

[80] M. Vittot, 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 1109977 | Zbl 0707.70015

[81] A. Voros, Asymptotic h-Expansions of Stationnary Quantum States, Ann. I.H.P., Vol. A26, 1977, pp. 343-403. | Numdam | MR 456138

[82] M. Wilkinson, Critical Properties of Electrons Eigenstates in Incommensurate Systems, Proc. R. Soc. London, Vol. A391, 1984, pp. 305-350. | MR 739684

[83] M. Wilkinson, Von Neumann Lattices of Wannier Functions for Bloch Electrons in a Magnetic Field, Proc. R. Soc. London, Vol. A403, 1986, pp. 135-166. | MR 828687

[84] M. Wilkinson, An Example of Phase Holonomy and WKB Theory, J. Phys., Vol. A17, 1984, pp. 3459-3476. | MR 772333