Equation de Schrödinger avec champ magnétique et équation de Harper

  • B. Helffer
  • J. Sjőstrand
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 345)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Références

  1. [AD]
    E.N.Adams,II Motion of an electron in a perturbed periodic potential Phys.Rev. Vo185 no1,Jan.1 1952Google Scholar
  2. [AV.S1]
    J.Avron-B.Simon: Stability of gaps for periodic potentials under variation of a magnetic field. J.Phys.A:Math.Gen. 18 (1985) p2199–2205Google Scholar
  3. [AZ]
    Ya Azbel: Energy spectrum of a conduction electron in a magnetic field Soviet Physics JETP vol.19, no3 Sept.1964Google Scholar
  4. [BEA]
    R.Beals: Characterization of PRO and applications Duke Math.Journal 44 (1977),p..45–57Google Scholar
  5. [BEL][1]
    J.Bellissard: Almost periodicity in solid state Physics and C*-algebras Harald Bohr Centennary conference on almost periodic functions Apri1,25 1987Google Scholar
  6. [BEL][2]
    C*-Algebras in solid State Physics-2D Electrons in a uniform magnetic field; Warwick conference on operator algebras (jul 1987)Google Scholar
  7. [BEL][3]
    : K-Theory of C*-algebras in solid state physics Lecture note in Physics 257 (1986), 99–156Google Scholar
  8. [BEL][4]
    : Ordinary quantum Hall effect and non-commutative cohomology conference in Bad Schandau, Dec. 1986 (Teubner Leipzig 1987)Google Scholar
  9. [BE-LI-TE]
    J.Bellissard, R.Lima, D.Testard: Almost periodic Schrödinger operators Math.Phys. Vol.1, p.1–64; Wordl Sci. Publishing, SingaporeGoogle Scholar
  10. [BE-SI]
    J.Bellissard-B.Simon: Cantor spectrum for the almost Matthieu equation Journal of functional analysis, vo1.48, no3, oct 1982Google Scholar
  11. [BL]
    E.I.Blount: Bloch Electrons in a magnetic field; Phys.Rev.126 (1962) p.1636–1653Google Scholar
  12. [B.G.H]
    L.Boutet de Monvel, A.Grigis, B.Helffer: Parametrixes d'opérateurspseudo-differentiels à caractèristiques multiples Astèrisque 34–35, p. 93–121Google Scholar
  13. [BU]
    V.S.Buslaev: développements semi-classiques pour des équations à coefficients pèriodiques; Yspehi Mat. Nayk no42,6 (258) (1987) en russeGoogle Scholar
  14. [CA]
    P.Cartier Quantum mechanical commutation relations and θ functions Proc.Symp.Pure Math., Boulder 1965,ppl83–186Google Scholar
  15. [CH]
    R.G.Chambers: The wave function of a Bloch electron in a Magnetic field Proc.Phys.Soc.89 (1966), 695–710Google Scholar
  16. [CL.WA]
    F.H.Claro-W.H.Wannier: Magnetic subband structure of electrons in Hexagonal lattices Phys.Rev. B19 (1979), 6068–74Google Scholar
  17. [C.D.V]
    Y.Colin de Verdière: Constructions de Laplaciens dont une partie du spectre est donnée Annales de PENS 4 eme série, t.20, 1987,p.599–613Google Scholar
  18. [CO.DU.SE]
    J.M.Combes-P.Duclos-R.Seiler: The Born-Oppenheimer approximation in Rigorous Atomic and Molecular Physics (G.Velo,A.Wightman editors) Plenum,New york, 1981Google Scholar
  19. [DU.NO][1]
    B.A.Dubrovin-S.P.Novikov: Ground states of a two-dimentional electron in a periodic magnetic field. Zh.Eksp.Teor.Fiz. 79;p.1006–1016Google Scholar
  20. [DU.NO][2]
    B.A.Dubrovin-S.P.Novikov: Ground states in a periodic field, Magnetic Bloch functions and vector bundles. Sov.Math.Dokl.Vol.22 (1980) no1Google Scholar
  21. [FE]
    H.Feschbach: Unified Theory of Nuclear reactions I et II Ann.Phys.5 (1958) (p.363) et 19 (1962) p.287-313Google Scholar
  22. [GR]
    V.V.Grushin: On a class of Hypoelliptic operators Math.USSR Sb.,12 (1970) p.458–476Google Scholar
  23. [GU.RA.TRU]
    J.C.Guillot-J.V.Ralston-E.Trubowitz: Semi-classical methods in solid state physics announced in Proc. VIII ELAM,Rio de Janeiro Juillet 86 and: comm.in Math.Physics 116, 401–415 (1988)Google Scholar
  24. [HE]
    B.Helffer: Sur l'hypoellipticite d'operateurs pseudodiff'erentiels à caractèristiques multiples avec perte de 3/2 dérivees Mémoires de la S.M.F n°51–52 (1977) p13–61Google Scholar
  25. [HE.Sj][1]
    B.Helffer,J.Sjostrand: Analyse semi-classique pour l'equation de Harper (avec application à f'ètude de l'equation de SchrÖdinger avec champ magnétique) à paraitre aux memoires de la SMF 1988, annoncé au Séminaire EDP de l'école Polytechnique 86–87 exposé 10 et aux Proceedings du colloque de st-jean de Monts Juin 87Google Scholar
  26. [HE.Sj][2]
    B.Helffer,J.Sjostrand: Analyse semi-classique pour l'équation de Harper II preprint octobre 1988Google Scholar
  27. [HE.Sj][3]
    B.Helffer,J.Sjostrand: Analyse semi-classique pour l'équation de Harper III preprint Orsay (avril 1988), à paraitre mémoires de la SMF annoncé au Seminaire EDP de l'école Polytechnique 87–88Google Scholar
  28. [HE.Sj][4]
    B.Helffer,J.Sjostrand: Equation de Schrödinger avec champ magnétique fost et équation de Harper; preprint juin 1988 (première version de la partie I )Google Scholar
  29. [HE.Sj][5]
    B.Helffer,J.Sjostrand: On the link between the spectrum of the Schrödinger equation with magnetic field and Harper's equation proceedings of the conference in Holzhau (mars 1988) (à paraître Editeur Teubner )Google Scholar
  30. [HO]
    D.Hofstadter: Energy Levels and Wave functions of Bloch electrons in rational and irrational magnetic fields. Phys.Rev.B 14 (1976), 2239–2249Google Scholar
  31. [KO]
    W.Kohn: Theory of Bloch functions in a magnetic field: the effective Hamiltonian; Phys.Rev. Vol. 115 no6 September 15 (1959)Google Scholar
  32. [LA]
    L.D.Landau: Zeitschrift. Physik 64,629 (1930)Google Scholar
  33. [LE]
    J.Leray: Lagrangian Analysis and Quantum Mechanics; MIT Press 1981Google Scholar
  34. [LU]
    J.M.Luttinger: The effect of a magnetic field on electrons in a periodic potential Phys.Rev.84 no4 (1951) p.814–817Google Scholar
  35. [LYS]
    A.S.Lyskova: Topological properties of the Schrödinger operator in a magneti field and with a weak potential. Usp.Mat.Nauk. 36 no5 p.181–182Google Scholar
  36. [MA]
    V.P.Maslov: Théorie des perturbations et Méthodes asymptotiques. Dunod (1972)Google Scholar
  37. [NE][1]
    G.Nenciu: Existence of the exponentially localised Wannier functions Comm.in Math.Phys. 91,81–85 (1983)Google Scholar
  38. [NE][2]
    G.Nenciu: Stability of energy gaps under variation of the magnetic field Letters in Mathematical Physics 11 (1986) p.127–132Google Scholar
  39. [NE][3]
    G.Nenciu: Bloch electrons in a magnetic field: rigorous justification o the Peierls-Onsager effective Hamiltonian (Preprint april 1988)Google Scholar
  40. [NO][1]
    S.P.Novikov: Magnetic Bloch functions and vector-bundles Typical dispersion laws and their quantum numbers Soviet Math.Dokl. Vol.23 (1981) no2Google Scholar
  41. [NO][1]
    S.P.Novikov: Two dimensional operators in periodic fields Journal of soviet Mathematics.Vol 28, no1,Janvier 1985Google Scholar
  42. [ON]
    L.Onsager: Interpretation of the de Haas-Von Halphen Effect Phil.Mag.43 (1952) 1006–1008Google Scholar
  43. [PE]
    R.Peierls Zur theory des diamagnetismus von Leitungelectronen Z. für Physik 80 (1933) 763–791Google Scholar
  44. [PI-VO]
    M.Pimsner-D.Voilescu: Imbedding the irrational rotation C*-algebra into an AF-algebra Journal of operator theory 4 (1980) p.211–218Google Scholar
  45. [RE.SI]
    M.Reed-B.Simon: Methods of modern Mathematical Physics; Academic pressGoogle Scholar
  46. [SJ]
    J.Sjöstrand: Parametrices for pseudo-differential operators with multiple characteristics. Arkiv for Mathematik vol 12 (1974) no1,85–130Google Scholar
  47. [WILC]
    C.Wilcox: Theory of Bloch Waves. J.Analyse Math. 33 (1978),146–167Google Scholar
  48. [WILK][l]
    M.Wilkinson: Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field Proc. R.Soc.Lond.A 403 p 135–166 (1986)Google Scholar
  49. [WILK][2]
    M.Wilkinson: An exact effective hamiltonian for a perturbed Landau Level A paraitre au journal of Physics AGoogle Scholar
  50. [Z]
    J.Zak: Magnetic Translation Group Physical Review Volume 134, No6A,Juin 1964Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • B. Helffer
    • 1
  • J. Sjőstrand
    • 2
  1. 1.Département de MathematiquesURA CNRS 758 Université de NantesNantes Cédex 03France
  2. 2.Département de MathématiquesURA CNRS 760 Université Paris SudOrsay CédexFrance

Personalised recommendations