Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum

B. Helffer; J. Sjöstrand

Mémoires de la Société Mathématique de France (1989)

  • Volume: 39, page 1-124
  • ISSN: 0249-633X

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Helffer, B., and Sjöstrand, J.. "Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum." Mémoires de la Société Mathématique de France 39 (1989): 1-124. <http://eudml.org/doc/94884>.

@article{Helffer1989,
author = {Helffer, B., Sjöstrand, J.},
journal = {Mémoires de la Société Mathématique de France},
keywords = {Harper's operator; microlocalization; renormalization; spectrum; periodic magnetic Schrödinger operator},
language = {eng},
pages = {1-124},
publisher = {Société mathématique de France},
title = {Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum},
url = {http://eudml.org/doc/94884},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Helffer, B.
AU - Sjöstrand, J.
TI - Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum
JO - Mémoires de la Société Mathématique de France
PY - 1989
PB - Société mathématique de France
VL - 39
SP - 1
EP - 124
LA - eng
KW - Harper's operator; microlocalization; renormalization; spectrum; periodic magnetic Schrödinger operator
UR - http://eudml.org/doc/94884
ER -

References

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  1. [AM] R. Abraham, J. Marsden, Foundations of Mechanics, Benjamin/Cumming publ. Co. (1978), Zbl0393.70001MR81e:58025
  2. [AuAn] S. Aubry, C. André, Proc. Israel Phys. Soc. Ed.C.G.Kuper 3 (Adam Hilger, Bristol) (1979), 133-, Zbl0943.82510
  3. [AvSi] J. Avron, B. Simon, Stability of gaps for periodic potentials under a variation of a magnetic field, J. Phys. A, 18 (1985), 2199-2205, Zbl0586.35084MR87e:81039
  4. [Az] Ya. Azbel, Energy spectrum of a conduction electron in a magnetic field, Soviet Physics JETP, 19, n°3, Sept 1964. 
  5. [Be] J. Bellissard, Schrödinger operators with almost periodic potentials, Springer Lecture Notes in Physics, 153, 
  6. [BeSi] J. Bellissard, B. Simon, Cantor spectrum for the almost Mathieu equation, J. Funct. An., 48, n°3, Oct. 1982, Zbl0516.47018MR84h:81019
  7. [GeS] C. Gérard, J. Sjöstrand, Semi-classical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys., 108 (1987), 391-421. Zbl0637.35027
  8. [GrS] A. Grigis, J. Sjöstrand, Front d'onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J., 52, n° 1 (1985), 35-51, Zbl0581.35009MR86h:58136
  9. [HR1] B. Helffer, D. Robert, Calcul fonctionnel par la transformée de Mellin et applications, J. Funct. An., 53, n°3, oct. 1983. MR85i:47052
  10. [HR2] B. Helffer, D. Robert, Puits de potentiel généralisés et asymptotique semiclassique, Ann. I.H.P. (section Phys. théorique), 41, n°3, (1984), 291-331, Zbl0565.35082MR86m:81049
  11. [HS1] B. Helffer, J. Sjöstrand, Analyse semi-classique pour l'équation de Harper, Preprint (1987) Zbl0634.35056
  12. [HS2] B. Helffer, J. Sjöstrand, Analyse semi-classique pour l'équation de Harper II Comportement semi-classique près d'un rationnel, Preprint. Zbl0714.34131
  13. [HS3] B. Helffer, J. Sjöstrand, Structure Cantorienne du spectre de l'opérateur de Harper, Sém. des Equations aux Dérivées partielles, Ecole polytechnique, March 1988, Zbl0666.34012
  14. [HS4] B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit I, Comm. PDE, 9(4) (1984), 337-408, Zbl0546.35053MR86c:35113
  15. [HS5] B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit II, Ann. I.H.P. (section Phys. théorique), 42, n°2, (1985), 127-212, Zbl0595.35031
  16. [HS6] B. Helffer, J. Sjöstrand, Effet tunnel pour l'équation de Schrödinger avec champ magnétique, Ann. Sc. Norm. Sup., to appear, Zbl0699.35205
  17. [HS7] B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Bull. de la SMF, 114, fasc. 3, (mémoire n°24-25). Zbl0631.35075
  18. [Ho] D. Hofstadter, Energy levels and wave functions for Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976), 2239-2249, 
  19. [L] J. Leray, Lagrangian analysis and quantum mechanics, MIT Press (1981). Zbl0483.35002MR83k:58081a
  20. [Mo] P. Van Mouche, The coexistence problem for the discrete Mathieu operator, Preprint (1988), 
  21. [No] S.P. Novikov, Two dimensional operators in periodic fields, J. Sov. Math., 28, n°1, January 1985, 
  22. [O] F.W.J. Olver, Asymptotics and special functions, Academic Press, New York, London, 1974. Zbl0308.41023MR55 #8655
  23. [Si] B. Simon, Almost periodic Schrödinger operators: a review, Advances in applied math. 3, (1982), 463-490, Zbl0545.34023MR85d:34030
  24. [S1] J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982), Zbl0524.35007MR84m:58151
  25. [S2] J. Sjöstrand, Resonances generated by non-degenerate critical points, Springer Lecture Notes in Mathematics, n°1256, 402-429, Zbl0627.35074
  26. [So] J.B. Sokoloff, Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials, Physics reports (review section of Physics letters), 126, n°4 (1985), 189-244, 
  27. [W1] M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems, Proc. R. Soc. London A391 (1984), 305-350, MR86b:81136
  28. [W2] M. Wilkinson, Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field, Proc. R. Soc. London A403 (1986), 135-166, MR87f:81178
  29. [W3] M. Wilkinson, An exact effective Hamiltonian for a perturbed Landau level, Journal of Phys. A, 20, n°7, 11-May 1987, 1761-, Zbl0639.47010MR88g:81157
  30. [W4] M. Wilkinson, An exact renormalisation group for Bloch electrons in a magnetic field, Journal of Physics A, to appear. 

Citations in EuDML Documents

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  1. Yves Colin de Verdière, Bernard Parisse, Équilibre instable en régime semi-classique. II. Conditions de Bohr-Sommerfeld
  2. Y. Colin de Verdière, B. Parisse, Équilibre instable en régime semi-classique
  3. A. Grigis, Résonances par correspondance
  4. Johannes Sjöstrand, Maciej Zworski, Elementary linear algebra for advanced spectral problems
  5. Michael Hitrik, Karel Pravda-Starov, Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics
  6. San Vũ Ngọc, Conditions de Bohr-Sommerfeld pour les singularités focus-focus et monodromie quantique
  7. Alexander Fedotov, Frédéric Klopp, Transitions d’Anderson pour des opérateurs de Schrödinger quasi-périodiques en dimension 1
  8. M. Rouleux, Résonances de Feschbach en limite semi-classique
  9. Mauricio D. Garay, Perturbative expansions in quantum mechanics
  10. Robert Berman, Johannes Sjöstrand, Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles

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