Le papillon de Hofstadter revisité
B. Helffer; P. Kerdelhue; J. Sjöstrand
Mémoires de la Société Mathématique de France (1990)
- Volume: 43, page 1-87
- ISSN: 0249-633X
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topHelffer, B., Kerdelhue, P., and Sjöstrand, J.. "Le papillon de Hofstadter revisité." Mémoires de la Société Mathématique de France 43 (1990): 1-87. <http://eudml.org/doc/94887>.
@article{Helffer1990,
author = {Helffer, B., Kerdelhue, P., Sjöstrand, J.},
journal = {Mémoires de la Société Mathématique de France},
keywords = {Harper operator; Hofstadter butterfly; spectral gaps},
language = {fre},
pages = {1-87},
publisher = {Société mathématique de France},
title = {Le papillon de Hofstadter revisité},
url = {http://eudml.org/doc/94887},
volume = {43},
year = {1990},
}
TY - JOUR
AU - Helffer, B.
AU - Kerdelhue, P.
AU - Sjöstrand, J.
TI - Le papillon de Hofstadter revisité
JO - Mémoires de la Société Mathématique de France
PY - 1990
PB - Société mathématique de France
VL - 43
SP - 1
EP - 87
LA - fre
KW - Harper operator; Hofstadter butterfly; spectral gaps
UR - http://eudml.org/doc/94887
ER -
References
top- [Au] S. Aubry : The new concept of transition by breaking of analyticity Solid State Sci.8 (1978), 264 MR80d:82034
- [Av - Se] J. Avron - R. Seiler : Quantization of the Hall conductance for general multiparticle Schrödinger Hamiltonians. Phys. Rev. Letters Vol. 54, n° 4, Janvier 1985, p. 259-262 MR86f:81175
- [Av - Si] J. Avron - B. Simon :
- J. Avron - B. Simon [1] Almost periodic Schrödinger operators, I. Limit Periodic potentials Comm. in Math. Physics. 82 p. 101-120 (1981) Zbl0484.35069
- J. Avron - B. Simon [2] Almost periodic Schrödinger operators, II The density of states Duke Math. Journal 50 (1983), 369-391 Zbl0544.35030MR85i:34009a
- J. Avron - B. Simon [3] Stability of gaps for periodic potentials under variation of a magnetic field J. Phys. A ; Math. Gen. 18 (1985) p 2199-2205 Zbl0586.35084MR87e:81039
- [Az] Ya Azbel : Energy spectrum of a conduction electron in a magnetic field Soviet Physics JETP vol. 19, n° 3 Sept. 1964
- [Be] J. Bellissard :
- J. Bellissard [1] Almost periodicity in solid state Physics and C*-algebras "The Harald Bohr Centennary" Proceedings of the symposium held in Copenhaguen conference on almost periodic functions April : 24-25 1987 ; C. Berg, B. Fuglede Ed. The Royal Academy of Sciences, Editions, Copenhaguen 1989
- J. Bellissard [2] C*-Algebras in solid State Physics-2D Electrons in a uniform magnetic field ; "operator algebras and applications" D.E. Evans and M. Takesaki Eds Cambridge University Press, Vol. 2 (1988) pp 49-76 Zbl0677.46055MR91c:46094
- [Be - Li - Te] J. Bellissard - R. Lima - D. Testard : A metal-insulator transition for the almost Mathieu model Comm. in Math. Phys. 88, 207-234 (1983) Zbl0542.35059MR85i:82055
- [Be - Si] J. Bellissard - B. Simon : Cantor Spectrum for the Almost Mathieu Equation Journal of functional Analysis, vol. 48, N° 3, Oct 1982 Zbl0516.47018MR84h:81019
- [Ben - Pas] Benderskii - L. Pasteur : On the spectrum of the one-dimensional Schrödinger equation with a random potential. Math. USSR Sb. 11, 245 (1970) Zbl0216.37301
- [Ch] R.G. Chambers : The wave function of a Bloch electron in a Magnetic field Proc. Phys. Soc. 89 (1966), 695-710
- [C - E - Y] Man Duen Choi, Georges A. Elliott, Noriko Yui : Gauss polynomials and the rotation algebra Preprint Septembre 1988, et Inventiones matematicae Vol 99 Fasc. 2 1990 Zbl0665.46051
- [Cl - Wa] F. H. Claro, W. H. Wannier : Magnetic subband structure of electrons in hexagonal lattices Phys. Rev. B19 (1979), 6068-74
- [C - F - K - S] H. L. Cycon - R. G. Froese - W. Kirsch - B. Simon : Schrödinger operators with applications to quantum mechanics and global geometry Texts and monographs in Physics, Springer Verlag Zbl0619.47005
- [El] G.A. Elliott : Gaps in the spectrum of an almost periodic Schrödinger operator C.R. Math. Rep. Acad. Sci. Canada 4 (1982) p. 255-259 Zbl0516.46048MR85b:47054
- [Gu - He - Tr] J.P. Guillement - B. Helffer - P. Treton : Walk inside Hofstadter's butterfly Journal de Physique France 50 (1989) 2019-2058
- [He - Ro] B. Helffer - D. Robert : Calcul fonctionnel par la transformée de Mellin et opérateurs admissibles Journal of Functional Analysis 53 (1983) p. 246-268 Zbl0524.35103MR85i:47052
- [He - Sj] B. Helffer, J. Sjostrand :
- B. Helffer, J. Sjostrand [1] Analyse semi-classique pour l'équation de Harper (avec application à l'étude de l'équation de Schrödinger avec champ magnétique) ; Mémoires de la SMF 1988 Zbl0714.34130
- B. Helffer, J. Sjostrand [2] Analyse semi-classique pour l'équation de Harper II preprint nov. 1988 ; à paraître aux mémoires de la SMF 1989 Zbl0714.34131
- B. Helffer, J. Sjostrand [3] Semi-classical analysis for Harper's equation III preprint Orsay (avril 1988), à paraître aux mémoires de la SMF 1989 et annoncé au Séminaire EDP de l'école Polytechnique 87-88 Zbl0725.34099
- B. Helffer, J. Sjostrand [4] Equation de Schrödinger avec champ magnétique et équation de Harper ; Preprint Dec. 1988 et paru dans "Schrödinger operators", Proceedings, Sønderborg, Denmark, 1988 ; Lecture Notes in Physics n° 345, Springer Verlag Zbl0699.35189
- B. Helffer, J. Sjostrand [5] On Diamagnetism and de Haas-Van Alphen effect Preprint Paris-sud, mai 1989 ; à paraître aux annales de l'IHP (section Physique théorique) Zbl0715.35070
- [Ho] D. Hofstadter : Energy Levels and Wave functions of Bloch electrons in rational and irrational magnetic fields Phys. Rev. B 14 (1976), 2239-2249
- [J - M] R. Johnson - J. Moser : The rotation number for almost periodic potentials Comm. in Math. Phys. 84, (1982), 403-438 ; 90, (1983), 317-318 Zbl0497.35026MR83h:34018
- [Kh] A. Ya Khinchin : Continued fractions Phoenix Science Series (1964)
- [Mo] P. (Van) Mouche :
- P. (Van) Mouche [1] Clustering and Nesting of Energy spectra Proceedings ICIAM 87, Paris la Villette, June 29-July 3 1987 MR89k:81050
- P. (Van) Mouche [2] The coexistence problem for the discrete Mathieu operator Preprint Février 88 et Comm. in Math. Physics Vol 122 n° 1, 1989, p. 23-34 Zbl0669.34016MR90e:47027
- [Pi-Vo] M. Pimsner, D. Voiculescu : Imbedding the irrational rotation C*-algebra into an AF-algebra Journal Operator theory 4, 1980 p. 211-218 Zbl0525.46031MR82d:46086
- [Sh] M.A. Shubin : The spectral theory and the index of elliptic operators with almost periodic coefficients Russian Math. Surveys, 34, (1979), 109-157 Zbl0448.47032MR81f:35090
- [Si] B. Simon : Almost periodic Schrödinger operators. A review Advances in applied Mathematics 3 p. 463-490 (1982) Zbl0545.34023MR85d:34030
- [T - K - N - N] D. Thouless, M. Kohmoto, M. Nightingale, M. de Nijs : Quantized Hall conductance in two dimensional periodic potential. Phys. Rev. Letters, 49, (1982), p. 405-408
- [Wa] G.H. Wannier : Phys. Status Solidi B 88, 757 (1978)
- [Wilk] M. Wilkinson :
- M. Wilkinson [1] Critical properties of electron eigenstates in incommensurate systems Proc. Royal Society of London A391 p 305-350 (1984) MR86b:81136
- M. Wilkinson [2] Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field Proc. R. Soc. Lond. A 403, p 135-166 (1986) MR87f:81178
- M. Wilkinson [3] An exact effective hamiltonian for a perturbed Landau Level Journal of Physics A, Vol. 20, n° 7, 11 May 1987, p. 1761 Zbl0639.47010MR88g:81157
- M. Wilkinson [4] An exact renormalization group for Bloch electrons in a magnetic fields. Journal of Physics A, Vol. 20, p. 4337-4354 MR88j:81087
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