Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum
Mémoires de la Société Mathématique de France, no. 39 (1989) , 132 p. 
@book{MSMF_1989_2_39__1_0,
     author = {Helffer, Bernard and Sj\"ostrand, Johannes},
     title = {Semi-classical analysis for {Harper's} equation. {III} : {Cantor} structure of the spectrum},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {39},
     year = {1989},
     doi = {10.24033/msmf.346},
     zbl = {0725.34099},
     language = {en},
     url = {http://archive.numdam.org/item/MSMF_1989_2_39__1_0/}
}
TY  - BOOK
AU  - Helffer, Bernard
AU  - Sjöstrand, Johannes
TI  - Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum
T3  - Mémoires de la Société Mathématique de France
PY  - 1989
DA  - 1989///
IS  - 39
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/MSMF_1989_2_39__1_0/
UR  - https://zbmath.org/?q=an%3A0725.34099
UR  - https://doi.org/10.24033/msmf.346
DO  - 10.24033/msmf.346
LA  - en
ID  - MSMF_1989_2_39__1_0
ER  - 
Helffer, B.; Sjöstrand, J. Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum. Mémoires de la Société Mathématique de France, Série 2, , no. 39 (1989), 132 p. doi : 10.24033/msmf.346. http://numdam.org/item/MSMF_1989_2_39__1_0/

[AM] R. Abraham, J. Marsden, Foundations of Mechanics, Benjamin/Cumming publ. Co. (1978), | MR 81e:58025 | Zbl 0393.70001

[AuAn] S. Aubry, C. André, Proc. Israel Phys. Soc. Ed.C.G.Kuper 3 (Adam Hilger, Bristol) (1979), 133-, | Zbl 0943.82510

[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, | MR 87e:81039 | Zbl 0586.35084

[Az] Ya. Azbel, Energy spectrum of a conduction electron in a magnetic field, Soviet Physics JETP, 19, n°3, Sept 1964.

[Be] J. Bellissard, Schrödinger operators with almost periodic potentials, Springer Lecture Notes in Physics, 153,

[BeSi] J. Bellissard, B. Simon, Cantor spectrum for the almost Mathieu equation, J. Funct. An., 48, n°3, Oct. 1982, | MR 84h:81019 | Zbl 0516.47018

[GeS] C. Gérard, J. Sjöstrand, Semi-classical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys., 108 (1987), 391-421. | Zbl 0637.35027

[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, | MR 86h:58136 | Zbl 0581.35009

[HR1] B. Helffer, D. Robert, Calcul fonctionnel par la transformée de Mellin et applications, J. Funct. An., 53, n°3, oct. 1983. | MR 85i:47052

[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, | Numdam | MR 86m:81049 | Zbl 0565.35082

[HS1] B. Helffer, J. Sjöstrand, Analyse semi-classique pour l'équation de Harper, Preprint (1987)

[HS2] B. Helffer, J. Sjöstrand, Analyse semi-classique pour l'équation de Harper II Comportement semi-classique près d'un rationnel, Preprint.

[HS3] B. Helffer, J. Sjöstrand, Structure Cantorienne du spectre de l'opérateur de Harper, Sém. des Equations aux Dérivées partielles, Ecole polytechnique, March 1988, | Numdam | Zbl 0666.34012

[HS4] B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit I, Comm. PDE, 9(4) (1984), 337-408, | MR 86c:35113 | Zbl 0546.35053

[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, | Numdam | Zbl 0595.35031

[HS6] B. Helffer, J. Sjöstrand, Effet tunnel pour l'équation de Schrödinger avec champ magnétique, Ann. Sc. Norm. Sup., to appear, | Numdam | Zbl 0699.35205

[HS7] B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Bull. de la SMF, 114, fasc. 3, (mémoire n°24-25). | Numdam | Zbl 0631.35075

[Ho] D. Hofstadter, Energy levels and wave functions for Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976), 2239-2249,

[L] J. Leray, Lagrangian analysis and quantum mechanics, MIT Press (1981). | MR 83k:58081a | Zbl 0483.35002

[Mo] P. Van Mouche, The coexistence problem for the discrete Mathieu operator, Preprint (1988),

[No] S.P. Novikov, Two dimensional operators in periodic fields, J. Sov. Math., 28, n°1, January 1985,

[O] F.W.J. Olver, Asymptotics and special functions, Academic Press, New York, London, 1974. | MR 55 #8655 | Zbl 0308.41023

[Si] B. Simon, Almost periodic Schrödinger operators: a review, Advances in applied math. 3, (1982), 463-490, | MR 85d:34030 | Zbl 0545.34023

[S1] J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982), | MR 84m:58151 | Zbl 0524.35007

[S2] J. Sjöstrand, Resonances generated by non-degenerate critical points, Springer Lecture Notes in Mathematics, n°1256, 402-429, | Zbl 0627.35074

[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,

[W1] M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems, Proc. R. Soc. London A391 (1984), 305-350, | MR 86b:81136

[W2] M. Wilkinson, Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field, Proc. R. Soc. London A403 (1986), 135-166, | MR 87f:81178

[W3] M. Wilkinson, An exact effective Hamiltonian for a perturbed Landau level, Journal of Phys. A, 20, n°7, 11-May 1987, 1761-, | MR 88g:81157 | Zbl 0639.47010

[W4] M. Wilkinson, An exact renormalisation group for Bloch electrons in a magnetic field, Journal of Physics A, to appear.

Cité par Sources :