×

Semi-classical analysis for Harper’s equation. III: Cantor structure of the spectrum. (English) Zbl 0725.34099

[For parts I, II see ibid. 34, 113 p. (1988; Zbl 0714.34130), 40, 139 p. (1990; Zbl 0714.34131).]
The authors continue their study of Harper’s operator cosh D\(+\cos x\) in \(L^ 2({\mathbb{R}})\) by means of microlocalization and renormalization. In the case when h/2\(\pi\) is irrational they prove that the spectrum is a Cantor set of measure zero. Application to the periodic magnetic Schrödinger operator on \({\mathbb{R}}^ 2\) is given.
Reviewer: D.Robert (Nantes)

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35P05 General topics in linear spectral theory for PDEs
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] R. Abraham , J. Marsden , Foundations of Mechanics , Benjamin/Cumming publ. Co. ( 1978 ), MR 81e:58025 | Zbl 0393.70001 · Zbl 0393.70001
[2] S. Aubry , C. André , Proc. Israel Phys. Soc. Ed.C.G.Kuper 3 (Adam Hilger, Bristol) ( 1979 ), 133-, Zbl 0943.82510 · Zbl 0943.82510
[3] 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 · Zbl 0586.35084 · doi:10.1088/0305-4470/18/12/017
[4] Ya. Azbel , Energy spectrum of a conduction electron in a magnetic field , Soviet Physics JETP, 19, n^\circ 3, Sept 1964 .
[5] 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^\circ 3, Oct. 1982 , MR 84h:81019 | Zbl 0516.47018 · Zbl 0516.47018 · doi:10.1016/0022-1236(82)90094-5
[6] C. Gérard , J. Sjöstrand , Semi-classical resonances generated by a closed trajectory of hyperbolic type , Comm. Math. Phys., 108 ( 1987 ), 391-421. Article | Zbl 0637.35027 · Zbl 0637.35027 · doi:10.1007/BF01212317
[7] A. Grigis , J. Sjöstrand , Front d’onde analytique et sommes de carrés de champs de vecteurs , Duke Math. J., 52, n^\circ 1 ( 1985 ), 35-51, Article | MR 86h:58136 | Zbl 0581.35009 · Zbl 0581.35009 · doi:10.1215/S0012-7094-85-05203-2
[8] B. Helffer , D. Robert , Calcul fonctionnel par la transformée de Mellin et applications , J. Funct. An., 53, n^\circ 3, oct. 1983 . MR 85i:47052 · Zbl 0524.35103
[9] B. Helffer , D. Robert , Puits de potentiel généralisés et asymptotique semiclassique , Ann. I.H.P. (section Phys. théorique), 41, n^\circ 3, ( 1984 ), 291-331, Numdam | MR 86m:81049 | Zbl 0565.35082 · Zbl 0565.35082
[10] 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. · Zbl 0634.35056
[11] 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 · Zbl 0666.34012
[12] B. Helffer , J. Sjöstrand , Multiple wells in the semiclassical limit I , Comm. PDE, 9(4) ( 1984 ), 337-408, MR 86c:35113 | Zbl 0546.35053 · Zbl 0546.35053 · doi:10.1080/03605308408820335
[13] B. Helffer , J. Sjöstrand , Multiple wells in the semiclassical limit II , Ann. I.H.P. (section Phys. théorique), 42, n^\circ 2, ( 1985 ), 127-212, Numdam | Zbl 0595.35031 · Zbl 0595.35031
[14] B. Helffer , J. Sjöstrand , Effet tunnel pour l’équation de Schrödinger avec champ magnétique , Ann. Sc. Norm. Sup., to appear, Numdam | Zbl 0699.35205 · Zbl 0699.35205
[15] B. Helffer , J. Sjöstrand , Résonances en limite semi-classique , Bull. de la SMF, 114, fasc. 3, (mémoire n^\circ 24-25). Numdam | Zbl 0631.35075 · Zbl 0631.35075
[16] 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 · Zbl 0483.35002
[17] 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^\circ 1, January 1985 , [O] F.W.J. Olver , Asymptotics and special functions , Academic Press, New York, London, 1974 . MR 55 #8655 | Zbl 0308.41023 · Zbl 0308.41023
[18] B. Simon , Almost periodic Schrödinger operators: a review , Advances in applied math. 3, ( 1982 ), 463-490, MR 85d:34030 | Zbl 0545.34023 · Zbl 0545.34023 · doi:10.1016/S0196-8858(82)80018-3
[19] J. Sjöstrand , Singularités analytiques microlocales , Astérisque 95 ( 1982 ), MR 84m:58151 | Zbl 0524.35007 · Zbl 0524.35007
[20] J. Sjöstrand , Resonances generated by non-degenerate critical points , Springer Lecture Notes in Mathematics, n^\circ 1256, 402-429, Zbl 0627.35074 · Zbl 0627.35074
[21] 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^\circ 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
[22] 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
[23] M. Wilkinson , An exact effective Hamiltonian for a perturbed Landau level , Journal of Phys. A, 20, n^\circ 7, 11-May 1987 , 1761-, MR 88g:81157 | Zbl 0639.47010 · Zbl 0639.47010 · doi:10.1088/0305-4470/20/7/022
[24] M. Wilkinson , An exact renormalisation group for Bloch electrons in a magnetic field , Journal of Physics A, to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.