메뉴 건너뛰기




Volumn 150, Issue 3, 1999, Pages 1159-1175

Metal-insulator transition for the almost Mathieu operator

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0033235963     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.2307/121066     Document Type: Article
Times cited : (399)

References (44)
  • 1
    • 0001215377 scopus 로고
    • Analyticity breaking and Anderson localisation in incommensurate lattices
    • S. AUBRY and G. ANDRÉ, Analyticity breaking and Anderson localisation in incommensurate lattices. Ann. Israel Phys. Soc. 3 (1980), 133-164.
    • (1980) Ann. Israel Phys. Soc. , vol.3 , pp. 133-164
    • Aubry, S.1    André, G.2
  • 2
    • 0000519980 scopus 로고
    • The new concept of transitions by breaking of analyticity in a crystallographic model
    • S. AUBRY, The new concept of transitions by breaking of analyticity in a crystallographic model, Solid State Sci. 8 (1978), 264-277.
    • (1978) Solid State Sci. , vol.8 , pp. 264-277
    • Aubry, S.1
  • 3
    • 84972566320 scopus 로고
    • Almost periodic schrödinger operators
    • II. The integrated density of states
    • J. AVRON AND B. SIMON, Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50 (1983), 369-391.
    • (1983) Duke Math. J. , vol.50 , pp. 369-391
    • Avron, J.1    Simon, B.2
  • 4
    • 84966202323 scopus 로고
    • Singular continuous spectrum for a class of almost periodic jacobi matrices
    • _, Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. A.M.S. 6 (1982), 81-85.
    • (1982) Bull. A.M.S. , vol.6 , pp. 81-85
  • 5
    • 0001539982 scopus 로고
    • On the measure of the spectrum for the almost Mathieu operator
    • J. AVRON, P. VAN MOUCHE, and B. SIMON, On the measure of the spectrum for the almost Mathieu operator, Comm. Math. Phys. 132 (1990), 103-118; Erratum: ibid. 139 (1991), 215.
    • (1990) Comm. Math. Phys. , vol.132 , pp. 103-118
    • Avron, J.1    Van Mouche, P.2    Simon, B.3
  • 6
    • 0001539982 scopus 로고
    • Erratum
    • J. AVRON, P. VAN MOUCHE, and B. SIMON, On the measure of the spectrum for the almost Mathieu operator, Comm. Math. Phys. 132 (1990), 103-118; Erratum: ibid. 139 (1991), 215.
    • (1991) Comm. Math. Phys. , vol.139 , pp. 215
  • 7
    • 0001723647 scopus 로고
    • A metal-insulator transition for the almost Mathieu model
    • J. BELLISARD, R. LIMA, and D. TESTAKD, A metal-insulator transition for the almost Mathieu model. Comm. Math. Phys. 88 (1983), 207-234.
    • (1983) Comm. Math. Phys. , vol.88 , pp. 207-234
    • Bellisard, J.1    Lima, R.2    Testakd, D.3
  • 8
    • 0003316646 scopus 로고
    • Expansions in eigenfunctions of selfadjoint operators
    • A.M.S., Providence. RI
    • Y. BEREZANSKII, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr., Vol. 17, A.M.S., Providence. RI, 1968.
    • (1968) Transl. Math. Monogr. , vol.17
    • Berezanskii, Y.1
  • 9
    • 0001180437 scopus 로고
    • Purely absolutely continuous spectrum for almost Mathieu operators
    • V. CHULAEVSKY and F. DELYON, Purely absolutely continuous spectrum for almost Mathieu operators, J. Statist. Phys. 55 (1989), 1279-1284.
    • (1989) J. Statist. Phys. , vol.55 , pp. 1279-1284
    • Chulaevsky, V.1    Delyon, F.2
  • 11
    • 84972546484 scopus 로고
    • Subharmonicity of the Lyaponov index
    • W. CRAIG and B. SIMON, Subharmonicity of the Lyaponov index, Duke Math. J. 50 (1983), 551-560.
    • (1983) Duke Math. J. , vol.50 , pp. 551-560
    • Craig, W.1    Simon, B.2
  • 13
    • 36149032983 scopus 로고
    • Absence of localisation in the almost Mathieu equation
    • F. DELYON, Absence of localisation in the almost Mathieu equation, J. Phys. A 20 (1987), L21-L23.
    • (1987) J. Phys. A , vol.20
    • Delyon, F.1
  • 14
    • 34250407352 scopus 로고
    • The one-dimensional schrödinger equation with a quasiperiodic potential
    • K. DINABURG and YA. SINAI, The one-dimensional Schrödinger equation with a quasiperiodic potential. Funct. Anal. Appl. 9 (1975), 279-289.
    • (1975) Funct. Anal. Appl. , vol.9 , pp. 279-289
    • Dinaburg, K.1    Sinai, Y.A.2
  • 15
    • 34249966896 scopus 로고
    • A new proof of localization in the Anderson tight binding model
    • H. VON DREIFUS and A. KLEIN, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), 285-299.
    • (1989) Comm. Math. Phys. , vol.124 , pp. 285-299
    • Von Dreifus, H.1    Klein, A.2
  • 16
    • 0001401431 scopus 로고
    • Floquet solutions for the 1-dimensional quasiperiodic Schrödinger equation
    • L. H. ELIASSON, Floquet solutions for the 1-dimensional quasiperiodic Schrödinger equation, Comm. Math. Phys. 146 (1992), 447-482.
    • (1992) Comm. Math. Phys. , vol.146 , pp. 447-482
    • Eliasson, L.H.1
  • 17
    • 0001699970 scopus 로고    scopus 로고
    • Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum
    • _, Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179 (1997), 153-196.
    • (1997) Acta Math. , vol.179 , pp. 153-196
  • 18
    • 0009371751 scopus 로고
    • The positivity of Lyapunov exponent and absence of the absolutely continuous spectrum for the almost-Mathieu equation
    • A. FIGOTIN and L. PASTUR, The positivity of Lyapunov exponent and absence of the absolutely continuous spectrum for the almost-Mathieu equation, J. Math. Phys. 25 (1984), 774-777.
    • (1984) J. Math. Phys. , vol.25 , pp. 774-777
    • Figotin, A.1    Pastur, L.2
  • 19
    • 0000284780 scopus 로고
    • Absence of diffusion in the Anderson tight binding model for large disorder or low energy
    • J. FRÖHLICH and T. SPENCER, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151-184.
    • (1983) Comm. Math. Phys. , vol.88 , pp. 151-184
    • Fröhlich, J.1    Spencer, T.2
  • 20
    • 0002119217 scopus 로고
    • Constructive proof of localization in the Anderson tight binding model
    • J. FRÖHLICH, F. MARTINELLI, E. SCOPPOLA, and T. SPENCER, Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), 21-46.
    • (1985) Comm. Math. Phys. , vol.101 , pp. 21-46
    • Fröhlich, J.1    Martinelli, F.2    Scoppola, E.3    Spencer, T.4
  • 21
    • 0002234361 scopus 로고
    • Localization for a class of one-dimensional quasi-periodic Schrödinger operators
    • J. FRÖHLICH, T. SPENCER, and P. WITTWER, Localization for a class of one-dimensional quasi-periodic Schrödinger operators, Comm. Math. Phys. 132 (1990), 5-25.
    • (1990) Comm. Math. Phys. , vol.132 , pp. 5-25
    • Fröhlich, J.1    Spencer, T.2    Wittwer, P.3
  • 22
    • 0040517213 scopus 로고    scopus 로고
    • The xi function
    • F. GESZTESY and B. SIMON, The xi function, Acta Math. 176 (1996), 49-71.
    • (1996) Acta Math. , vol.176 , pp. 49-71
    • Gesztesy, F.1    Simon, B.2
  • 24
    • 0033246857 scopus 로고    scopus 로고
    • Dynamical localization II with an application to the almost Mathieu operator
    • F. GERMINET, Dynamical localization II with an application to the almost Mathieu operator. J. Statist. Phys. 95(1999), 273-286.
    • (1999) J. Statist. Phys. , vol.95 , pp. 273-286
    • Germinet, F.1
  • 25
    • 0000425895 scopus 로고
    • The point spectrum of the one-dimensional Schrödinger operator
    • A. GORDON, The point spectrum of the one-dimensional Schrödinger operator, Usp. Math. Nauk. 31 (1976), 257-258.
    • (1976) Usp. Math. Nauk. , vol.31 , pp. 257-258
    • Gordon, A.1
  • 26
    • 0002638383 scopus 로고    scopus 로고
    • Duality and singular continuous spectrum in the almost Mathieu equation
    • A. GORDON.,S. JITOMIRSKAYA, Y. LAST, and B. SIMON, Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178 (1997), 169-183.
    • (1997) Acta Math. , vol.178 , pp. 169-183
    • Gordon, A.1    Jitomirskaya, S.2    Last, Y.3    Simon, B.4
  • 27
    • 0002330387 scopus 로고
    • Semiclassical analysis for Harper's equation. III. Cantor structure of the spectrum
    • B. HELFFER and J. SJÖSTRAND, Semiclassical analysis for Harper's equation. III. Cantor structure of the spectrum, Mém. Soc. Math. France (N.S.) 39 (1989), 1-124.
    • (1989) Mém. Soc. Math. France (N.S.) , vol.39 , pp. 1-124
    • Helffer, B.1    Sjöstrand, J.2
  • 28
    • 51249181642 scopus 로고
    • Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local dun théorème d'Arnol'd et de Moser sur le tore en dimension 2
    • M. HERMAN, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local dun théorème d'Arnol'd et de Moser sur le tore en dimension 2, Comment. Math. Helv. 58 (1983), 453-502.
    • (1983) Comment. Math. Helv. , vol.58 , pp. 453-502
    • Herman, M.1
  • 29
    • 21844515681 scopus 로고
    • Anderson localization for the almost Mathieu equation: A nonperturbative proof
    • S. JITOMIRSKAYA, Anderson localization for the almost Mathieu equation: a nonperturbative proof, Comm. Math. Phys. 165 (1994), 49-57.
    • (1994) Comm. Math. Phys. , vol.165 , pp. 49-57
    • Jitomirskaya, S.1
  • 30
    • 0039738784 scopus 로고
    • Anderson localization for the almost Mathieu equation. II. Point spectrum for λ > 2
    • _, Anderson localization for the almost Mathieu equation. II. Point spectrum for λ > 2, Comm. Math. Phys. 168 (1995), 563-570.
    • (1995) Comm. Math. Phys. , vol.168 , pp. 563-570
  • 31
    • 0001885983 scopus 로고
    • Almost everything about the almost Mathieu operator II
    • Paris, Internat. Press, Cambridge, MA
    • _, Almost everything about the almost Mathieu operator II, 373-382, Proc., XI Internat. Congr. of Math. Phys. (Paris, 1994), Internat. Press, Cambridge, MA, 1995.
    • (1994) Proc., XI Internat. Congr. of Math. Phys. , pp. 373-382
  • 33
    • 0000938201 scopus 로고    scopus 로고
    • Dimensional Hausdorff properties of singular continuous spectra
    • S. JITOMIRSKAYA and Y. LAST, Dimensional Hausdorff properties of singular continuous spectra. Phys. Rev. Lett. 76 (1996), 1765-1769.
    • (1996) Phys. Rev. Lett. , vol.76 , pp. 1765-1769
    • Jitomirskaya, S.1    Last, Y.2
  • 34
    • 0032121409 scopus 로고    scopus 로고
    • Anderson localization for the almost Mathieu equation. III. Semi-uniform localization, continuity of gaps, and measure of the spectrum
    • _, Anderson localization for the almost Mathieu equation. III. Semi-uniform localization, continuity of gaps, and measure of the spectrum, Comm. Math. Phys. 195 (1998), 1-14.
    • (1998) Comm. Math. Phys. , vol.195 , pp. 1-14
  • 36
    • 21844505868 scopus 로고
    • Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators
    • S. JITOMIRSKAYA and B. SIMON, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Comm. Math. Phys. 165 (1994), 201-205.
    • (1994) Comm. Math. Phys. , vol.165 , pp. 201-205
    • Jitomirskaya, S.1    Simon, B.2
  • 37
    • 33846873343 scopus 로고
    • A relation between absolutely continuous spectrum of ergodic Jacobi matrices and the spectra of periodic approximants
    • Y. LAST, A relation between absolutely continuous spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Comm. Math. Phys. 151 (1993), 183-192.
    • (1993) Comm. Math. Phys. , vol.151 , pp. 183-192
    • Last, Y.1
  • 38
    • 0001885983 scopus 로고
    • Almost everything about the almost Mathieu operator I
    • Paris, Internat. Press, Cambridge, MA
    • _, Almost everything about the almost Mathieu operator I, 366-372, Proc. XI Internat. Congr. Math. Phys. (Paris, 1994), Internat. Press, Cambridge, MA, 1995.
    • (1994) Proc. XI Internat. Congr. Math. Phys. , pp. 366-372
  • 39
    • 0033074693 scopus 로고    scopus 로고
    • Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
    • Y. LAST and B. SIMON, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329-367.
    • (1999) Invent. Math. , vol.135 , pp. 329-367
    • Last, Y.1    Simon, B.2
  • 40
    • 33744664346 scopus 로고
    • Zur theorie des diamagnetismus von leitungselektronen
    • R. PEIRLS, Zur Theorie des Diamagnetismus von Leitungselektronen, Z. Phys. 80 (1933), 763-791.
    • (1933) Z. Phys. , vol.80 , pp. 763-791
    • Peirls, R.1
  • 41
    • 0001004549 scopus 로고
    • Discrete magnetic Laplacia
    • M. SHUBIN, Discrete magnetic Laplacia,. Comm. Math. Phys. 164 (1994), 259-275.
    • (1994) Comm. Math. Phys. , vol.164 , pp. 259-275
    • Shubin, M.1
  • 42
    • 85016600983 scopus 로고
    • Schrödinger semigroups
    • B. SIMON, Schrödinger semigroups, Bull. A.M.S. 7 (1982), 447-526.
    • (1982) Bull. A.M.S. , vol.7 , pp. 447-526
    • Simon, B.1
  • 43
    • 0000763917 scopus 로고
    • Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential
    • YA. SINAI, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Stat. Phys. 46 (1987), 861-909.
    • (1987) J. Stat. Phys. , vol.46 , pp. 861-909
    • Sinai, Ya.1
  • 44
    • 0033245196 scopus 로고    scopus 로고
    • On the singular spectrum of the almost Mathieu operator Arithmetics and Cantor spectra of integrable models
    • P. B. WIEGMANN, On the singular spectrum of the almost Mathieu operator Arithmetics and Cantor spectra of integrable models, Prog. Theor. Phys. Suppl. 134 (1999), 171-181.
    • (1999) Prog. Theor. Phys. Suppl. , vol.134 , pp. 171-181
    • Wiegmann, P.B.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.