Abstract
We prove the conjecture (known as the “Ten Martini Problem” after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies.
-
[AK]
A. Avila and R. Krikorian, "Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles," Ann. of Math., vol. 164, iss. 3, pp. 911-940, 2006.
@article{AK, MRKEY = {2259248},
AUTHOR = {Avila, Artur and Krikorian, Rapha{ë}l},
TITLE = {Reducibility or nonuniform hyperbolicity for quasiperiodic {S}chrödinger cocycles},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {164},
YEAR = {2006},
NUMBER = {3},
PAGES = {911--940},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {81Q10 (37A20 37D25 39A10 47B36 47B80 47N50)},
MRNUMBER = {2008h:81044},
URL = {http://projecteuclid.org/getRecord?id=euclid.annm/1172614615},
ZBLNUMBER = {1138.47033},
MRREVIEWER = {Jean-Michel Ghez},
DOI = {10.4007/annals.2006.164.911},
} -
[AK2] A. Avila and R. Krikorian, "Quasiperiodic $\operatorname{SL}(2,\mathbb{R})$ cocycles," , in preparation.
@techreport{AK2,
author = {Avila, A. and Krikorian, R.},
TITLE = {Quasiperiodic {$\operatorname{SL}(2,\mathbb{R})$} cocycles},
TYPE = {in preparation},
marginpar = {please supply any updates},
} -
[AvMS]
J. Avron, P. H. M. van Mouche, and B. Simon, "On the measure of the spectrum for the almost Mathieu operator," Comm. Math. Phys., vol. 132, iss. 1, pp. 103-118, 1990.
@article{AvMS, MRKEY = {1069202},
AUTHOR = {Avron, J. and van Mouche, P. H. M. and Simon, B.},
TITLE = {On the measure of the spectrum for the almost {M}athieu operator},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {132},
YEAR = {1990},
NUMBER = {1},
PAGES = {103--118},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {39A70 (39A12 47B39 47N50 82B05)},
MRNUMBER = {92d:39014a},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104201031},
ZBLNUMBER = {0724.47002},
} -
[AS]
J. Avron and B. Simon, "Almost periodic Schrödinger operators. II: The integrated density of states," Duke Math. J., vol. 50, iss. 1, pp. 369-391, 1983.
@article{AS, MRKEY = {700145},
AUTHOR = {Avron, Joseph and Simon, Barry},
TITLE = {Almost periodic {S}chrödinger operators. {II}: {T}he integrated density of states},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {50},
YEAR = {1983},
NUMBER = {1},
PAGES = {369--391},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {34B25 (35P05 81C10)},
MRNUMBER = {85i:34009a},
DOI = {10.1215/S0012-7094-83-05016-0},
ZBLNUMBER = {0544.35030},
MRREVIEWER = {W. O. Amrein},
} -
[Az] Y. M. Azbel, "Energy spectrum of a conduction electron in a magnetic field," Sov. Phys. JETP, vol. 19, pp. 634-645, 1964.
@article{Az,
author = {Azbel, M. Ya.},
TITLE = {Energy spectrum of a conduction electron in a magnetic field},
JOURNAL = {Sov. Phys. JETP},
VOLUME = {19},
YEAR = {1964},
PAGES = {634--645},
} -
[ber] J. M. Berezanscprimekiui, Expansions in Eigenfunctions of Selfadjoint Operators, Providence, R.I.: Amer. Math. Soc., 1968.
@book{ber, MRKEY = {0222718},
AUTHOR = {Berezans{\cprime}ki{\u\i},
Ju. M.},
TITLE = {Expansions in Eigenfunctions of Selfadjoint Operators},
SERIES = {Transl. Math. Monogr.},
NUMBER = {17},
FSERIES = {Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, Vol. 17},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1968},
PAGES = {ix+809},
MRCLASS = {47.65 (34.00)},
MRNUMBER = {36 \#5768},
} -
[BJ1]
J. Bourgain and S. Jitomirskaya, "Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential," J. Statist. Phys., vol. 108, iss. 5-6, pp. 1203-1218, 2002.
@article{BJ1, MRKEY = {1933451},
AUTHOR = {Bourgain, J. and Jitomirskaya, S.},
TITLE = {Continuity of the {L}yapunov exponent for quasiperiodic operators with analytic potential},
INVNOTE = {Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays},
JOURNAL = {J. Statist. Phys.},
FJOURNAL = {Journal of Statistical Physics},
VOLUME = {108},
YEAR = {2002},
NUMBER = {5-6},
PAGES = {1203--1218},
ISSN = {0022-4715},
CODEN = {JSTPSB},
MRCLASS = {47B39 (47B36 47N50 81Q10 82B44)},
MRNUMBER = {2004c:47073},
DOI = {10.1023/A:1019751801035},
ZBLNUMBER = {1039.81019},
} -
[BS]
J. Béllissard and B. Simon, "Cantor spectrum for the almost Mathieu equation," J. Funct. Anal., vol. 48, iss. 3, pp. 408-419, 1982.
@article{BS, MRKEY = {678179},
AUTHOR = {B{é}llissard, J. and Simon, B.},
TITLE = {Cantor spectrum for the almost {M}athieu equation},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {48},
YEAR = {1982},
NUMBER = {3},
PAGES = {408--419},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {81C10 (34B25 39A12 47B39)},
MRNUMBER = {84h:81019},
DOI = {10.1016/0022-1236(82)90094-5},
ZBLNUMBER = {0516.47018},
MRREVIEWER = {Evans M. Harrell},
} -
[CEY]
M. D. Choi, G. A. Elliott, and N. Yui, "Gauss polynomials and the rotation algebra," Invent. Math., vol. 99, iss. 2, pp. 225-246, 1990.
@article{CEY, MRKEY = {1031901},
AUTHOR = {Choi, Man Duen and Elliott, George A. and Yui, Noriko},
TITLE = {Gauss polynomials and the rotation algebra},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {99},
YEAR = {1990},
NUMBER = {2},
PAGES = {225--246},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {46L55 (05A10 16W10 58F19 82B23)},
MRNUMBER = {91b:46067},
DOI = {10.1007/BF01234419},
ZBLNUMBER = {0665.46051},
MRREVIEWER = {Palle E. T. Jorgensen},
} -
[fur]
A. Furman, "On the multiplicative ergodic theorem for uniquely ergodic systems," Ann. Inst. H. Poincaré Probab. Statist., vol. 33, iss. 6, pp. 797-815, 1997.
@article{fur, MRKEY = {1484541},
AUTHOR = {Furman, Alex},
TITLE = {On the multiplicative ergodic theorem for uniquely ergodic systems},
JOURNAL = {Ann. Inst. H. Poincaré Probab. Statist.},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Probabilités et Statistiques},
VOLUME = {33},
YEAR = {1997},
NUMBER = {6},
PAGES = {797--815},
ISSN = {0246-0203},
CODEN = {AHPBAR},
MRCLASS = {28D05},
MRNUMBER = {98i:28018},
DOI = {10.1016/S0246-0203(97)80113-6},
ZBLNUMBER = {0892.60011},
MRREVIEWER = {Oliver Knill},
} -
[HS] B. Helffer and J. Sjöstrand, "Semiclassical analysis for Harper’s equation. III: Cantor structure of the spectrum," Mém. Soc. Math. France, iss. 39, pp. 1-124, 1989.
@article{HS, MRKEY = {1041490},
AUTHOR = {Helffer, B. and Sj{ö}strand, J.},
TITLE = {Semiclassical analysis for {H}arper's equation. {III}: {C}antor structure of the spectrum},
JOURNAL = {Mém. Soc. Math. France},
FJOURNAL = {Mémoires de la Société Mathématique de France. Nouvelle Série},
NUMBER = {39},
YEAR = {1989},
PAGES = {1--124},
ISSN = {0037-9484},
MRCLASS = {35A27 (35J10 35P05 47B39 47G30 47N50 58G07 81Q20)},
MRNUMBER = {92m:35011},
ZBLNUMBER = {0699.35273},
} -
[H]
M. Herman, "Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol\cprime d et de Moser sur le tore de dimension $2$," Comment. Math. Helv., vol. 58, iss. 3, pp. 453-502, 1983.
@article{H, MRKEY = {727713},
AUTHOR = {Herman, Michael-R.},
TITLE = {Une méthode pour minorer les exposants de {L}yapounov et quelques exemples montrant le caractère local d'un théorème d'{A}rnol\cprime d et de {M}oser sur le tore de dimension {$2$}},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {58},
YEAR = {1983},
NUMBER = {3},
PAGES = {453--502},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {58F11 (58F15)},
MRNUMBER = {85g:58057},
DOI = {10.1007/BF02564647},
ZBLNUMBER = {0554.58034},
MRREVIEWER = {Dietrich Flockerzi},
} -
[Ho] L. Hörmander, Notions of Convexity, Boston: Birkhäuser, 1994.
@book{Ho, MRKEY = {1301332},
AUTHOR = {H{ö}rmander, Lars},
TITLE = {Notions of {C}onvexity},
SERIES = {Progr. Math.},
FSERIES = {Progress in Mathematics},
NUMBER = {127},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston},
YEAR = {1994},
PAGES = {viii+414},
ISBN = {0-8176-3799-0},
MRCLASS = {00A05 (26B25 32F05 32F15 35A27 52A40)},
MRNUMBER = {95k:00002},
MRREVIEWER = {Steven George Krantz},
} -
[J]
S. Y. Jitomirskaya, "Metal-insulator transition for the almost Mathieu operator," Ann. of Math., vol. 150, iss. 3, pp. 1159-1175, 1999.
@article{J, MRKEY = {1740982},
AUTHOR = {Jitomirskaya, Svetlana Ya.},
TITLE = {Metal-insulator transition for the almost {M}athieu operator},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {150},
YEAR = {1999},
NUMBER = {3},
PAGES = {1159--1175},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {81Q10 (39A70 47B36 47B39 47N50)},
MRNUMBER = {2000k:81084},
DOI = {10.2307/121066},
ZBLNUMBER = {0946.47018},
MRREVIEWER = {Jean-Michel Ghez},
} -
[jks]
S. Jitomirskaya, D. A. Koslover, and M. S. Schulteis, "Localization for a family of one-dimensional quasiperiodic operators of magnetic origin," Ann. Henri Poincaré, vol. 6, iss. 1, pp. 103-124, 2005.
@article{jks, MRKEY = {2121278},
AUTHOR = {Jitomirskaya, S. and Koslover, D. A. and Schulteis, M. S.},
TITLE = {Localization for a family of one-dimensional quasiperiodic operators of magnetic origin},
JOURNAL = {Ann. Henri Poincaré},
FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
VOLUME = {6},
YEAR = {2005},
NUMBER = {1},
PAGES = {103--124},
ISSN = {1424-0637},
MRCLASS = {81Q10 (47B36 47N50)},
MRNUMBER = {2005j:81057},
DOI = {10.1007/s00023-005-0200-5},
ZBLNUMBER = {1062.81029},
MRREVIEWER = {David Damanik},
} -
[JK] Y. S. Jitomirskaya and I. V. Krasovsky, "Continuity of the measure of the spectrum for discrete quasiperiodic operators," Math. Res. Lett., vol. 9, iss. 4, pp. 413-421, 2002.
@article{JK, MRKEY = {1928861},
AUTHOR = {Jitomirskaya, S. Ya. and Krasovsky, I. V.},
TITLE = {Continuity of the measure of the spectrum for discrete quasiperiodic operators},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {9},
YEAR = {2002},
NUMBER = {4},
PAGES = {413--421},
ISSN = {1073-2780},
MRCLASS = {81Q10 (39A12 39A70 82B44)},
MRNUMBER = {2004b:81051},
ZBLNUMBER = {1020.47002},
MRREVIEWER = {Kalyan B. Sinha},
} -
[JM]
R. Johnson and J. Moser, "The rotation number for almost periodic potentials," Comm. Math. Phys., vol. 84, iss. 3, pp. 403-438, 1982.
@article{JM, MRKEY = {667409},
AUTHOR = {Johnson, R. and Moser, J.},
TITLE = {The rotation number for almost periodic potentials},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {84},
YEAR = {1982},
NUMBER = {3},
PAGES = {403--438},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {34B25 (58F07 58F19)},
MRNUMBER = {83h:34018},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103921211},
ZBLNUMBER = {0497.35026},
MRREVIEWER = {H. Hochstadt},
} -
[L]
Y. Last, "Zero measure spectrum for the almost Mathieu operator," Comm. Math. Phys., vol. 164, iss. 2, pp. 421-432, 1994.
@article{L, MRKEY = {1289331},
AUTHOR = {Last, Y.},
TITLE = {Zero measure spectrum for the almost {M}athieu operator},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {164},
YEAR = {1994},
NUMBER = {2},
PAGES = {421--432},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {47N50 (39A12 47A10 47B39 82D30)},
MRNUMBER = {95f:47096},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104270838},
ZBLNUMBER = {0814.11040},
MRREVIEWER = {Russell A. Johnson},
} -
[L3] Y. Last, "Spectral theory of Sturm-Liouville operators on infinite intervals: A review of recent developments," in Sturm-Liouville Theory, Amrein, W. O., Hinz, A. M., and Pearson, D. B., Eds., Basel: Birkhäuser, 2005, pp. 99-120.
@incollection{L3, MRKEY = {2145079},
AUTHOR = {Last, Yoram},
EDITOR = {Amrein, Werner O. and Hinz, Andreas M. and Pearson, David B.},
TITLE = {Spectral theory of {S}turm-{L}iouville operators on infinite intervals: {A} review of recent developments},
BOOKTITLE = {Sturm-{L}iouville Theory},
PAGES = {99--120},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2005},
MRCLASS = {47E05 (34B40 34L40)},
MRNUMBER = {2145079},
ZBLNUMBER = {1098.39011},
} -
[P]
J. Puig, "Cantor spectrum for the almost Mathieu operator," Comm. Math. Phys., vol. 244, iss. 2, pp. 297-309, 2004.
@article{P, MRKEY = {2031032},
AUTHOR = {Puig, Joaquim},
TITLE = {Cantor spectrum for the almost {M}athieu operator},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {244},
YEAR = {2004},
NUMBER = {2},
PAGES = {297--309},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {11K60 (39A70)},
MRNUMBER = {2004k:11129},
DOI = {10.1007/s00220-003-0977-3},
ZBLNUMBER = {1075.39021},
MRREVIEWER = {David Damanik},
} -
[Sim2]
B. Simon, "Kotani theory for one-dimensional stochastic Jacobi matrices," Comm. Math. Phys., vol. 89, iss. 2, pp. 227-234, 1983.
@article{Sim2, MRKEY = {709464},
AUTHOR = {Simon, Barry},
TITLE = {Kotani theory for one-dimensional stochastic {J}acobi matrices},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {89},
YEAR = {1983},
NUMBER = {2},
PAGES = {227--234},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {60H25 (35P99 47B99 58F19 82A57)},
MRNUMBER = {85d:60122},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103922682},
ZBLNUMBER = {0534.60057},
MRREVIEWER = {Ren{é} Carmona},
} -
[Sim1] B. Simon, "Schrödinger operators in the twenty-first century," in Mathematical Physics 2000, Fokas, A., Grigoryan, A., Kibble, T., and Zegarlinski, B., Eds., London: Imp. Coll. Press, 2000, pp. 283-288.
@incollection{Sim1, MRKEY = {1773049},
AUTHOR = {Simon, Barry},
EDITOR = {Fokas, A. and Grigoryan, A. and Kibble, T. and Zegarlinski, B.},
TITLE = {Schrödinger operators in the twenty-first century},
BOOKTITLE = {Mathematical {P}hysics 2000},
PAGES = {283--288},
PUBLISHER = {Imp. Coll. Press},
ADDRESS = {London},
YEAR = {2000},
MRCLASS = {81Q10 (81-02)},
MRNUMBER = {2001g:81071},
MRREVIEWER = {Hiroshi Isozaki},
ZBLNUMBER = {1074.81521},
} -
[Sin] Y. G. Sinaui, "Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential," J. Statist. Phys., vol. 46, iss. 5-6, pp. 861-909, 1987.
@article{Sin, MRKEY = {893122},
AUTHOR = {Sina{\u\i},
Ya. G.},
TITLE = {Anderson localization for one-dimensional difference {S}chrödinger operator with quasiperiodic potential},
JOURNAL = {J. Statist. Phys.},
FJOURNAL = {Journal of Statistical Physics},
VOLUME = {46},
YEAR = {1987},
NUMBER = {5-6},
PAGES = {861--909},
ISSN = {0022-4715},
CODEN = {JSTPSB},
MRCLASS = {82A05 (39A12 58F30)},
MRNUMBER = {88h:82016},
MRREVIEWER = {Eugen Belokolos},
ZBLNUMBER = {0682.34023},
} -
[vM]
P. van Mouche, "The coexistence problem for the discrete Mathieu operator," Comm. Math. Phys., vol. 122, iss. 1, pp. 23-33, 1989.
@article{vM, MRKEY = {994494},
AUTHOR = {van Mouche, Pierre},
TITLE = {The coexistence problem for the discrete {M}athieu operator},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {122},
YEAR = {1989},
NUMBER = {1},
PAGES = {23--33},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {47B39 (47A10 82A42 82A57)},
MRNUMBER = {90e:47027},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104178315},
ZBLNUMBER = {0669.34016},
} -
[dCJ87] C. De Concini and R. A. Johnson, "The algebraic-geometric AKNS potentials," Ergodic Theory Dynam. Systems, vol. 7, iss. 1, pp. 1-24, 1987.
@article {dCJ87, MRKEY = {MR886368},
AUTHOR = {De Concini, Corrado and Johnson, Russell A.},
TITLE = {The algebraic-geometric {AKNS} potentials},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {7},
YEAR = {1987},
NUMBER = {1},
PAGES = {1--24},
ISSN = {0143-3857},
MRCLASS = {58F07 (35Q20)},
MRNUMBER = {88f:58065},
ZBLNUMBER = {0636.35077},
MRREVIEWER = {Eugen Belokolos},
}