On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations

Abstract

We consider one-dimensional difference Schrödinger equations $$ \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n) = E\varphi(n) , $$ $n \in \mathbb{Z}$, $x,\omega \in [0, 1]$ with real-analytic potential function $V(x)$. If $L(E,\omega_0)$ is greater than $0$ for all $E\in(E’, E”)$ and some Diophantine $\omega_0$, then the integrated density of states is absolutely continuous for almost every $\omega$ close to $\omega_0$, as shown by the authors in earlier work. In this paper we establish the formation of a dense set of gaps in $\mathop{\rm spec}( H(x,\omega))\cap (E’,E”)$. Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfunctions of one scale lead to “pre-gaps” at a larger scale. To pass to actual gaps in the spectrum, we show that these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, one relates a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle. Amongst other things, we establish a nonperturbative version of the co-variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of Sinai’s (perturbative) description of the spectrum via his function $\Lambda$. This allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase. Our infinite volume theorems hold for all Diophantine frequencies $\omega$ up to a set of Hausdorff dimension zero.

  • [AJ] Go to document A. Avila and S. Jitomirskaya, "Solving the ten martini problem," in Mathematical Physics of Quantum Mechanics, New York: Springer-Verlag, 2006, vol. 690, pp. 5-16.
    @incollection {AJ, MRKEY = {2238822},
      AUTHOR = {Avila, Artur and Jitomirskaya, Svetlana},
      TITLE = {Solving the ten martini problem},
      BOOKTITLE = {Mathematical {P}hysics of {Q}uantum {M}echanics},
      SERIES = {Lecture Notes in Phys.},
      VOLUME = {690},
      PAGES = {5--16},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2006},
      MRCLASS = {47B36 (39A70 47A10 47B39 47N50 81-02 81Q10)},
      MRNUMBER = {2238822},
      MRREVIEWER = {David Damanik},
      DOI = {10.1007/3-540-34273-7_2},
      ZBLNUMBER = {1166.47303},
      }
  • [AvrSim] Go to document 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 {AvrSim, 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 = {700145},
      MRREVIEWER = {W. O. Amrein},
      ZBLNUMBER={0544.35030},
      DOI = {10.1215/S0012-7094-83-05016-0},
      }
  • [BelSim] Go to document J. Béllissard and B. Simon, "Cantor spectrum for the almost Mathieu equation," J. Funct. Anal., vol. 48, iss. 3, pp. 408-419, 1982.
    @article {BelSim, 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},
      MRREVIEWER = {Evans M. Harrell},
      DOI = {10.1016/0022-1236(82)90094-5},
      ZBLNUMBER = {0516.47018},
      }
  • [Bhat] R. Bhatia, Perturbation Bounds for Matrix Eigenvalues, Harlow: Longman Scientific & Technical, 1987, vol. 162.
    @book {Bhat, MRKEY = {925418},
      AUTHOR = {Bhatia, Rajendra},
      TITLE = {Perturbation Bounds for Matrix Eigenvalues},
      SERIES = {Pitman {R}esearch {N}otes in {M}ath. {S}eries},
      VOLUME = {162},
      PUBLISHER = {Longman Scientific \& Technical},
      ADDRESS = {Harlow},
      YEAR = {1987},
      PAGES = {viii+129},
      ISBN = {0-582-01379-8},
      MRCLASS = {15A42 (15A60 47A55 65F15)},
      MRNUMBER = {88k:15020},
      MRREVIEWER = {A. R. Amir-Mo{é}z},
      ZBLNUMBER = {0696.15013},
      }
  • [BouKAM] Go to document J. Bourgain, "Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime," Lett. Math. Phys., vol. 51, iss. 2, pp. 83-118, 2000.
    @article {BouKAM, MRKEY = {1774640},
      AUTHOR = {Bourgain, J.},
      TITLE = {Hölder regularity of integrated density of states for the almost {M}athieu operator in a perturbative regime},
      JOURNAL = {Lett. Math. Phys.},
      FJOURNAL = {Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics},
      VOLUME = {51},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {83--118},
      ISSN = {0377-9017},
      CODEN = {LMPHDY},
      MRCLASS = {47B39 (39A12 47N50 82B44)},
      MRNUMBER = {1774640},
      MRREVIEWER = {David Damanik},
      ZBLNUMBER={0960.34071},
      DOI = {10.1023/A:1007641323456},
      }
  • [Boupos] Go to document J. Bourgain, "On the spectrum of lattice Schrödinger operators with deterministic potential," J. Anal. Math., vol. 87, pp. 37-75, 2002.
    @article {Boupos, MRKEY = {1945277},
      AUTHOR = {Bourgain, J.},
      TITLE = {On the spectrum of lattice {S}chrödinger operators with deterministic potential},
      NOTE = {dedicated to the memory of Thomas H. Wolff},
      JOURNAL = {J. Anal. Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {87},
      YEAR = {2002},
      PAGES = {37--75},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {47B39 (35P20 47A10 47N50 81Q10)},
      MRNUMBER = {1945277},
      MRREVIEWER = {David Damanik},
      ZBLNUMBER={1022.47024},
      DOI = {10.1007/BF02868469},
      }
  • [Bou] J. Bourgain, Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Princeton, NJ: Princeton Univ. Press, 2005, vol. 158.
    @book {Bou, MRKEY = {2100420},
      AUTHOR = {Bourgain, J.},
      TITLE = {Green's {F}unction {E}stimates for {L}attice {S}chrödinger {O}perators and {A}pplications},
      SERIES = {Annals of Math. Studies},
      VOLUME = {158},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {2005},
      PAGES = {x+173},
      ISBN = {0-691-12098-6},
      MRCLASS = {35Q40 (35A08 35Q55 37N20 47B80 47N50 81Q10 82B44)},
      MRNUMBER = {2005j:35184},
      MRREVIEWER = {David Damanik},
      }
  • [BouGol] Go to document J. Bourgain and M. Goldstein, "On nonperturbative localization with quasi-periodic potential," Ann. of Math., vol. 152, iss. 3, pp. 835-879, 2000.
    @article {BouGol, MRKEY = {1815703},
      AUTHOR = {Bourgain, J. and Goldstein, M.},
      TITLE = {On nonperturbative localization with quasi-periodic potential},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {152},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {835--879},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {39A70 (35J10 47N20 60F10 60H25 82B05)},
      MRNUMBER = {1815703},
      MRREVIEWER = {Kalyan B. Sinha},
      DOI = {10.2307/2661356},
      ZBLNUMBER = {1053.39035},
      }
  • [BourJit] Go to document 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 {BourJit, MRKEY = {1933451},
      AUTHOR = {Bourgain, J. and Jitomirskaya, S.},
      TITLE = {Continuity of the {L}yapunov exponent for quasiperiodic operators with analytic potential},
      NOTE = {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 = {1933451},
      DOI = {10.1023/A:1019751801035},
      ZBLNUMBER = {1039.81019},
      }
  • [Chan] Go to document J. Chan, "Method of variations of potential of quasi-periodic Schrödinger equations," Geom. Funct. Anal., vol. 17, iss. 5, pp. 1416-1478, 2008.
    @article {Chan, MRKEY = {2377494},
      AUTHOR = {Chan, Jackson},
      TITLE = {Method of variations of potential of quasi-periodic {S}chrödinger equations},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {17},
      YEAR = {2008},
      NUMBER = {5},
      PAGES = {1416--1478},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {82B44 (39A70 47B39 47B80 47N55 81Q10)},
      MRNUMBER = {2377494},
      MRREVIEWER = {David Damanik},
      ZBLNUMBER={1165.82014},
      DOI = {10.1007/s00039-007-0633-8},
      }
  • [CEY] Go to document 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 = {1031901},
      MRREVIEWER = {Palle E. T. Jorgensen},
      DOI = {10.1007/BF01234419},
      ZBLNUMBER = {0665.46051},
      }
  • [Falc] K. J. Falconer, The Geometry of Fractal Sets, Cambridge: Cambridge Univ. Press, 1986, vol. 85.
    @book {Falc, MRKEY = {867284},
      AUTHOR = {Falconer, K. J.},
      TITLE = {The {G}eometry of {F}ractal {S}ets},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {85},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1986},
      PAGES = {xiv+162},
      ISBN = {0-521-25694-1; 0-521-33705-4},
      MRCLASS = {28-02 (28A99 60J65)},
      MRNUMBER = {867284},
      MRREVIEWER = {K. E. Hirst},
      ZBLNUMBER = {0587.28004},
      }
  • [fed] H. Federer, Geometric Measure Theory, New York: Springer-Verlag, 1969.
    @book {fed, MRKEY = {0257325},
      AUTHOR = {Federer, Herbert},
      TITLE = {Geometric {M}easure {T}heory},
      SERIES = {Die Grundlehren der mathematischen Wissenschaften, Band 153},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1969},
      PAGES = {xiv+676},
      MRCLASS = {28.80 (26.00)},
      MRNUMBER = {41 \#1976},
      MRREVIEWER = {J. E. Brothers},
      ADDRESS = {New York},
      ZBLNUMBER = {0176.00801},
      }
  • [FS1] Go to document J. Fröhlich and T. Spencer, "Absence of diffusion in the Anderson tight binding model for large disorder or low energy," Comm. Math. Phys., vol. 88, iss. 2, pp. 151-184, 1983.
    @article {FS1, MRKEY = {696803},
      AUTHOR = {Fr{ö}hlich, J{ü}rg and Spencer, Thomas},
      TITLE = {Absence of diffusion in the {A}nderson tight binding model for large disorder or low energy},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {88},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {151--184},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82A05 (60H25 82A57)},
      MRNUMBER = {85c:82004},
      MRREVIEWER = {W. Kirsch},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103922279},
      ZBLNUMBER = {0519.60066},
      }
  • [FS2] Go to document J. Fröhlich and T. Spencer, "A rigorous approach to Anderson localization," Phys. Rep., vol. 103, iss. 1-4, pp. 9-25, 1984.
    @article {FS2, MRKEY = {839666},
      AUTHOR = {Fr{ö}hlich, J{ü}rg and Spencer, Thomas},
      TITLE = {A rigorous approach to {A}nderson localization},
      JOURNAL = {Phys. Rep.},
      FJOURNAL = {Physics Reports. A Review Section of Physics Letters},
      VOLUME = {103},
      YEAR = {1984},
      NUMBER = {1-4},
      PAGES = {9--25},
      ISSN = {0370-1573},
      CODEN = {PRPLCM},
      MRCLASS = {82A05 (82A57)},
      MRNUMBER = {87e:82003},
      DOI = {10.1016/0370-1573(84)90061-9},
      }
  • [FSW] Go to document J. Fröhlich, T. Spencer, and P. Wittwer, "Localization for a class of one-dimensional quasi-periodic Schrödinger operators," Comm. Math. Phys., vol. 132, iss. 1, pp. 5-25, 1990.
    @article {FSW, MRKEY = {1069198},
      AUTHOR = {Fr{ö}hlich, J. and Spencer, T. and Wittwer, P.},
      TITLE = {Localization for a class of one-dimensional quasi-periodic {S}chrödinger operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {132},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {5--25},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35J10 (34L05 34L40 35P05 81Q10)},
      MRNUMBER = {91h:35095},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104201027},
      }
  • [Furdes] Go to document H. Furstenberg and H. Kesten, "Products of random matrices," Ann. Math. Statist., vol. 31, pp. 457-469, 1960.
    @article {Furdes, MRKEY = {0121828},
      AUTHOR = {Furstenberg, H. and Kesten, H.},
      TITLE = {Products of random matrices},
      JOURNAL = {Ann. Math. Statist.},
      FJOURNAL = {Annals of Mathematical Statistics},
      VOLUME = {31},
      YEAR = {1960},
      PAGES = {457--469},
      ISSN = {0003-4851},
      MRCLASS = {60.00},
      MRNUMBER = {22 \#12558},
      MRREVIEWER = {R. E. Bellman},
      DOI = {10.1214/aoms/1177705909},
      ZBLNUMBER = {0137.35501},
      }
  • [GolSch1] Go to document M. Goldstein and W. Schlag, "Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions," Ann. of Math., vol. 154, iss. 1, pp. 155-203, 2001.
    @article {GolSch1, MRKEY = {1847592},
      AUTHOR = {Goldstein, Michael and Schlag, Wilhelm},
      TITLE = {Hölder continuity of the integrated density of states for quasi-periodic {S}chrödinger equations and averages of shifts of subharmonic functions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {154},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {155--203},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {82B44 (39A70 47B39 47N55 81Q10)},
      MRNUMBER = {1847592},
      MRREVIEWER = {G{ü}nter Stolz},
      ZBLNUMBER={0990.39014},
      DOI = {10.2307/3062114},
      }
  • [GolSch2] Go to document M. Goldstein and W. Schlag, "Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues," Geom. Funct. Anal., vol. 18, iss. 3, pp. 755-869, 2008.
    @article {GolSch2, MRKEY = {2438997},
      AUTHOR = {Goldstein, Michael and Schlag, Wilhelm},
      TITLE = {Fine properties of the integrated density of states and a quantitative separation property of the {D}irichlet eigenvalues},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {18},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {755--869},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {47B80 (39A70 47B36 47N50 81Q10)},
      MRNUMBER = {2438997},
      DOI = {10.1007/s00039-008-0670-y},
      ZBLNUMBER = {1171.82011},
      }
  • [GolSch3] M. Goldstein and W. Schlag, "On the formation of gaps in the spectrum of Schrödinger operators with quasi-periodic potentials," in in Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday, Providence, RI: Amer. Math. Soc., 2007, vol. 76, pp. 591-611.
    @incollection {GolSch3, MRKEY = {2307749},
      AUTHOR = {Goldstein, Michael and Schlag, Wilhelm},
      TITLE = {On the formation of gaps in the spectrum of {S}chrödinger operators with quasi-periodic potentials},
      BOOKTITLE = {{\rm {i}n} Spectral {T}heory and {M}athematical {P}hysics: a {F}estschrift in {H}onor of {B}arry {S}imon's 60th {B}irthday},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {76},
      PAGES = {591--611},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2007},
      MRCLASS = {47B36 (39A70 47B39 47B80 47N50 81Q10 82B44)},
      MRNUMBER = {2307749},
      ZBLNUMBER={pre05217632},
      }
  • [GolSch4] M. Goldstein and W. Schlag, "On Schrödinger operators with dynamically defined potentials," Mosc. Math. J., vol. 5, iss. 3, pp. 577-612, 742, 2005.
    @article {GolSch4, MRKEY = {2241813},
      AUTHOR = {Goldstein, Michael and Schlag, Wilhelm},
      TITLE = {On {S}chrödinger operators with dynamically defined potentials},
      JOURNAL = {Mosc. Math. J.},
      FJOURNAL = {Moscow Mathematical Journal},
      VOLUME = {5},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {577--612, 742--743},
      ISSN = {1609-3321},
      MRCLASS = {47B36 (47B80 81Q10)},
      ZBLNUMBER={1143.47031},
      MRNUMBER = {2241813},
      }
  • [herman] Go to document 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 {herman, 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},
      MRREVIEWER = {Dietrich Flockerzi},
      DOI = {10.1007/BF02564647},
      }
  • [Joh] Go to document R. Johnson, "Cantor spectrum for the quasi-periodic Schrödinger equation," J. Differential Equations, vol. 91, iss. 1, pp. 88-110, 1991.
    @article {Joh, MRKEY = {1106119},
      AUTHOR = {Johnson, Russell},
      TITLE = {Cantor spectrum for the quasi-periodic {S}chrödinger equation},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {91},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {88--110},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {34L40 (34D05 34L05 47E05 81Q10 82B44)},
      MRNUMBER = {92i:34118},
      MRREVIEWER = {Nariyuki Minami},
      DOI = {10.1016/0022-0396(91)90133-T},
      }
  • [Lan] S. Lang, Algebra, Addison-Wesley Publishing Co., Reading, MA, 1965.
    @book {Lan, MRKEY = {0197234},
      AUTHOR = {Lang, Serge},
      TITLE = {Algebra},
      PUBLISHER = {Addison-Wesley Publishing Co., Reading, MA},
      YEAR = {1965},
      PAGES = {xvii+508},
      MRCLASS = {00.00},
      MRNUMBER = {33 \#5416},
      MRREVIEWER = {R. S. Pierce},
      ZBLNUMBER = {0193.34701},
      }
  • [Last2] Go to document Y. Last, "Zero measure spectrum for the almost Mathieu operator," Comm. Math. Phys., vol. 164, iss. 2, pp. 421-432, 1994.
    @article {Last2, 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},
      MRREVIEWER = {Russell A. Johnson},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104270838},
      ZBLNUMBER = {0814.11040},
      }
  • [levin] Y. B. Levin, Lectures on Entire Functions, Providence, RI: Amer. Math. Soc., 1996, vol. 150.
    @book {levin, MRKEY = {1400006},
      AUTHOR = {Levin, B. Ya.},
      TITLE = {Lectures on {E}ntire {F}unctions},
      SERIES = {Translations of Math. Monographs},
      VOLUME = {150},
      NOTE = {in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1996},
      PAGES = {xvi+248},
      ISBN = {0-8218-0282-8},
      MRCLASS = {30-02 (30D15 30D20 31-02 46E99 46J15)},
      MRNUMBER = {1400006},
      MRREVIEWER = {Vladimir S. Azarin},
      ZBLNUMBER = {0856.30001},
      }
  • [Puig] Go to document J. Puig, "Cantor spectrum for the almost Mathieu operator," Comm. Math. Phys., vol. 244, iss. 2, pp. 297-309, 2004.
    @article {Puig, 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 = {2031032},
      MRREVIEWER = {David Damanik},
      DOI = {10.1007/s00220-003-0977-3},
      ZBLNUMBER = {1075.39021},
      }
  • [RS4] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, Second ed., New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1980.
    @book {RS4, MRKEY = {751959},
      AUTHOR = {Reed, Michael and Simon, Barry},
      TITLE = {Methods of {M}odern {M}athematical {P}hysics. {I}},
      EDITION = {Second},
      NOTE = {Functional analysis},
      PUBLISHER = {Academic Press [Harcourt Brace Jovanovich Publishers]},
      ADDRESS = {New York},
      YEAR = {1980},
      PAGES = {xv+400},
      ISBN = {0-12-585050-6},
      MRCLASS = {46-01 (47-01)},
      MRNUMBER = {85e:46002},
      ZBLNUMBER = {0459.46001},
      }
  • [Sin1] Y. G. Sinai, "Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential," J. Statist. Phys., vol. 46, iss. 5-6, pp. 861-909, 1987.
    @article {Sin1, MRKEY = {893122},
      AUTHOR = {Sinai, 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 = {893122},
      MRREVIEWER = {Eugen Belokolos},
      }
  • [To] M. Toda, Theory of Nonlinear Lattices, Second ed., New York: Springer-Verlag, 1989, vol. 20.
    @book {To, MRKEY = {971987},
      AUTHOR = {Toda, Morikazu},
      TITLE = {Theory of {N}onlinear {L}attices},
      SERIES = {Springer Series in Solid-State Sciences},
      VOLUME= {20},
      EDITION = {Second},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      PAGES = {x+225},
      ISBN = {3-540-18327-2},
      MRCLASS = {58F07 (70F99 70K99 82A99)},
      MRNUMBER = {971987},
      ZBLNUMBER = {0694.70001},
      }

Authors

Michael Goldstein

Department of Mathematics
University of Toronto
Toronto, Ontario
Canada M5S 2E4

Wilhelm Schlag

Department of Mathematics
The University of Chicago
Chicago IL 60637