The Ten Martini Problem

被引：234

作者：

Avila, Artur ^{[1
,2
]}

Jitomirskaya, Svetlana ^{[3
]}

机构：

[1] Univ Paris 06, CNRS, UMR 7599, Lab Probabilites & Modeles Aleatoires, F-75252 Paris 05, France

[2] IMPA, BR-22460320 Rio De Janeiro, Brazil

[3] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA

来源：

ANNALS OF MATHEMATICS | 2009年 / 170卷 / 01期

关键词：

QUASI-PERIODIC OPERATORS; MATHIEU OPERATOR; CANTOR SPECTRUM; LOCALIZATION; CONTINUITY; POTENTIALS; THEOREM;

D O I：

10.4007/annals.2009.170.303

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页码：303 / 342

页数：40

共 24 条

[1]

[Anonymous], COMMUN MATH PHYS

[2]

[Anonymous], PROGR MATH

[3] Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles [J].

Avila, Artur ;

Krikorian, Raphael .

ANNALS OF MATHEMATICS, 2006, 164 (03) :911-940

[4] ON THE MEASURE OF THE SPECTRUM FOR THE ALMOST MATHIEU OPERATOR [J].

AVRON, J ;

VONMOUCHE, PHM ;

SIMON, B .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1990, 132 (01) :103-118

[5] ALMOST PERIODIC SCHRODINGER-OPERATORS .2. THE INTEGRATED DENSITY OF STATES [J].

AVRON, J ;

SIMON, B .

DUKE MATHEMATICAL JOURNAL, 1983, 50 (01) :369-391

[6]

AZBEL MY, 1964, SOV PHYS JETP-USSR, V19, P634

[7] CANTOR SPECTRUM FOR THE ALMOST MATHIEU EQUATION [J].

BELLISSARD, J ;

SIMON, B .

JOURNAL OF FUNCTIONAL ANALYSIS, 1982, 48 (03) :408-419

[8]

BEREZANSKIL JM, 1968, TRANSL MATH MONOGR, V17

[9] Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential [J].

Bourgain, J ;

Jitomirskaya, S .

JOURNAL OF STATISTICAL PHYSICS, 2002, 108 (5-6) :1203-1218

[10] GAUSS POLYNOMIALS AND THE ROTATION ALGEBRA [J].

CHOI, MD ;

ELLIOTT, GA ;

YUI, NK .

INVENTIONES MATHEMATICAE, 1990, 99 (02) :225-246

← 1 2 3 →