Almost Mathieu operator: Difference between revisions

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*For <math>\lambda > 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has almost surely pure point spectrum and exhibits [[Anderson localization]].<ref>{{cite journal |last=Jitomirskaya |first=Svetlana Ya. |title=Metal-insulator transition for the almost Mathieu operator |journal=[[Annals of Mathematics|Ann. of Math.]] |volume=150 |year=1999 |issue=3 |pages=1159–1175 |doi= 10.2307/121066|jstor=121066 |arxiv=math/9911265|bibcode=1999math.....11265J |s2cid=10641385 }}</ref> (It is known that almost surely can not be replaced by surely.)<ref>{{cite journal |first1=J. |last1=Avron |first2=B. |last2=Simon |title=Singular continuous spectrum for a class of almost periodic Jacobi matrices |journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc.]] |volume=6 |year=1982 |issue=1 |pages=81–85 |doi= 10.1090/s0273-0979-1982-14971-0|zbl=0491.47014 |doi-access=free }}</ref><ref>{{cite journal |first1=S. |last1=Jitomirskaya |first2=B. |last2=Simon |title=Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators |journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]] |volume=165 |year=1994 |issue=1 |pages=201–205 |zbl=0830.34074 |doi=10.1007/bf02099743|bibcode=1994CMaPh.165..201J |url=http://www.math.caltech.edu/papers/bsimon/p235.pdf|citeseerx=10.1.1.31.4995 |s2cid=16267690 }}</ref>

That the spectral measures are singular when <math> \lambda \geq 1 </math> follows (through the work of Yoram Last and Simon)

<ref>{{cite journal |first1=Y. |last1=Last |first2=B. |last2=Simon |title=Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators |journal=[[Inventiones Mathematicae|Invent. Math.]] |volume=135 |year=1999 |issue=2 |pages=329–367 |doi=10.1007/s002220050288 |arxiv=math-ph/9907023 |bibcode=1999InMat.135..329L |s2cid=9429122 }}</ref>

from the lower bound on the [[Lyapunov exponent]] <math>\gamma(E)</math> given by

: <math> \gamma(E) \geq \max \{0,\log(\lambda)\}. \, </math>

This lower bound was proved independently by Joseph Avron, [[Barry Simon]] and [[Michael Herman (mathematician)|Michael Herman]], after an earlier almost rigorous argument of Serge Aubry and Gilles André. In fact, when <math> E </math> belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), proved by [[Jean Bourgain]] and [[Svetlana Jitomirskaya]].<ref>{{cite journal |first1=J. |last1=Bourgain |first2=S. |last2=Jitomirskaya |title=Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential |journal=[[Journal of Statistical Physics]] |volume=108 |year=2002 |issue=5–6 |pages=1203–1218 |doi=10.1023/A:1019751801035 |s2cid=14062549 }}</ref>

==The structure of the spectrum==