Almost Mathieu operator: Difference between revisions

Revision as of 15:00, 8 May 2014

Revision as of 14:02, 21 September 2014

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[[Image:Hofstadter's_butterfly.png|thumb|Hofstadter's Butterfly]]

Another striking characteristic of the almost Mathieu operator is that its spectrum is a [[Cantor set]] for all irrational <math>\alpha</math> and <math>\lambda > 0</math>. This was shown by [[Artur Avila|Avila]] and [[Svetlana Jitomirskaya|Jitomirskaya]] solving the by-then famous "Ten Martini Problem"<ref>{{cite journal |first=A. |last=Avila |first2=S. |last2=Jitomirskaya |title=The Ten Martini problem |work=Preprint |year=2005 |idarxiv={{ArXiv|math|/0503363}} }}</ref> (also one of Simon's problems) after several earlier results (including generically<ref>{{cite journal |first=J. |last=Bellissard |first2=B. |last2=Simon |title=Cantor spectrum for the almost Mathieu equation |journal=[[Journal of Functional Analysis|J. Funct. Anal.]] |volume=48 |year=1982 |issue=3 |pages=408–419 |doi=10.1016/0022-1236(82)90094-5 }}</ref> and almost surely<ref>{{cite journal |last=Puig |first=Joaquim |title=Cantor spectrum for the almost Mathieu operator |journal=Comm. Math. Phys. |volume=244 |year=2004 |issue=2 |pages=297–309 |doi=10.1007/s00220-003-0977-3 }}</ref> with respect to the parameters).

Furthermore, the measure of the spectrum of the almost Mathieu operator is known to be