Almost Mathieu operator: Difference between revisions

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Since the transformation <math> \omega \mapsto \omega + \alpha </math> is minimal, it follows that the spectrum of <math>H^{\lambda,\alpha}_\omega</math> does not depend on <math> \omega </math>. On the other hand, by ergodicity, the supports of absolutely continuous, singular continuous, and pure point parts of the spectrum are almost surely independent of <math> \omega </math>.

It is now known, that

*For <math>0 < \lambda < 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has surely purely absolutely continuous spectrum. <ref>{{cite journal |first=A. |last=Avila |year=2008 |title=The absolutely continuous spectrum of the almost Mathieu operator |workjournal=Preprint |arxiv=0810.2965 |bibcode=2008arXiv0810.2965A }}</ref> (This was one of Simon's problems.)

*For <math>\lambda= 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has almost surely purely singular continuous spectrum.<ref>{{cite journal |last=Gordon |first=A. Y. |last2=Jitomirskaya |first2=S. |last3=Last |first3=Y. |last4=Simon |first4=B. |title=Duality and singular continuous spectrum in the almost Mathieu equation |journal=[[Acta Mathematica|Acta Math.]] |volume=178 |year=1997 |issue=2 |pages=169–183 |doi=10.1007/BF02392693 }}</ref> (It is not known whether eigenvalues can exist for exceptional parameters.)

*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= |jstor=121066 }}</ref> (It is known that almost surely can not be replaced by surely.)<ref>{{cite journal |first=J. |last=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 }}</ref><ref>{{cite journal |first=S. |last=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 }}</ref>

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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 |workjournal=Preprint |year=2005 |arxiv=math/0503363 |bibcode=2006LNP...690....5A |doi=10.1007/3-540-34273-7_2 }}</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 |arxiv=math-ph/0309004 |bibcode=2004CMaPh.244..297P }}</ref> with respect to the parameters).

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