Almost Mathieu operator: Difference between revisions

Revision as of 03:58, 2 August 2018

Revision as of 03:58, 2 August 2018

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: <math> [H^{\lambda,\alpha}_\omega u](n) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, </math>

acting as a [[self-adjoint operator]] on the Hilbert space <math>\ell^2(\mathbb{Z})</math>. Here <math>\alpha,\omega \in\mathbb{T}, \lambda > 0</math> are parameters. In [[pure mathematics]], its importance comes from the fact of being one of the best-understood examples of an [[ergodic]] [[Schrödinger operator]]. For example, three problems (now all solved) of [[Barry Simon]]'s fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.<ref>{{cite book |first=Barry |last=Simon |chapter=Schrödinger operators in the twenty-first century |title=Mathematical Physics 2000 |pages=283–288 |publisher=Imp. Coll. Press |___location=London |year=2000 |isbn=186094230X978-1860942303 }}</ref>

For <math>\lambda = 1</math>, the almost Mathieu operator is sometimes called '''Harper's equation'''.

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*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 |journal= |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= 10.2307/121066|jstor=121066 |arxiv=math/9911265 |class=math.SP }}</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 |url=http://www.math.caltech.edu/papers/bsimon/p235.pdf |format=Submitted manuscript }}</ref>

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

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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 journalCite book|first=A. |last=Avila |first2=S. |last2=Jitomirskaya |title=The Ten Martini problem |journal= |volume=690 |pages=5–16 |year=2005 |arxiv=math/0503363 |bibcode=2006LNP...690....5A |doi=10.1007/3-540-34273-7_2 |chapter=Solving the Ten Martini Problem |series=Lecture Notes in Physics |isbn=978-3-540-31026-6 }}</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