Almost Mathieu operator: Difference between revisions

Revision as of 16:49, 13 April 2016

Revision as of 19:56, 10 January 2017

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

== The spectral type ==

If <math>\alpha</math> is a [[rational number]], then <math>H^{\lambda,\alpha}_\omega</math>

is a periodic operator and by [[Floquet theory]] its [[spectrum (functional analysis)|spectrum]] is purely [[absolutely continuous]].

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This lower bound was proved independently by Avron, Simon and [[Michael Herman (mathematician)|Michael Herman]], after an earlier almost rigorous argument of Aubry and 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 |first=J. |last=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 }}</ref>

== The structure of the spectrum ==

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

[[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 "Tenten Martinimartini Problemproblem"<ref>{{cite journal |first=A. |last=Avila |first2=S. |last2=Jitomirskaya |title=The Ten Martini problem |work=Preprint |year=2005 |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 [[Lebesgue measure]] of the spectrum of the almost Mathieu operator is known to be

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The study of the spectrum for <math> \lambda =1 </math> leads to the [[Hofstadter's butterfly]], where the spectrum is shown as a set.

== References ==

{{reflist|2}}