Almost Mathieu operator: Difference between revisions

Revision as of 06:26, 24 November 2024

Revision as of 12:48, 17 June 2025

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

== The 'Ten Martinimartini Problem'problem ==

The structure of this operator's spectrum was first conjectured by [[Mark Kac]], who offered ten martinis for the first proof of the following conjecture:

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==The spectralSpectral 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 Joseph Avron, 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 structureStructure of the spectrum==

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