Almost Mathieu operator: Difference between revisions

Revision as of 19:56, 10 January 2017

Revision as of 03:34, 9 March 2017

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

: <math> \operatorname{Leb}(\sigma(H^{\lambda,\alpha}_\omega)) = |4 - 4 \lambda| \, </math>

for all <math>\lambda > 0</math>. For <math> \lambda = 1 </math> this means that the spectrum has zero measure (this was first proposed by [[Douglas Hofstadter]] and later became one of Simon's problems<ref>{{cite journal |first=A. |last=Avila |first2=R. |last2=Krikorian |title=Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles |journal=[[Annals of Mathematics]] |volume=164 |year=2006 |issue=3 |pages=911–940 |doi=10.4007/annals.2006.164.911 }}</ref>). For <math> \lambda \neq 1 </math>, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky.