Almost Mathieu operator: Difference between revisions

Revision as of 22:05, 8 October 2020

Revision as of 12:33, 16 October 2020

Line 32:

: <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 |arxiv=math/0306382}}</ref>. For <math> \lambda \neq 1 </math>, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky. Earlier Last <ref>{{cite journal |first=Y. |last=Last|title= A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants|journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]] |volume=151|year=1993|issue=1 |pages=183–192 |doi=10.1007/BF02096752}}</ref> <ref>{{cite journal |first=Y. |last=Last|title=Zero measure spectrum for the almost Mathieu operator|journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]] |volume=164|year=1994|issue=2 |pages=421-432 |doi=10.1007/BF02096752}}</ref> had proven this formula for most values of the parameters.

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