Almost Mathieu operator: Difference between revisions

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Revision as of 06:49, 22 November 2020

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Since the transformation <math> \omega \mapsto \omega + \alpha </math> is minimal, it follows that the spectrum of <math>H^{\lambda,\alpha}_\omega</math> does not depend on <math> \omega </math>. On the other hand, by ergodicity, the supports of absolutely continuous, singular continuous, and pure point parts of the spectrum are almost surely independent of <math> \omega </math>.

It is now known, that

*For <math>0 < \lambda < 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has surely purely absolutely continuous spectrum.<ref>{{cite arxiv |first=A. |last=Avila |year=2008 |title=The absolutely continuous spectrum of the almost Mathieu operator |journal= |eprint=0810.2965|class=math.DS }}</ref> (This was one of Simon's problems.)

*For <math>\lambda= 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has surely purely singular continuous spectrum for any irrational <math>\alpha</math>.<ref>{{cite journal |last1=Jitomirskaya |first1=S. |title=On point spectrum of critical almost Mathieu operators| url=https://www.math.uci.edu/~mathphysics/preprints/point.pdf}}</ref>

*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}}</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 |doi-access=free }}</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|citeseerx=10.1.1.31.4995 }}</ref>

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: <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=421-432421–432 |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.