Almost Mathieu operator: Difference between revisions

Revision as of 03:58, 2 August 2018

Revision as of 03:59, 2 August 2018

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*For <math>0 < \lambda < 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has surely purely absolutely continuous spectrum.<ref>{{cite journal |first=A. |last=Avila |year=2008 |title=The absolutely continuous spectrum of the almost Mathieu operator |journal= |arxiv=0810.2965 |bibcode=2008arXiv0810.2965A }}</ref> (This was one of Simon's problems.)

*For <math>\lambda= 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has almost surely purely singular continuous spectrum.<ref>{{cite journal |last=Gordon |first=A. Y. |last2=Jitomirskaya |first2=S. |last3=Last |first3=Y. |last4=Simon |first4=B. |title=Duality and singular continuous spectrum in the almost Mathieu equation |journal=[[Acta Mathematica|Acta Math.]] |volume=178 |year=1997 |issue=2 |pages=169–183 |doi=10.1007/BF02392693 }}</ref> (It is not known whether eigenvalues can exist for exceptional parameters.)

*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 |class=math.SP }}</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 }}</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 |format=Submitted manuscript }}</ref>

That the spectral measures are singular when <math> \lambda \geq 1 </math> follows (through the work of Last and Simon)