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{{Short description|Self-adjoint operator that arises in physical transition problems}}
In [[mathematical physics]], the '''almost Mathieu operator''', named for its similarity to the [[Mathieu function|Mathieu operator]]<ref name=simon1982almost>{{cite journal |last1=Simon |first1=Barry |year=1982 |title=Almost periodic Schrodinger operators: a review |journal=Advances in Applied Mathematics |volume=3 |issue=4 |pages=463-490}}</ref> introduced by [[Émile Léonard Mathieu]],<ref>{{cite web |title=Mathieu equation |url=https://encyclopediaofmath.org/wiki/Mathieu_equation |website=Encyclopedia of Mathematics |publisher=Springer |access-date=February 9, 2024}}</ref>
: <math> [H^{\lambda,\alpha}_\omega u](n) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, </math>
acting as a [[self-adjoint operator]] on the [[Hilbert_space#Sequence_spaces|Hilbert space <math>\ell^2(\mathbb{Z})</math>]]. Here <math>\alpha,\omega \in\mathbb{T}, \lambda > 0</math> are parameters. In [[pure mathematics]], its importance comes from the fact of being one of the best-understood examples of an [[ergodic]] [[Schrödinger operator]]. For example, three problems (now all solved) of [[Barry Simon]]'s fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.<ref>{{cite book |first=Barry |last=Simon |chapter=Schrödinger operators in the twenty-first century |title=Mathematical Physics 2000 |pages=283–288 |publisher=Imp. Coll. Press |___location=London |year=2000 |isbn=978-1860942303 }}</ref> In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the [[Aubry–André model]].
For <math>\lambda = 1</math>, the almost Mathieu operator is sometimes called '''Harper's equation'''.
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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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This problem was named the 'Dry Ten Martini Problem' by [[Barry Simon]] as it was [[Martini (cocktail)#Preparation|'stronger']] than the weaker problem which became known as the 'Ten Martini Problem':<ref name=simon1982almost></ref>
{{blockquote|text=
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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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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 |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
| last | doi = 10.1016/j.aim.2021.107997 | journal = [[Advances in Mathematics]] | page = 6 | title = On point spectrum of critical almost Mathieu operators | url = https://www.math.uci.edu/~mathphysics/preprints/point.pdf | volume = 392
| year = 2021}}</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|bibcode=1999math.....11265J |s2cid=10641385 }}</ref> (It is known that almost surely can not be replaced by surely.)<ref>{{cite journal |first1=J. |last1=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 |first1=S. |last1=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 |s2cid=16267690 }}</ref>
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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>
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[[Image:Hofstadter's_butterfly.png|thumb|[[Hofstadter's butterfly]]]]
Another striking characteristic of the almost Mathieu operator is that its spectrum is a [[Cantor set]] for all irrational <math>\alpha</math> and <math>\lambda > 0</math>. This was shown by [[Artur Avila|Avila]] and [[Svetlana Jitomirskaya|Jitomirskaya]] solving the by-then famous 'Ten Martini Problem' <ref>{{Cite book|first1=A. |last1=Avila |first2=S. |last2=Jitomirskaya |title=The Ten Martini problem |volume=690 |pages=5–16 |year=2005 |arxiv=math/0503363 |bibcode=2006LNP...690....5A |doi=10.1007/3-540-34273-7_2 |chapter=Solving the Ten Martini Problem |series=Lecture Notes in Physics |isbn=978-3-540-31026-6 |s2cid=55259301 }}</ref> (also one of Simon's problems) after several earlier results (including generically<ref>{{cite journal |first1=J. |last1=Bellissard |first2=B. |last2=Simon |title=Cantor spectrum for the almost Mathieu equation |journal=[[Journal of Functional Analysis|J. Funct. Anal.]] |volume=48 |year=1982 |issue=3 |pages=408–419 |doi=10.1016/0022-1236(82)90094-5 |doi-access=free }}</ref> and almost surely<ref>{{cite journal |last=Puig |first=Joaquim |title=Cantor spectrum for the almost Mathieu operator |journal=Comm. Math. Phys. |volume=244 |year=2004 |issue=2 |pages=297–309 |doi=10.1007/s00220-003-0977-3 |arxiv=math-ph/0309004 |bibcode=2004CMaPh.244..297P |s2cid=120589515 }}</ref> with respect to the parameters).
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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 |first1=A. |last1=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|s2cid=14625584 }}</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|bibcode=1993CMaPh.151..183L|s2cid=189834787|url=http://projecteuclid.org/euclid.cmp/1104252049}}</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/
The study of the spectrum for <math> \lambda =1 </math> leads to the [[Hofstadter's butterfly]], where the spectrum is shown as a set.
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