Almost Mathieu operator: Difference between revisions

Revision as of 08:01, 24 October 2022

Revision as of 11:36, 9 February 2024

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{{Short description|Self-adjoint operator that arises in physical transition problems}}

In [[mathematical physics]], the '''almost Mathieu operator'''<!--, named afterfor 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>, arises in the study of the [[quantum Hall effect]]. It is given by

: <math> [H^{\lambda,\alpha}_\omega u](n) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, </math>

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For <math>\lambda = 1</math>, the almost Mathieu operator is sometimes called '''Harper's equation'''.

== The 'Ten Martini Problem' ==

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:

{{blockquote|text=

For all <math>\lambda \neq 0</math>, all irrational <math>a</math>, and all integers <math>n_l, n_2</math>, with <math>0 < n_l+ n_2a < 1</math>, there is a gap for the almost Mathieu operator on which <math>k(E) = n_l + n_2a</math>, where <math>k(E)</math> is the integrated [[density of states]].

}}

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=

For all <math>\lambda \neq 0</math>, all irrational <math>a</math>, and all <math>\omega</math>, the spectrum of the almost Mathieu operator is a [[Cantor set]].

}}

==The spectral type==

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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'Ten martiniMartini Problem' 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).

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