Inmathematical physics, thealmost Mathieu operator, named for its similarity to theMathieu operator^[1] introduced byÉmile Léonard Mathieu,^[2] arises in the study of thequantum Hall effect. It is given by

${\displaystyle [H_{\omega }^{\lambda ,\alpha }u](n)=u(n+1)+u(n-1)+2\lambda \cos (2\pi (\omega +n\alpha ))u(n),}$

acting as aself-adjoint operator on theHilbert space${\displaystyle {\ell }^2(\mathbb{Z})}$. Here${\displaystyle \alpha ,\omega \in \mathbb{T},\lambda >0}$ are parameters. Inpure mathematics, its importance comes from the fact of being one of the best-understood examples of anergodicSchrödinger operator. For example, three problems (now all solved) ofBarry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.^[3] In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in theAubry–André model.

For${\displaystyle \lambda =1}$, the almost Mathieu operator is sometimes calledHarper's equation.

The structure of this operator's spectrum was first conjectured byMark Kac, who offered ten martinis for the first proof of the following conjecture:

For all${\displaystyle \lambda \ne 0}$, all irrational${\displaystyle a}$, and all integers${\displaystyle n_1,n_2}$, with, there is a gap for the almost Mathieu operator on which${\displaystyle k(E)=n_1+n_{2}a}$, where${\displaystyle k(E)}$ is the integrateddensity of states.

This problem was named the 'Dry Ten Martini Problem' byBarry Simon as it was'stronger' than the weaker problem which became known as the 'Ten Martini Problem':^[1]

For all${\displaystyle \lambda \ne 0}$, all irrational${\displaystyle a}$, and all${\displaystyle \omega }$, the spectrum of the almost Mathieu operator is aCantor set.

If${\displaystyle \alpha }$ is arational number, then${\displaystyle H_{\omega }^{\lambda ,\alpha }}$is a periodic operator and byFloquet theory itsspectrum is purelyabsolutely continuous.

Now to the case when${\displaystyle \alpha }$ isirrational.Since the transformation${\displaystyle \omega \mapsto \omega +\alpha }$ is minimal, it follows that the spectrum of${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ does not depend on${\displaystyle \omega }$. 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${\displaystyle \omega }$.It is now known, that

• For,${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has surely purely absolutely continuous spectrum.^[4] (This was one of Simon's problems.)
• For${\displaystyle \lambda =1}$,${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has surely purely singular continuous spectrum for any irrational${\displaystyle \alpha }$.^[5]
• For${\displaystyle \lambda >1}$,${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has almost surely pure point spectrum and exhibitsAnderson localization.^[6] (It is known that almost surely can not be replaced by surely.)^[7][8]

That the spectral measures are singular when${\displaystyle \lambda \ge 1}$ follows (through the work of Yoram Last and Simon)^[9]from the lower bound on theLyapunov exponent${\displaystyle \gamma (E)}$ given by

${\displaystyle \gamma (E)\ge max\{0,\log (\lambda )\}.}$

This lower bound was proved independently by Joseph Avron, Simon andMichael Herman, after an earlier almost rigorous argument of Serge Aubry and Gilles André. In fact, when${\displaystyle E}$ belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), proved byJean Bourgain andSvetlana Jitomirskaya.^[10]

Another striking characteristic of the almost Mathieu operator is that its spectrum is aCantor set for all irrational${\displaystyle \alpha }$ and${\displaystyle \lambda >0}$. This was shown byAvila andJitomirskaya solving the by-then famous 'Ten Martini Problem'^[11] (also one of Simon's problems) after several earlier results (including generically^[12] and almost surely^[13] with respect to the parameters).

Furthermore, theLebesgue measure of the spectrum of the almost Mathieu operator is known to be

${\displaystyle \mathrm{Leb}(\sigma (H_{\omega }^{\lambda ,\alpha }))=|4-4\lambda |}$

for all${\displaystyle \lambda >0}$. For${\displaystyle \lambda =1}$ this means that the spectrum has zero measure (this was first proposed byDouglas Hofstadter and later became one of Simon's problems).^[14] For${\displaystyle \lambda \ne 1}$, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky. Earlier Last^[15][16] had proven this formula for most values of the parameters.

The study of the spectrum for${\displaystyle \lambda =1}$ leads to theHofstadter's butterfly, where the spectrum is shown as a set.

• Spectral theory
• Mathematical physics

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