Almost Mathieu operator

In mathematical physics, the almost Mathieu operator arises in the study of the quantum 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 a self-adjoint operator on the Hilbert space ${\displaystyle \ell ^{2}(\mathbb {Z} )}$. Here ${\displaystyle \alpha ,\omega \in \mathbb {T} ,\lambda >0}$ 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.^[1]

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

The spectral type[]

If ${\displaystyle \alpha }$ is a rational number, then ${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous.

Now to the case when ${\displaystyle \alpha }$ is irrational. 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 ${\displaystyle 0<\lambda <1}$, ${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has surely purely absolutely continuous spectrum.^[2] (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 ${\displaystyle \alpha }$.^[3]
• For ${\displaystyle \lambda >1}$, ${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has almost surely pure point spectrum and exhibits Anderson localization.^[4] (It is known that almost surely can not be replaced by surely.)^[5][6]

That the spectral measures are singular when ${\displaystyle \lambda \geq 1}$ follows (through the work of Last and Simon) ^[7] from the lower bound on the Lyapunov exponent ${\displaystyle \gamma (E)}$ given by

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

This lower bound was proved independently by Avron, Simon and Michael Herman, after an earlier almost rigorous argument of Aubry and André. In fact, when ${\displaystyle E}$ belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), proved by Jean Bourgain and Svetlana Jitomirskaya.^[8]

The structure of the spectrum[]

Hofstadter's butterfly

Another striking characteristic of the almost Mathieu operator is that its spectrum is a Cantor set for all irrational ${\displaystyle \alpha }$ and ${\displaystyle \lambda >0}$. This was shown by Avila and Jitomirskaya solving the by-then famous "ten martini problem"^[9] (also one of Simon's problems) after several earlier results (including generically^[10] and almost surely^[11] with respect to the parameters).

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

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

for all ${\displaystyle \lambda >0}$. For ${\displaystyle \lambda =1}$ this means that the spectrum has zero measure (this was first proposed by Douglas Hofstadter and later became one of Simon's problems).^[12] For ${\displaystyle \lambda \neq 1}$, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky. Earlier Last ^[13][14] had proven this formula for most values of the parameters.

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

References[]