Almost Mathieu operator


In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by
acting as a self-adjoint operator on the Hilbert space. Here 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 of Barry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.
For, the almost Mathieu operator is sometimes called Harper's equation.

The spectral type

If is a rational number, then
is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous.
Now to the case when is irrational.
Since the transformation is minimal, it follows that the spectrum of does not depend on. 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.
It is now known, that
That the spectral measures are singular when follows
from the lower bound on the Lyapunov exponent given by
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 belongs to the spectrum, the inequality becomes an equality, proved by Jean Bourgain and Svetlana Jitomirskaya.

The structure of the spectrum

Another striking characteristic of the almost Mathieu operator is that its spectrum is a Cantor set for all irrational and. This was shown by Avila and Jitomirskaya solving the by-then famous "ten martini problem" after several earlier results.
Furthermore, the Lebesgue measure of the spectrum of the almost Mathieu operator is known to be
for all. For this means that the spectrum has zero measure. For, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky.
The study of the spectrum for leads to the Hofstadter's butterfly, where the spectrum is shown as a set.