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 formula_2. Here formula_3 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.
For formula_4, the almost Mathieu operator is sometimes called Harper's equation.
The spectral type.
If formula_5 is a rational number, then formula_6
is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous.
Now to the case when formula_5 is irrational.
Since the transformation formula_8 is minimal, it follows that the spectrum of formula_6 does not depend on formula_10. 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 formula_10.
It is now known, that
That the spectral measures are singular when formula_19 follows (through the work of Last and Simon)
from the lower bound on the Lyapunov exponent formula_20 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 formula_22 belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), 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 formula_5 and formula_24. This was shown by Avila and Jitomirskaya solving the by-then famous "ten martini problem" (also one of Simon's problems) after several earlier results (including generically and almost surely with respect to the parameters).
Furthermore, the Lebesgue measure of the spectrum of the almost Mathieu operator is known to be
for all formula_24. For formula_27 this means that the spectrum has zero measure (this was first proposed by Douglas Hofstadter and later became one of Simon's problems). For formula_28, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky. Earlier Last had proven this formula for most values of the parameters.
The study of the spectrum for formula_29 leads to the Hofstadter's butterfly, where the spectrum is shown as a set.