Almost Mathieu operator
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 (now all solved) 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
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
- For , has surely purely absolutely continuous spectrum. (This was one of Simon's problems.)
- For , has surely purely singular continuous spectrum for any irrational .
- For , has almost surely pure point spectrum and exhibits Anderson localization. (It is known that almost surely can not be replaced by surely.)
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 (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 and . 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 . For this means that the spectrum has zero measure (this was first proposed by Douglas Hofstadter and later became one of Simon's problems). For , 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 leads to the Hofstadter's butterfly, where the spectrum is shown as a set.
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