Almost Mathieu operator

In mathematical physics, the almost Mathieu operator, named for its similarity to the Mathieu operator[1] introduced by Émile Léonard Mathieu,[2] arises in the study of the quantum Hall effect.

It is given by acting as a self-adjoint operator on the Hilbert space

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.

[3] In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the Aubry–André model.

, the almost Mathieu operator is sometimes called Harper's equation.

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

, there is a gap for the almost Mathieu operator on which

is the integrated density of states.This problem was named the 'Dry Ten Martini Problem' by Barry Simon as it was 'stronger' than the weaker problem which became known as the 'Ten Martini Problem':[1] For all

, the spectrum of the almost Mathieu operator is a Cantor set.If

λ , α

is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous.

λ , α

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 (through the work of Yoram Last and Simon) [9] from the lower bound on the Lyapunov exponent

given by This lower bound was proved independently by Joseph Avron, Simon and Michael Herman, after an earlier almost rigorous argument of Serge Aubry and Gilles André.

belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), proved by Jean Bourgain and Svetlana Jitomirskaya.

[10] Another striking characteristic of the almost Mathieu operator is that its spectrum is a Cantor set for all irrational

This was shown by Avila and Jitomirskaya 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, the Lebesgue measure of the spectrum of the almost Mathieu operator is known to be for all

this means that the spectrum has zero measure (this was first proposed by Douglas Hofstadter and later became one of Simon's problems).

, 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.

leads to the Hofstadter's butterfly, where the spectrum is shown as a set.