Almost Mathieu operator

In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by

$[H^{\lambda,\alpha}_\omega u](n) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \,$

acting as a self-adjoint operator on the Hilbert space $\ell^2(\mathbb{Z})$. Here $\alpha,\omega \in\mathbb{T}, \lambda > 0$ 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 $\lambda = 1$, the almost Mathieu operator is sometimes called Harper's equation.

The spectral type

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

Now to the case when $\alpha$ is irrational. Since the transformation $\omega \mapsto \omega + \alpha$ is minimal, it follows that the spectrum of $H^{\lambda,\alpha}_\omega$ does not depend on $\omega$. 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 $\omega$. It is now known, that

• For $0 < \lambda < 1$, $H^{\lambda,\alpha}_\omega$ has surely purely absolutely continuous spectrum. ^[2] (This was one of Simon's problems.)
• For $\lambda= 1$, $H^{\lambda,\alpha}_\omega$ has almost surely purely singular continuous spectrum.^[3] (It is not known whether eigenvalues can exist for exceptional parameters.)
• For $\lambda > 1$, $H^{\lambda,\alpha}_\omega$ 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 $\lambda \geq 1$ follows (through the work of Last and Simon) ^[7] from the lower bound on the Lyapunov exponent $\gamma(E)$ given by

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

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 $E$ 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 $\alpha$ and $\lambda > 0$. 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

$Leb(\sigma(H^{\lambda,\alpha}_\omega)) = |4 - 4 \lambda| \,$

for all $\lambda > 0$. For $\lambda = 1$ 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 $\lambda \neq 1$, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky.

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

References

1. Simon, Barry (2000). "Schrödinger operators in the twenty-first century". Mathematical Physics 2000. London: Imp. Coll. Press. pp. 283–288. ISBN 186094230X.
2. Avila, A. (2008). "The absolutely continuous spectrum of the almost Mathieu operator". Preprint. arXiv:0810.2965.
3. Gordon, A. Y.; Jitomirskaya, S.; Last, Y.; Simon, B. (1997). "Duality and singular continuous spectrum in the almost Mathieu equation". Acta Math. 178 (2): 169–183. doi:10.1007/BF02392693.
4. Jitomirskaya, Svetlana Ya. (1999). "Metal-insulator transition for the almost Mathieu operator". Ann. of Math. 150 (3): 1159–1175. JSTOR 121066.
5. Avron, J.; Simon, B. (1982). "Singular continuous spectrum for a class of almost periodic Jacobi matrices". Bull. Amer. Math. Soc. 6 (1): 81–85. doi:10.1090/s0273-0979-1982-14971-0. Zbl 0491.47014.
6. Jitomirskaya, S.; Simon, B. (1994). "Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators". Comm. Math. Phys. 165 (1): 201–205. doi:10.1007/bf02099743. Zbl 0830.34074.
7. Last, Y.; Simon, B. (1999). "Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators". Invent. Math. 135 (2): 329–367. doi:10.1007/s002220050288.
8. Bourgain, J.; Jitomirskaya, S. (2002). "Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential". Journal of Statistical Physics 108 (5–6): 1203–1218. doi:10.1023/A:1019751801035.
9. Avila, A.; Jitomirskaya, S. (2005). "The Ten Martini problem". Preprint. arXiv:math/0503363.
10. Bellissard, J.; Simon, B. (1982). "Cantor spectrum for the almost Mathieu equation". J. Funct. Anal. 48 (3): 408–419. doi:10.1016/0022-1236(82)90094-5.
11. Puig, Joaquim (2004). "Cantor spectrum for the almost Mathieu operator". Comm. Math. Phys. 244 (2): 297–309. doi:10.1007/s00220-003-0977-3.
12. Avila, A.; Krikorian, R. (2006). "Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles". Annals of Mathematics 164 (3): 911–940. doi:10.4007/annals.2006.164.911.