Almost Mathieu operator

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

[H_{\omega }^{{\lambda ,\alpha }}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 $H_{\omega }^{{\lambda ,\alpha }}$ 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 $H_{\omega }^{{\lambda ,\alpha }}$ 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 $0 < \lambda < 1$ , $H_{\omega }^{{\lambda ,\alpha }}$ has surely purely absolutely continuous spectrum. ^[2] (This was one of Simon's problems.)
For $\lambda =1$ , $H_{\omega }^{{\lambda ,\alpha }}$ has almost surely purely singular continuous spectrum.^[3] (It is not known whether eigenvalues can exist for exceptional parameters.)
For $\lambda >1$ , $H_{\omega }^{{\lambda ,\alpha }}$ 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

\operatorname {Leb} (\sigma (H_{\omega }^{\lambda ,\alpha }))=|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

↑ Simon, Barry (2000). "Schrödinger operators in the twenty-first century". Mathematical Physics 2000. London: Imp. Coll. Press. pp. 283–288. ISBN 186094230X.
↑ Avila, A. (2008). "The absolutely continuous spectrum of the almost Mathieu operator". Preprint. arXiv:0810.2965 .
↑ 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.
↑ Jitomirskaya, Svetlana Ya. (1999). "Metal-insulator transition for the almost Mathieu operator". Ann. of Math. 150 (3): 1159–1175. JSTOR 121066.
↑ Avron, J.; Simon, B. (1982). "Singular continuous spectrum for a class of almost periodic Jacobi matrices". Bull. Amer. Math. Soc. 6 (1): 81–85. Zbl 0491.47014. doi:10.1090/s0273-0979-1982-14971-0.
↑ Jitomirskaya, S.; Simon, B. (1994). "Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators". Comm. Math. Phys. 165 (1): 201–205. Zbl 0830.34074. doi:10.1007/bf02099743.
↑ 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.
↑ 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.
↑ Avila, A.; Jitomirskaya, S. (2005). "The Ten Martini problem". Preprint. arXiv:math/0503363 .
↑ 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.
↑ Puig, Joaquim (2004). "Cantor spectrum for the almost Mathieu operator". Comm. Math. Phys. 244 (2): 297–309. doi:10.1007/s00220-003-0977-3.
↑ 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.

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