Almost Mathieu operator

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

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

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

The spectral type

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

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

• For ${\displaystyle 0<\lambda <1}$, ${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has surely purely absolutely continuous spectrum.[2] (This was one of Simon's problems.)
• For ${\displaystyle \lambda =1}$, ${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ has surely purely singular continuous spectrum for any irrational ${\displaystyle \alpha }$.[3]
• For ${\displaystyle \lambda >1}$, ${\displaystyle H_{\omega }^{\lambda ,\alpha }}$ 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 ${\displaystyle \lambda \geq 1}$ follows (through the work of Last and Simon) [7] from the lower bound on the Lyapunov exponent ${\displaystyle \gamma (E)}$ given by

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

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 ${\displaystyle E}$ 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 ${\displaystyle \alpha }$ and ${\displaystyle \lambda >0}$. 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

${\displaystyle \operatorname {Leb} (\sigma (H_{\omega }^{\lambda ,\alpha }))=|4-4\lambda |\,}$

for all ${\displaystyle \lambda >0}$. For ${\displaystyle \lambda =1}$ 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 ${\displaystyle \lambda \neq 1}$, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky. Earlier Last [13][14] had proven this formula for most values of the parameters.

The study of the spectrum for ${\displaystyle \lambda =1}$ 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 978-1860942303.
2. Avila, A. (2008). "The absolutely continuous spectrum of the almost Mathieu operator". arXiv:0810.2965 [math.DS].
3. Jitomirskaya, S. "On point spectrum of critical almost Mathieu operators" (PDF). Cite journal requires |journal= (help)
4. Jitomirskaya, Svetlana Ya. (1999). "Metal-insulator transition for the almost Mathieu operator". Ann. of Math. 150 (3): 1159–1175. arXiv:math/9911265. doi:10.2307/121066. 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" (PDF). Comm. Math. Phys. 165 (1): 201–205. Bibcode:1994CMaPh.165..201J. CiteSeerX 10.1.1.31.4995. 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. arXiv:math-ph/9907023. Bibcode:1999InMat.135..329L. 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). "Solving the Ten Martini Problem". The Ten Martini problem. Lecture Notes in Physics. 690. pp. 5–16. arXiv:math/0503363. Bibcode:2006LNP...690....5A. doi:10.1007/3-540-34273-7_2. ISBN 978-3-540-31026-6.
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. arXiv:math-ph/0309004. Bibcode:2004CMaPh.244..297P. 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. arXiv:math/0306382. doi:10.4007/annals.2006.164.911.
13. Last, Y. (1993). "A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants". Comm. Math. Phys. 151 (1): 183–192. doi:10.1007/BF02096752.
14. Last, Y. (1994). "Zero measure spectrum for the almost Mathieu operator". Comm. Math. Phys. 164 (2): 421–432. doi:10.1007/BF02096752.