Almost Mathieu operator

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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 of Barry Simon's fifteen problems (all solved by now!) in mathematical physics featured the almost Mathieu operator.[1]

For λ = 1, the almost Mathieu operator is sometimes called Harper's equation.

[edit] The spectral type

If α 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 α is irrational. Since the transformation \omega \mapsto \omega + \alpha is ergodic, it follows that the spectrum of H^{\lambda,\alpha}_\omega is constant for almost every ω. The same is true for the absolutely continuous, singular continuous, and pure point spectrum. It is now known, that

  • For 0 < λ < 1, H^{\lambda,\alpha}_\omega has surely purely absolutely continuous spectrum. [2] [3]
  • For λ = 1, H^{\lambda,\alpha}_\omega has surely purely singular continuous spectrum. [4]
  • For λ > 1, H^{\lambda,\alpha}_\omega has almost surely pure point spectrum and exhibits Anderson localization. [5] [6]

For example that the spectrum is singular continuous follows from Michael Herman's lower bound on the Lyapunov exponent γ(E) given by

\gamma(E) \geq \log(\lambda). \,

[edit] The structure of the spectrum

Hofstadter's Butterfly
Hofstadter's Butterfly

Another striking characteristic of the almost Mathieu operator is that its spectrum is a Cantor set for all irrational α and λ > 0. This was first shown by Puig[7] solving the by-then famous "Ten Martini Problem".

Furthermore, the measure of the spectrum of the almost Mathieu operator is known to be

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

for all λ > 0. The study of the spectrum for λ = 1 leads to the Hofstadter's butterfly, where the spectrum is shown as a set.

[edit] References

  1. ^ Simon, Barry Fifteen problems in mathematical physics. Perspectives in mathematics, 423--454.
  2. ^ Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999).
  3. ^ Gordon, A. Y.; Jitomirskaya, S.; Last, Y.; Simon, B. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178 (1997), no. 2, 169--183.
  4. ^ Gordon, A. Y.; Jitomirskaya, S.; Last, Y.; Simon, B. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178 (1997), no. 2, 169--183.
  5. ^ Jitomirskaya, Svetlana Ya. Metal-insulator transition for the almost Mathieu operator. Ann. of Math. (2) 150 (1999), no. 3, 1159--1175.
  6. ^ S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Comm. Math. Phys. 165 (1994), 201--205.
  7. ^ Puig, Joaquim Cantor spectrum for the almost Mathieu operator. Comm. Math. Phys. 244 (2004), no. 2, 297–309.