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Consider the discrete magnetic Laplacian on $\mathbb Z^2.$

$$(\Delta_{\alpha,\lambda}\psi)(n_1,n_2) = e^{-i \pi \alpha n_2} \psi(n_1+1,n_2) + e^{i\pi \alpha n_2} \psi(n_1-1,n_2) + \lambda \left(e^{i \pi \alpha n_1} \psi(n_1,n_2+1)+e^{-i\pi \alpha n_1} \psi(n_1,n_2-1) \right)$$

We consider $\alpha$ irrational. It is known (to me) that the spectrum of this operator as a set always coincides with the spectrum of the Almost Mathieu operator and that this operator has no point spectrum. But I am curious whether it is known when the spectrum (depending on $\lambda$) of this operator is absolutely continuous/singular continuous. Apparently, it is singular continuous if $\lambda =1$ as the measure of the spectrum is zero in this case.

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Metal-insulator transition for the almost Mathieu operator (1999) proves that the spectrum is purely absolutely continuous for $\lambda<1$. (They write $\lambda<2$, but their $\lambda$ is different by a factor of two.) The spectrum is singular-continuous if $\lambda>1$ and $\alpha$ is a Liouville number. For a more extensive discussion, see the article by Y. Last (page 102 and following) in Sturm-Liouville theory.

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    $\begingroup$ These references are about one-dimensional operators and have little to do with the OP's question, which is about a two-dimensional operator. $\endgroup$
    – Christian Remling
    Jan 21, 2018 at 18:43
  • $\begingroup$ @ChristianRemling --- correct me if I'm wrong, please, but the papers I referred to address the Almost Mathieu operator on $\mathbb{Z}$ (one-dimensional), which is equivalent to the discrete magnetic Laplacian on $\mathbb{Z}^2$ (two-dimensional). See for example eqs. 1.1 and 1.2 from this paper. This is why the Mathieu eq. is used to study the spectrum of electrons in a two-dimensional lattice in a perpendicular magnetic field (Hofstadter problem). The parameter $\alpha$ in the Mathieu operator is the magnetic flux through a unit cell. $\endgroup$
    – Carlo Beenakker
    Jan 21, 2018 at 19:00
  • $\begingroup$ Thanks, I wasn't aware of this, this is relevant. However, my original comment still seems correct because the correspondence between the two problems (apparently) is not very strong: it just concerns the spectrum as a set and the DOS; the papers you quote prove much more refined results on the type of the spectrum of Almost Mathieu, which will not give anything on the 2d problem. $\endgroup$
    – Christian Remling
    Jan 21, 2018 at 20:39

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