Quasiperiodic operators with monotone potentials

The talk is based on joint works [KPS], [KKPS] with S. Krymskii, L. Parnovski, and R. Shterenberg.
We consider quasiperiodic operators on $\mathbb Z^d$ of the form
$$ (H(x)\psi)_{\mathbf n}=(\Delta \psi)_{\mathbf n}+\varepsilon f(x+{\mathbf n}\cdot\omega)\psi_{\mathbf n}, $$
where $f$ is a monotone function that maps the interval $(0,1)$ onto $(-\infty,+\infty)$ and is extended into $\mathbb R$ by $1$-periodicity. The frequency vector $\omega$ is assumed to satisfy a Diophantine property (and, in particular, have rationally independent components). For small $\varepsilon>0$ and under some additional monotonicity assumptions on $f$, we construct a diagonalization of such operator by direct analysis of the perturbation series.
[KPS] Kachkovskiy I., Parnovski L., Shterenberg R., Convergence of perturbation series for unbounded monotone quasiperiodic operators, https://arxiv.org/abs/2006.00346. [KKPS] Kachkovskiy I., Krymski S., Parnovski L., Shterenberg R., Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces, J. Math. Phys. 62 (2021), no. 6, 063509.