All couplings localization for quasiperiodic operators with Lipschitz monotone potentials

We establish Anderson localization for quasiperiodic operator families of the form $(H(x)ψ)(m)=ψ(m+1)+ψ(m-1)+λv(x+mα)ψ(m)$ for all $λ>0$ and all Diophantine $α$, provided that $v$ is a $1$-periodic function satisfying a Lipschitz monotonicity condition on $[0,1)$. The localization is uniform on any energy interval on which Lyapunov exponent is bounded from below.

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