Stark–Wannier type operators with purely singular spectrum

We consider the one-dimensional Stark–Wannier type operators $\[H=-\dfrac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-Fx-q(x)+v(x),\quad F>0,\]$ where q is a smooth function slowly growing at infinity and v is periodic, $\[$v\in L_{1}(\mathbb{T})$$ , with the Fourier coefficients of the form (ln |n|)^−β, 0<β<1/2, as n→∞. We show that for suitable q and F the spectrum of the corresponding operator is purely singular continuous. This proves the sharpness of the a.c. spectrum stability result obtained in Comm. Math. Phys. 234 (2003), 359–381.