  Motivated by applications for non-perturbative topological strings in toric
Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting
modular conjugate (in the sense of Faddeev) Harper type operators,
corresponding to a special case of the quantized mirror curve of local
$\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We
illustrate our analytical results by numerical calculations.
