On minimal entire solutions of the one-dimensional difference Schrödinger equation with the potential $v(z)=e^{-2\pi iz}$

Let $z\in\mathbb C$ be the complex variable, and let $h\in(0,1)$ and $p\in\mathbb C$ be parameters. For the equation
$$ \psi(z+h)+\psi(z-h)+e^{-2\pi iz}\psi(z)=2\cos(2\pi p)\psi(z), $$
we study its entire solutions that have the minimal possible growth both as $\operatorname{Im}z\to+\infty$ and as $\operatorname{Im}z\to-\infty$. In particular, we showed that they satisfy one more difference equation:
$$ \psi(z+1)+\psi(z-1)+e^{-2\pi iz/h}\psi(z)=2\cos(2\pi p/h)\psi(z). $$