Geodesic
On minimal entire solutions of the one-dimensional difference Schr\"odinger equation with the potential $v(z)=e^{-2\pi iz}$
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 279-297

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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: $$ \psi(z+1)+\psi(z-1)+e^{-2\pi iz/h}\psi(z)=2\cos(2\pi p/h)\psi(z). $$ 
@article{ZNSL_2017_461_a15,
     author = {A. A. Fedotov},
     title = {On minimal entire solutions of the one-dimensional difference {Schr\"odinger} equation with the potential~$v(z)=e^{-2\pi iz}$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {279--297},
     publisher = {mathdoc},
     volume = {461},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a15/}
}
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A. A. Fedotov. On minimal entire solutions of the one-dimensional difference Schr\"odinger equation with the potential $v(z)=e^{-2\pi iz}$. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 279-297. http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a15/