Geodesic
On monodromy matrices for a difference Schr\"odinger equation on the real line with a small periodic potential
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 223-244

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In this paper one considers a one-dimensional difference Schrödinger equation $\psi(z+h) + \psi(z-h) + \lambda v(z) \psi(z) = E \psi(z) $ with a periodic potential $v$. In the case when the potential is real analytic, as well as in the case when, in a neighborhood of $\mathbb{R}$, the potential has a finite number of simple poles per period, for small values of the coupling constant $\lambda$, we describe the asymptotics of a monodromy matrix. 
@article{ZNSL_2021_506_a14,
     author = {K. S. Sedov and A. A. Fedotov},
     title = {On monodromy matrices for a difference {Schr\"odinger} equation on the real line with a small periodic potential},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {223--244},
     publisher = {mathdoc},
     volume = {506},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a14/}
}
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K. S. Sedov; A. A. Fedotov. On monodromy matrices for a difference Schr\"odinger equation on the real line with a small periodic potential. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 223-244. http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a14/