Difference equations in the complex plane: quasiclassical asymptotics and Berry phase

@article{Fedotov2019DifferenceEI,
  title={Difference equations in the complex plane: quasiclassical asymptotics and Berry phase},
  author={A. Fedotov and E. Shchetka},
  journal={arXiv: Mathematical Physics},
  year={2019}
}
We study solutions to the difference equation $\Psi(z+h)=M(z)\Psi(z)$ where $z$ is a complex variable, $h>0$ is a parameter, and $M:\mathbb{C}\mapsto SL(2,\mathbb{C})$ is a given analytic function. We describe the asymptotics of its analytic solutions as $h\to 0$. The asymptotic formulas contain an analog of the geometric (Berry) phase well-known in the quasiclassical analysis of differential equations. 

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