Difference equations in the complex plane: quasiclassical asymptotics and Berry phase
Abstract
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 wellknown in the quasiclassical analysis of differential equations.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.09445
 Bibcode:
 2019arXiv191009445F
 Keywords:

 Mathematical Physics;
 39A45;
 81Q20
 EPrint:
 21 pages