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

2019/10/21 by Fedotov, Alexander, Shchetka, Ekaterina
#39A45 #81Q20 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1910.09445

Abstract

We study solutions to the difference equation Ψ(z+h)=M(z)Ψ(z) where z is a complex variable, h>0 is a parameter, and M:ℂ↦ SL(2,ℂ) is a given analytic function. We describe the asymptotics of its analytic solutions as h→ 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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