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Difference equations in the complex plane: quasiclassical asymptotics and Berry phase. (English) Zbl 1484.39018

Summary: We consider the equation \(\Psi (z+h) = M(z) \Psi (z)\), where \(z \in \mathbb{C}\), \(h>0\) is a parameter, and \(M: \mathbb{C} \mapsto \mathrm{SL}(2,\mathbb{C}\) is a given analytic function. We get asymptotics of its analytic solutions as \(h \rightarrow 0\). The asymptotic formulas contain an analog of the geometric (Berry) phase well known in the quasiclassical analysis of differential equations.

MSC:

39A45 Difference equations in the complex domain
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