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

Abstract

We consider the equation $\mathrm{\Psi }(z+h)=M\left(z\right)\mathrm{\Psi }\left(z\right)$, where $z\in \mathbb{C}$, h>0 is a parameter, and $M:\mathbb{C}\mapsto \mathrm{S}\mathrm{L}(2,\mathbb{C})$ is a given analytic function. We get 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.

COMMUNICATED BY:

• Grigory Panasenko

Keywords:

• Difference equation
• complex plane
• quasiclassical asymptotics
• geometric phase

AMS Classifications:

• 39A45
• 81Q20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of A.F. was supported by Russian Foundation for Basic Research [grant number 17-01-00668]. The work of E.S. was supported by ‘Native towns’, a social investment program of PJSC ‘Gazprom Neft’.