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 well-known in the quasiclassical analysis of differential equations.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2019
- DOI:
- 10.48550/arXiv.1910.09445
- arXiv:
- arXiv:1910.09445
- Bibcode:
- 2019arXiv191009445F
- Keywords:
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- Mathematical Physics;
- 39A45;
- 81Q20
- E-Print:
- 21 pages