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

@article{Fedotov2019DifferenceEI, title={Difference equations in the complex plane: quasiclassical asymptotics and Berry phase}, author={A. Fedotov and E. Shchetka}, journal={arXiv: Mathematical Physics}, year={2019} }

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.

#### References

SHOWING 1-10 OF 39 REFERENCES

Berry phase for difference equations

- Physics
- 2017 Days on Diffraction (DD)
- 2017

We study solutions to the difference equation Ψ(z + h) = M(z)Ψ(z), z ∊ C, where h > 0 is a small constant parameter and M : C ↦ SL(2, C) is a given analytic matrix valued function. We describe the… Expand

The Complex WKB Method for Difference Equations and Airy Functions

- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 2019

In an h-independent neighborhood of such a point, uniform asymptotic expansions for analytic solutions to the difference equation are derived from the natural analogues of the simple turning points defined for the differential equation. Expand

WKB asymptotics of meromorphic solutions to difference equations

- Physics, Mathematics
- Applicable Analysis
- 2019

We consider the difference Schrödinger equation where z is a complex variable, E is a spectral parameter, and h is a small positive parameter. If the potential v is an analytic function, then, for h… Expand

On the difference equations with periodic coefficients

- Mathematics, Physics
- 2001

In this paper, we study entire solutions of the difference equation $\psi(z+h)=M(z)\psi(z)$, $z\in{\mathbb C}$, $\psi(z)\in {\mathbb C}^2$. In this equation, $h$ is a fixed positive parameter and $M:… Expand

The Complex WKB Method for Difference Equations in Bounded Domains

- Mathematics
- 2016 Days on Diffraction (DD)
- 2016

The difference Schrӧdinger equation ψ(z+h)+ψ(z−h)+v(z)ψ(z) = Eψ(z), z ∈ ℂ, is considered, where h > 0 and E ∈ ℂ are parameters and v is a function analytic in a bounded domain D ⊂ ℂ. An asymptotic… Expand

Anderson Transitions for a Family of Almost Periodic Schrödinger Equations in the Adiabatic Case

- Mathematics
- 2002

Abstract: This work is devoted to the study of a family of almost periodic one-dimensional Schrödinger equations. Using results on the asymptotic behavior of a corresponding monodromy matrix in the… Expand

Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase

- Physics
- 1983

It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem… Expand

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

- Mathematics
- 2018

We discuss two approaches that can be used to obtain the asymptotics of Hermite polynomials. The first, well-known approach is based on the representation of Hermite polynomials as solutions of a… Expand

Semi-classical approximation in quantum mechanics

- Mathematics
- 1981

I Quantization of Velocity Field (the Canonical Operator).- 1. The method of Stationary phase. The Legendre Transformation.- 2. Pseudodifferential Operators.- 3. The Hamilton-Jacobi Equation. The… Expand

Quantal phase factors accompanying adiabatic changes

- Mathematics
- Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1984

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar… Expand