WKB asymptotics of meromorphic solutions to difference equations

Abstract

We consider the difference Schrödinger equation $\psi (z+h)+\psi (z-h)+v\left(z\right)\psi \left(z\right)=E\psi \left(z\right)$ 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 sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for differential equations. We assume that v has a simple pole and, in its neighborhood, we study the asymptotics of meromorphic solutions to the difference Schrödinger equation.

Keywords:

• Semiclassical asymptotics
• complex WKB method
• difference equations
• meromorphic coefficients

AMS CLASSIFICATIONS:

• 34M60
• 39A45

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Alexander Fedotov  http://orcid.org/0000-0002-8010-3004

Additional information

Funding

The work was supported by PRC CNRS France, and the Russian Foundation for Basic Research under the grant No 17-51-150008.