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Pré-Publication, Document De Travail Année : 2018

The complex WKB method for difference equations and Airy functions

Résumé

We consider the difference Schrödinger equation ψ(z + h) + ψ(z − h) + v(z)ψ(z) = 0 where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h → 0 analytic solutions to this equation have a standard quasiclassical behavior near the points where v(z) = ±2. We study analytic solutions near the points z 0 satisfying v(z 0) = ±2 and v (z 0) = 0. For the finite difference equation, these points are the natural analogues of the simple turning points defined for the differential equation −ψ (z) + v(z)ψ(z) = 0. In an h-independent neighborhood of such a point, we derive uniform asymptotic expansions for analytic solutions to the difference equation.
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Dates et versions

hal-01892639 , version 1 (10-10-2018)
hal-01892639 , version 2 (26-10-2018)
hal-01892639 , version 3 (21-11-2018)
hal-01892639 , version 4 (15-05-2019)

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Frédéric Klopp, Alexander Fedotov. The complex WKB method for difference equations and Airy functions. 2018. ⟨hal-01892639v3⟩
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