TY - JOUR

T1 - The complex WKB method for difference equations and airy functions

AU - Федотов, Александр Александрович

AU - Klopp, Frédéric

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 simple WKB behavior near the points where v(z) ≠ ± 2. We study analytic solutions near the points z0 satisfying v(z0) = ± 2 and v′ (z0) ≠ 0. These points play the same role as simple turning points for the differential equation ψ″ (z) + v(z)ψ (z) = 0. In an h-independent complex neighborhood of such a point, we derive uniform asymptotic expansions for analytic solutions to the difference equation.

AB - 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 simple WKB behavior near the points where v(z) ≠ ± 2. We study analytic solutions near the points z0 satisfying v(z0) = ± 2 and v′ (z0) ≠ 0. These points play the same role as simple turning points for the differential equation ψ″ (z) + v(z)ψ (z) = 0. In an h-independent complex neighborhood of such a point, we derive uniform asymptotic expansions for analytic solutions to the difference equation.

KW - Difference equations

KW - Turning point

KW - WKB

UR - http://www.scopus.com/inward/record.url?scp=85076461383&partnerID=8YFLogxK

U2 - 10.1137/18M1228694

DO - 10.1137/18M1228694

M3 - Article

AN - SCOPUS:85076461383

VL - 51

SP - 4413

EP - 4447

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 6

ER -