Year: 2018
Type: preprint
Abstract: We consider the difference Schr{ö}dinger 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 $\rightarrow$ 0 analytic solutions to this equation have a standard quasiclassical behavior near the points where v(z) = $\pm$2. We study analytic solutions near the points z 0 satisfying v(z 0) = $\pm$2 and v (z 0) = 0. For the finite difference equation, these points are the natural analogues of the simple turning points defined for the differential equation --$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.
Source: arXiv (Cornell University)
Authors: Klopp, Frédéric, Fedotov, Alexander
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G7511101140
OpenAlex ID: W2895921780