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The complex WKB method for difference equations in bounded domains. (English. Russian original) Zbl 1476.39025

J. Math. Sci., New York 224, No. 1, 157-169 (2017); translation from Zap. Nauchn. Semin. POMI 438, 236-256 (2015).
Summary: The difference Schrödinger equation \(\psi(z+h)+\psi(z-h)+v(z)\psi(z) = E\psi(z)\), \(z\in\mathbb{C}\), is considered, where \(h > 0\) and \(E\in\mathbb{C}\) are parameters and \(v\) is a function analytic in a bounded domain \(D\subset \mathbb{C}\). An asymptotic method is developed for studying its solutions in the domain \(D\) for small positive \(h\).

MSC:

39A70 Difference operators
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
39A12 Discrete version of topics in analysis
39B32 Functional equations for complex functions
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References:

[1] V. Buslaev and A. Fedotov, “The complex WKB method for the Harper equation,” Algebra Analiz, 6, No. 3, 59-83 (1994). · Zbl 0839.34066
[2] M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations [in Russian], Librokom, Moscow (2009).
[3] A. Fedotov and F. Klopp, “A complex WKB method for adiabatic problems,” Asymptotic Analysis, 27, 219-264 (2001). · Zbl 1001.34082
[4] A. A. Fedotov, “The method of monodromization in the theory of quasiperiodic equations,” Algebra Analiz, 25, No. 2, 203-235 (2013). · Zbl 1001.34082
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