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Complex WKB method for Harper’s equation. (Russian) Zbl 0821.34062

The authors study the spectral properties of Harper’s linear operator \((H \psi) (x) = 1/2 [\psi (x + h) + \psi (x - h)] + (\cos x) \psi (x)\), \(h > 0\) in the complex Hilbert space \(L_ 2 (\mathbb{R})\). With the complex WKB method they construct solutions of the equation \(H \psi = E \psi\) for \(E \in \sigma (H) = [- 2,2]\), that have a simple asymptotic behavior on some canonical domains in the complex plane \(\mathbb{C}\). All geometric constructions of the complex WKB method are described in terms of the complex impulse \(p(z)\), defined by the equation \(\cos p(z) + \cos z = E\).

MSC:

34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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