## 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:

 3.4e+21 Singular perturbations, turning point theory, WKB methods for ordinary differential equations