## Complex WKB method for adiabatic perturbations of a periodic Schrödinger operator.(English. Russian original)Zbl 1237.34148

J. Math. Sci., New York 173, No. 3, 320-339 (2011); translation from Zap. Nauchn. Semin. POMI 379, 142-178 (2010).
The paper considers the periodic Schrödinger operator $-\frac{d^{2}\psi }{dx^{2}}+(V(x)+W(\varepsilon x))\psi =E\psi ,\;x\in\mathbb{R},$ where $$E$$ is the spectral parameter, $$\varepsilon$$ is a small parameter, $$V$$ is a real valued $$1$$-periodic function from $$L_{loc}^{2}(\mathbb{R})$$ and $$W$$ is an analytic function in the strip of the form $$\delta =\left\{ \xi \in\mathbb{C}:y_{1}<\text{Im}\xi <y_{2}\right\}$$. The author presents an analog of the complex WKB method developed for studying effects of adiabatic perturbations of the periodic Schrödinger operator.

### MSC:

 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) 47E05 General theory of ordinary differential operators
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### References:

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