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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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