## A complex WKB method for adiabatic problems.(English)Zbl 1001.34082

Here, the authors give a new version of the complex WKB method suited for adiabatic perturbations of one-dimensional periodic Schrödinger operators. They introduce in the Schrödinger equation $-\frac{d^{2}}{dx^{2}}\psi(x)+(V(x)+ W(\varepsilon x))\psi(x) = E\psi(x)$ the additional term $$\varphi$$ so that it becomes $-\frac{d^{2}}{dx^{2}}\psi(x)+(V(x)+ W(\varepsilon x + \varphi))\psi(x)= E\psi(x).\tag{1}$ The authors consider solutions to (1) analytic in the parameter $$\varphi$$ and introduce a consistency condition for the basis $$\psi_{\pm}$$ of (1), complex momentum and canonical domains. Then they construct solutions to the Schrödinger equation with a standard asymptotic behavior.

### MSC:

 34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain