A complex WKB method for adiabatic problems (Q2774115)

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 NEWLINE\[NEWLINE-\frac{d^{2}}{dx^{2}}\psi(x)+(V(x)+ W(\varepsilon x))\psi(x) = E\psi(x)NEWLINE\]NEWLINE the additional term \(\varphi\) so that it becomes NEWLINE\[NEWLINE-\frac{d^{2}}{dx^{2}}\psi(x)+(V(x)+ W(\varepsilon x + \varphi))\psi(x)= E\psi(x).\tag{1}NEWLINE\]NEWLINE 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.