Adiabatic Evolution Generated by a Schrödinger Operator with a Continuous Spectrum

For a model nonstationary Schrödinger equation with a potential depending on time $\tau$ and with a small parameter $\varepsilon$ in front of the derivative with respect to $\tau$, we investigate a solution that in the case of a potential independent of time has the form $e^{-iE\tau/\varepsilon} \psi(x,E)$, where $\psi(\cdot,E)$ is a generalized eigenfunction of the stationary Schrödinger operator corresponding to a value $E$ of the spectral parameter, $E$ being in the continuous spectrum.