On the Delocalization of a Quantum Particle under the Adiabatic Evolution Generated by a One-Dimensional Schrödinger Operator

The one-dimensional nonstationary Schrödinger equation is discussed in the adiabatic approximation. The corresponding stationary operator $H$, depending on time as a parameter, has a continuous spectrum $\sigma_c=[0,+\infty)$ and finitely many negative eigenvalues. In time, the eigenvalues approach the edge of $\sigma_c$ and disappear one by one. The solution under consideration is close at some moment to an eigenfunction of $H$. As long as the corresponding eigenvalue $\lambda$ exists, the solution is localized inside the potential well. Its delocalization with the disappearance of $\lambda$ is described.