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

The one-dimensional nonstationary Schrodinger equation is discussed in the adiabatic approximation. The corresponding stationary operator H depending on time as a parameter has a continuous spectrum sigma(e) = [0, +infinity) and finitely many negative eigenvalues. In time, the eigenvalues approach the edge of sigma(e) 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.