Adiabatic Evolution Generated by a One-Dimensional Schrödinger Operator with Decreasing Number of Eigenvalues

We study a one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. The corresponding stationary operator depends on time as on a parameter. It has finitely many negative eigenvalues and absolutely continuous spectrum filling $$[0,+\infty)$$ . The eigenvalues move with time to the edge of the continuous spectrum and, having reached it, disappear one after another. We describe the asymptotic behavior of a solution close at some moment to an eigenfunction of the stationary operator, and, in particular, the phenomena occurring when the corresponding eigenvalue approaches the absolutely continuous spectrum and disappears.