Adiabatic evolution generated by a one-dimensional Schrödinger operator with decreasing number of eigenvalues

Abstract

We study a one-dimensional non-stationary Schrödinger equation with a potential slowly depending on time. The corresponding stationary operator depends on time as on a parameter. It has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. The eigenvalues move with time to the edge of the continuous spectrum and, having reached it, disappear one after another. We study 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.

25 pages