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We consider a time dependent trap externally manipulated in such a way thatone of its bound states is brought into an instant contact with the continuumthreshold, and then down again. It is shown that, in the limit of slowevolution, the probability to remain in the bound state, $P^{stay}$ tends to auniversal limit, and is determined only by the manner in which the adiabaticbound state approaches and leaves the threshold. The task of evaluating the$P^{stay}$ in the adiabatic limit can be reduced to studying the loss from azero range well, and is performed numerically. Various types of trappingpotentials are considered. Applications of the theory to cold atoms in traps,and to propagation of traversal modes in tapered wave guides are proposed.

Adiabaticity in a time-dependent trap: a universal limit for the loss by touching the continuum