Adiabaticity in a time dependent trap: a passage near continuum threshold

We consider a time dependent trap externally manipulated in such a way that one of its bound states is brought up towards the continuum threshold, and then down again. We evaluate the probability $P^{stay}$ for a particle, initially in a bound state of the trap, to continue in it at the end of the passage. We use the Sturmian representation, whereby the problem is reduced to evaluating the reflecting coefficient of an absorbing potential. In the slow passage limit, $P^{stay}$ goes to $1$ for a state turning before reaching the continuum threshold, and vanishes if the bound state crosses into the continuum. For a slowly moving state just "touching" the threshold $P^{stay}$ tends to a universal value of about $38\%$, for a broad class of potentials. In the rapid passage limit, $P^{stay}$ depends on the choice of the potential. Various types of trapping potentials are considered, with an analytical solution obtained in the special case of a zero-range well.