  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.
