  A remarkable feature of the Landau-Zener transition is insensitivity of the
survival probability to the decay rate, of the excited state. Namely, the
probability for a particle, which is initially in the ground state, to remain
in the same state is insensitive to decay, which is due to e.g. coupling to
continuum [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)]. This
insensitivity was demonstrated for the case when the density of states in the
continuum is energy-independent. We study the opposite limit when the density
of states in the continuum is a step-like function of energy. As a result of
this step-like behavior of the density of states, the decay rate of a driven
excited level experiences a jump as a function of time at certain moment t_0.
We take advantage of the fact that the analytical solution at t<t_0 and at
t>t_0 is known. We show that the decay enters the survival probability when t_0
is comparable to the transition time.
