  Solutions to explicit time-dependent problems in quantum mechanics are rare.
In fact, all known solutions are coupled to specific properties of the
Hamiltonian and may be divided into two categories: One class consists of
time-dependent Hamiltonians which are not higher than quadratic in the position
operator, like i.e the driven harmonic oscillator with time-dependent
frequency. The second class is related to the existence of additional
invariants in the Hamiltonian, which can be used to map the solution of the
time-dependent problem to that of a related time-independent one.
  In this article we discuss and develop analytic methods for solving
time-dependent tunneling problems, which cannot be addressed by using quadratic
Hamiltonians. Specifically, we give an analytic solution to the problem of
tunneling from an attractive time-dependent potential which is embedded in a
long-range repulsive potential.
  Recent progress in atomic physics makes it possible to observe experimentally
time-dependent phenomena and record the probability distribution over a long
range of time. Of special interest is the observation of macroscopical
quantum-tunneling phenomena in Bose-Einstein condensates with time-dependent
trapping potentials. We apply our model to such a case in the last section.
