  Consider a system governed by the time-dependent Schr\"odinger equation in
its ground state. When subjected to weak (size $\epsilon$) parametric forcing
by an "ionizing field" (time-varying), the state decays with advancing time due
to coupling of the bound state to radiation modes. The decay-rate of this
metastable state is governed by {\it Fermi's Golden Rule}, $\Gamma[V]$, which
depends on the potential $V$ and the details of the forcing. We pose the
potential design problem: find $V_{opt}$ which minimizes $\Gamma[V]$ (maximizes
the lifetime of the state) over an admissible class of potentials with fixed
spatial support. We formulate this problem as a constrained optimization
problem and prove that an admissible optimal solution exists. Then, using
quasi-Newton methods, we compute locally optimal potentials. These have the
structure of a truncated periodic potential with a localized defect. In
contrast to optimal structures for other spectral optimization problems, our
optimizing potentials appear to be interior points of the constraint set and to
be smooth. The multi-scale structures that emerge incorporate the physical
mechanisms of energy confinement via material contrast and interference
effects.
  An analysis of locally optimal potentials reveals local optimality is
attained via two mechanisms: (i) decreasing the density of states near a
resonant frequency in the continuum and (ii) tuning the oscillations of
extended states to make $\Gamma[V]$, an oscillatory integral, small. Our
approach achieves lifetimes, $\sim (\epsilon^2\Gamma[V])^{-1}$, for locally
optimal potentials with $\Gamma^{-1}\sim\mathcal{O}(10^{9})$ as compared with
$\Gamma^{-1}\sim \mathcal{O}(10^{2})$ for a typical potential. Finally, we
explore the performance of optimal potentials via simulations of the
time-evolution.
