  A class of above-barrier quantum-scattering problems is shown to provide an
experimentally-accessible platform for studying $\mathcal{PT}$-symmetric
Schr\"odinger equations that exhibit spontaneous $\mathcal{PT}$ symmetry
breaking despite having purely real potentials. These potentials are
one-dimensional, inverted, and unstable and have the form $V(x) = - \lvert
x\rvert^p$ ($p>0$), terminated at a finite length or energy to a constant value
as $x\to \pm\infty$. The signature of unbroken $\mathcal{PT}$ symmetry is the
existence of reflectionless propagating states at discrete real energies up to
arbitrarily high energy. In the $\mathcal{PT}$-broken phase, there are no such
solutions. In addition, there exists an intermediate mixed phase, where
reflectionless states exist at low energy but disappear at a fixed finite
energy, independent of termination length. In the mixed phase exceptional
points (EPs) occur at specific $p$ and energy values, with a quartic dip in the
reflectivity in contrast to the quadratic behavior away from EPs.
$\mathcal{PT}$-symmetry-breaking phenomena have not been previously predicted
in a quantum system with a real potential and no reservoir coupling. The
effects predicted here are measurable in standard cold-atom experiments with
programmable optical traps. The physical origin of the symmetry-breaking
transition is elucidated using a WKB force analysis that identifies the spatial
location of the above-barrier scattering.
