Abstract
A class of above-barrier quantum-scattering problems is shown to provide an experimentally-accessible platform for studying PT-symmetric Schr\"odinger equations that exhibit spontaneous PT symmetry breaking despite having purely real potentials. These potentials are one-dimensional, inverted, and unstable and have the form V(x) = - x^p (p>0), terminated at a finite length or energy to a constant value as x→ ±∞. The signature of unbroken PT symmetry is the existence of reflectionless propagating states at discrete real energies up to arbitrarily high energy. In the 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. 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.