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A class of above-barrier quantum-scattering problems is shown to provide anexperimentally-accessible platform for studying $\mathcal{PT}$-symmetricSchr\"odinger equations that exhibit spontaneous $\mathcal{PT}$ symmetrybreaking despite having purely real potentials. These potentials areone-dimensional, inverted, and unstable and have the form $V(x) = - \lvertx\rvert^p$ ($p>0$), terminated at a finite length or energy to a constant valueas $x\to \pm\infty$. The signature of unbroken $\mathcal{PT}$ symmetry is theexistence of reflectionless propagating states at discrete real energies up toarbitrarily high energy. In the $\mathcal{PT}$-broken phase, there are no suchsolutions. In addition, there exists an intermediate mixed phase, wherereflectionless states exist at low energy but disappear at a fixed finiteenergy, independent of termination length. In the mixed phase exceptionalpoints (EPs) occur at specific $p$ and energy values, with a quartic dip in thereflectivity in contrast to the quadratic behavior away from EPs.$\mathcal{PT}$-symmetry-breaking phenomena have not been previously predictedin a quantum system with a real potential and no reservoir coupling. Theeffects predicted here are measurable in standard cold-atom experiments withprogrammable optical traps. The physical origin of the symmetry-breakingtransition is elucidated using a WKB force analysis that identifies the spatiallocation of the above-barrier scattering.

Experimentally-realizable $\mathcal{PT}$ phase transitions in reflectionless quantum scattering