Asymptotically tunable quantum states and threshold scattering anomalies

Quantum systems can support irregular eigenstates at their threshold, which can be bound, loosely bound, or half bound. Such states are physically significant, and for instance half-bound states are known to lead to anomalous quantum scattering, where the reflection coefficient vanishes at the threshold rather than approach unity. Here, we present irregular threshold states which are generalizations of the above cases. The asymptotic behavior of these states can be tuned arbitrarily by precise control of the potential; hence, they are denoted "asymptotically tunable."We provide exact analytical prescriptions on how to generate and control these systems. We explore several examples in 1D, including states that exhibit a power-law-like asymptotic scaling, and hybrid states that exhibit asymmetric boundary conditions (e.g., are fully bounded for x→∞ but unbounded for x→-∞, etc.). We numerically explore the scattering properties of these systems and find a close connection between the asymptotic behavior of the threshold states, and the appearance of anomalous scattering. We show that the threshold reflection coefficient can exhibit both discontinuities and derivative discontinuities as the system transitions from regular to irregular, which persist even under perturbations, and thus seem to not be quantum critical. These states could be useful for quantum system engineering, and for potential applications in optical systems for manipulating light.