Repeated quantum backflow and overflow

Quantum particles moving in one dimension with rightwards momentum can exhibit the surprising phenomenon of quantum backflow: a net probability transfer to the left-hand half-line over a finite time interval. We generalise this phenomenon by considering the sum of probability differences for M disjoint time intervals. In classical mechanics, the total backflow lies in the interval [-1,0] for all M, indicating rightwards probability transfer. By contrast, we show that the maximum M-fold quantum backflow is positive and unbounded from above as M increases, demonstrating the existence of repeated backflow. For M≥ 2, a new phenomenon of `quantum overflow' is discovered: there are states whose total backflow is below -1, which is impossible for classical particles. The extent of the backflow and overflow effects is described by a hierarchy of backflow and overflow functions and constants, of which the M=1 backflow constant was first studied by Bracken and Melloy. Limiting cases of the backflow and overflow functions are studied, including cases in which two disjoint intervals merge. The analytical results are supported by detailed numerical investigations. Using numerical acceleration methods, we obtain a new estimate of the Bracken--Melloy constant of 0.0384506, slightly lower than the previously accepted value of 0.038452.