How superoscillating tunneling waves can overcome the step potential

We consider the Cauchy problem for the Schrödinger equation with step potential with jump V0 at the origin and whose initial datum is a superoscillatory function Fn that is traveling from −∞ in the direction of the barrier V0. We assume that the energies En,k of all the traveling waves eiλn,kx with |λn,k|≤1, are strictly less then the potential V0 (n and k are suitable natural numbers), so that all the waves eiλn,kx are subjected to the tunnel effect for x≥0. We prove that in the limit, as n tends to infinity, from the infinitely many tunneling waves eiλn,kx emerges a wave that passes through the potential barrier. In other words, we prove that superoscillations overcome the potential barrier of the step potential even though none of its constituent waves can.