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 F_n that is traveling from −∞ in the direction of the barrier V₀. We assume that the energies E_n,k of all the traveling waves e^iλ_n,kx with |λ_n,k|≤1, are strictly less then the potential V₀ (n and k are suitable natural numbers), so that all the waves e^iλ_n,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 e^iλ_n,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.