Quantum harmonic oscillator with superoscillating initial datum

In this paper we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schr"odinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.