Wave Function Shredding by Sparse Quantum Barriers

We discuss a model in which a quantum particle passes through $\delta$ potentials arranged in an increasingly sparse way. For infinitely many barriers we derive conditions, expressed in terms ergodic properties of wave function phases, which ensure that the point and absolutely continuous parts are absent leaving a purely singularly continuous spectrum. For a finite number of barriers, the transmission coefficient shows extreme sensitivity to the particle momentum with fluctuation in many different scales. We discuss a potential application of this behavior for erasing the information carried by the wave function.