Unusual properties of superoscillating particles

It has been found that differentiable functions can locally oscillate on length scales that are much smaller than the smallest wavelength contained in their Fourier spectrum-a phenomenon called superoscillation. Here, we consider the case of superoscillations in quantum mechanical wavefunctions. We find that superoscillations in wavefunctions lead to unusual phenomena that are of measurement theoretic, thermodynamic and information theoretic interest. We explicitly determine the wavefunctions with the most pronounced superoscillations, together with their scaling behaviour. We also briefly address the question of how superoscillating wavefunctions might be produced experimentally.