Time evolution of quantum fractals

We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density P-t(x) = [Psi (x, t)](2) is shown not to change during the time evolution. We prove a universal relation D-t = 1 + D-x/2 linking the dimensions of space cross sections D-x and time cross sections D-t of the fractal quantum carpets.