Time-dependent scattering on fractal measures

In this paper we study the time evolution for the Schrodinger equation and the wave equation on the line when the interaction term is a fractal measure. First, we extend the usual one-dimensional potential scattering formalism to interactions defined as measures, Then we show how to retrieve information on the fractality of the interaction term from time-dependent scattering data. In the case of the Schrodinger equation we shall obtain the wavelet correlation dimension of the scatterer. For the wave equation the whole set of generalized multifractal dimensions can be recovered, provided the scatterer actually is fractal (nonsmooth). Zn this latter case, we also show how the reflected wave packets can be interpreted in terms of wavelet transform of the interaction. (C) 1998 American Institute of Physics. [S0022-2488(98)01108-6].