Scattering on fractal measures

We study the one-dimensional potential-scattering problem when the potential is a Radon measure with compact support. We show that the usual reflection and transmission amplitude r(p) and t(p) of an incoming wave e(ipx) are well defined. We also show that the scattering problem on fractal potentials can be obtained as a limit case of scattering on smooth potentials. We then explain how to retrieve the fractal 2-wavelet dimension and/or the correlation dimension of the potential by means of the reflexion amplitude r(p). We study the particular case of self-similar measures and show that, under some general conditions, r(p) has a large-scale renormalization. A numerical application is presented.