Scattering on fractal measures

Abstract. 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 eipx 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.