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 obtainded 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 mean 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 renormalisation. A numerical application is presented. Key-Words : potential scattering, wavelet-dimension, correlation dimension, large scale renormalisation. 1991 MSC:28,42A,45 Number of figures: 10 March 1996 CPT-96/P.3328 anonymous ftp or gopher: cpt.univ-mrs.fr Unit'e Propre de Recherche 7061 1 e-mail: guerin@cpt.univ-mrs.fr 2 e-mail: ho..