Potential Scattering on Fractals in One Dimension

Abstract

Scattering is known to be a powerful method of analysing fractal systems. A classical heuristic result in small-angle X-ray or neutron scattering relates the scattered intensity I(q) in the high-frequency regime to some power-law of the momentum transfer q

$$I\left( q \right) \sim {\left| q \right|^{ - D}},$$

where D is some fractal dimension of the scatterer. In the last decade, the case of one-dimensional systems has been studied more rigorously, in the framework of optical and quantum scattering ([1], [2],[13],[19],[14]). For deterministic fractals such as Cantor-like features, the heuristic argument needs to be refined. As was first pointed out in ([1]), some scaling law actually exists if one performs frequency averages on the scattering data. Indeed as figure 1.1 shows, the reflection coefficient for quantum scattering on a triadic Cantor measure gives a “wrong” scaling. Only after some suitable averages have been performed, the right scaling behavior is recovered (figure 1.2). Recently, a general method was proposed ([7]) to retrieve a fractal dimension, namely the correlation dimension, of a potential barrier from the reflection amplitude. In this paper, we wish to give further developments to this method. In particular, we show how the same kind of results can be extended to the scattering phase for half-line scattering. Moreover we shall show how to recover the self-similarity (if there is any) of the potential through a large scale renormalization of the scattering amplitude and scattering phase, respectively.