SOME OBSERVATIONS ON QUANTUM-MECHANICS IN DISORDERED-SYSTEMS

The localization principle for vibrations on a percolation cluster and, more generally, on disordered systems leads to a modification of the Schrodinger equation by the addition of one term depending on the characteristic relaxation related to anomalous diffusion. As a consequence, quantum particles in disordered structures are subjected to an extra potential level resulting from the heterogeneous topology and geometry. This phenomenon is illustrated by a boundary condition problem at the interface between Euclidean and disordered media. A time dependent Schrodinger equation involving fractional operators is derived from the operational calculus of quantum mechanics, and from the dependence of the relaxation length on frequency.