and Molchanov [@AM93] and called the Fractional-Moment Method (FMM), when applicable, leads directly to the proof of dynamical localization; the latter, as it is well-known, implies spectral localization. Despite striking differences between the MSA and the FMM, both approaches to the localization share a certain similarity: the analysis of decay properties of the resolvents in finite volumes *precedes* the study of the eigenfunction correlators. In the method described below, the analysis of eigenfunctions is the dominant component. As a result, the simplified Scaling Analysis allows to prove localization bounds in finite volumes in the course of the scale induction. We would like to emphasize that it is close in spirit to an elementary method used earlier by Spencer [@Spe88] to prove exponential decay of Green functions at a fixed energy. The idea of a direct analysis of eigenfunctions is inspired by a work of Sinai [@Si87] where it was implemented in a different context (quasi-periodic 1D lattice Schrödinger operators) and in a different way (with the help of a KAM-type scale induction).

In a forthcoming work [@C11] our method is adapted to the multi-particle setting.

The structure of this manuscript is as follows:

1.  Basic notions and notations are introduced 