  We introduce a new approach for proving localization (pure point spectrum
with exponentially decaying eigenfunctions, dynamical localization) for the
Anderson model at high disorder. In contrast to the usual strategy, we do not
study finite volume Green's functions. Instead, we perform a multiscale
analysis based on finite volume eigensystems (eigenvalues and eigenfunctions).
Information about eigensystems at a given scale is used to derive information
about eigensystems at larger scales. This eigensystem multiscale analysis
treats all energies of the finite volume operator at the same time,
establishing level spacing and localization of eigenfunctions in a fixed box
with high probability. A new feature is the labeling of the eigenvalues and
eigenfunctions by the sites of the box.
