  We use a bootstrap argument to enhance the eigensystem multiscale analysis,
introduced by Elgart and Klein for proving localization for the Anderson model
at high disorder. The eigensystem multiscale analysis studies finite volume
eigensystems, not finite volume Green's functions. It yields pure point
spectrum with exponentially decaying eigenfunctions and dynamical localization.
The starting hypothesis for the eigensystem bootstrap multiscale analysis only
requires the verification of polynomial decay of the finite volume
eigenfunctions, at some sufficiently large scale, with some minimal probability
independent of the scale. It yields exponential localization of finite volume
eigenfunctions in boxes of side $L$, with the eigenvalues and eigenfunctions
labeled by the sites of the box, with probability higher than
$1-\mathrm{e}^{-L^\xi}$, for any desired $0<\xi<1$.
