Open Access Publications from the University of California

## Topics on Schrödinger Operators

• Author(s): Tsang, Chi Shing Sidney;
2. We develop an eigensystem bootstrap multiscale analysis 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 $0<\xi<1$.