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Topics on Schrödinger Operators


We study two topics in the theory of Schr\"odinger operators:

1. We establish bounds on the density of states measures for Schr\"odinger operators with singular potentials. We obtain log-H\"older continuity for the density of states outer-measure in one, two, and three dimensions for Schr\"odinger operators with singular potentials, results that hold for the density of states measure when it exists. To do this, we study the local behavior of solutions of the stationary Schr\"odinger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term, and, we prove a quantitative unique continuation principle for Schr\"odinger operators with singular potentials.

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 $0<\xi<1$.

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