02-19 Francois Germinet, Abel Klein

A characterization of the Anderson metal-insulator transport transition (version in .ps format) (548K, .ps) Jan 14, 02

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Abstract. This is file 01-486 in .ps format. Since several persons complained they did not manage to download correctly file 01-486, which is in .pdf format, compressed by mp_arc, we decided to post the same file, but in .ps this time. We investigate the Anderson metal-insulator transition for random Schr\"odinger operators. We define the \emph{strong insulator region} to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent $\beta(E)$, and set the \emph{metallic transport region} to be the part of the spectrum with nontrivial transport (i.e., $\beta(E)>0$). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator, and that the local transport exponent $\beta(E)$ provides a characterization of the \emph{metal-insulator transport transition}. Moreover, we show that if there is such a transition, then $\beta(E)$ has to be discontinuous at a \emph{transport mobility edge}. More precisely, we show that if the transport is nontrivial then $\beta(E)\ge \frac 1{2d}$, where $d$ is the space dimension. These results follow from a proof that slow time evolution of quantum waves irandom media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge.

Files: 02-19.src( 02-19.keywords , transitionmparc.ps )