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A characterization of the Anderson metal-insulator transport transition. (English) Zbl 1062.82020
This paper considers Schrödinger operators with random potential, in space dimension \(d\). One expects, for \(d\geq 3\), the occurrence of a metal-insulator transition, first predicted by Anderson. In other words, if \(E_0\) is the ground state energy, the eigenfunctions corresponding to energies in the range \([E_0,E_{me}]\) should be exponentially localized in space, while they should be extended in \([E_{me},+\infty)\). The corresponding physical picture is that the system behaves as an insulator at low energies, and as a metal with nonzero conductivity at high energies. The value \(E_{me}\) is called mobility edge. From a rigorous point of view, it is known that the system is an insulator at very low energies, but the proof of the existence of a metallic phase (and the analysis of the transition at the mobility edge) is still missing.
There are various possible mathematical definitions of conductor and insulator phase. In most of the literature, one takes the point of view of spectral theory and associates the insulator phase to the pure point part of the spectrum, and the metallic phase to the absolutely continuous part. The present paper takes a different point of view, based on transport (i.e., dynamical) rather than spectral properties. The authors introduce a local transport exponent \(\beta(E)\), and define the weak metallic transport region to be the part of the spectrum where \(\beta(E)>0\). On the other hand, they define the strong insulator region as the part of the spectrum where the random Schrödinger operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. An important point is that these two regions are shown to be complementary sets in the spectrum. The vanishing of \(\beta(E)\) therefore signals the occurrence of a metal-insulator transition. It is important to remark that, assuming that the transition exists, it is of discontinuous nature. Indeed, the authors show that, if \(\beta(E)>0\), then \(\beta(E)>1/(2d)\).
The proofs are based on the “bootstrap multiscale analysis” previously developed by the authors, which is an extension of the multiscale analysis developed for instance by J. Fröhlich and T. Spencer [Commun. Math. Phys. 88, 151–184 (1983; Zbl 0519.60066)] and by J. Fröhlich, F. Martinelli, E. Scoppola and T. Spencer [Commun. Math. Phys. 101, 21–46 (1985; Zbl 0573.60096)] as a tool to prove exponential localization at low energies.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80 Random linear operators
60H25 Random operators and equations (aspects of stochastic analysis)
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