Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 153, Issue 6, Pages 1022-1038Publisher
SPRINGER
DOI: 10.1007/s10955-013-0879-5
Keywords
Anderson localization; Discrete random Schrodinger operator; Extended states conjecture
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Funding
- [NSF-DMS-1101477]
- [NSF-DMS-1261687]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1261687] Funding Source: National Science Foundation
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We develop a rather explicit approach concerning the extended states conjecture for the discrete random Schrodinger operator, or more generally for the so-called Anderson-type Hamiltonian. Our work is based on deep mathematical results by JakiA double dagger-Last (Duke Math. J. 133(1):185-204, 2006). Concretely, we suggest two new directions of research: We provide a formula which may lead the way to a rigorous proof of the conjecture, and an implementation of the proposed approach which yields numerical evidence in favor of the conjecture being true for the discrete random Schrodinger operator in dimension two. Not being based on scaling theory, this method eliminates problems due to boundary conditions, common to previous numerical methods in the field. At the same time, as with any numerical experiment, one cannot exclude finite-size effects with complete certainty. We numerically track the bulk distribution (here: the distribution of where we most likely find an electron) of a wave packet initially located at the origin, after iterative application of the discrete random Schrodinger operator.
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