Delocalization in random polymer models
TL;DR: In this article, it was shown that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there, and the level spacing is shown to be regular at the critical energy.
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Abstract: A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.
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Citations
An Invitation to Random Schr¨ odinger operators
Werner Kirsch
- 24 Sep 2007
TL;DR: The authors essayent de presenter les bases de la theorie des operateurs de Schrodinger aleatoires and present a demonstration complete des asymptotiques de Lifshitz and de la localisation d'Anderson.
237
Schrödinger Operators with Dynamically Defined Potentials: A Survey
TL;DR: In this article, the authors discuss spectral and quantum properties of discrete one-dimensional Schrodinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation.
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Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators
Ivan Veselić
- 30 Nov 2007
TL;DR: The theory of random Schrodinger operators is devoted to the mathematical analysis of quantum mechanical Hamiltonians modeling disordered solids as mentioned in this paper, and it is a multifaceted subject in its own right, drawing on ideas and methods from various mathematical disciplines like functional analysis, selfadjoint operators, PDE, stochastic processes and multiscale methods.
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Upper Bounds in Quantum Dynamics
TL;DR: In this paper, the authors derived upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies, and showed that at sufficiently large coupling, all transport exponents take values strictly between zero and one.
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Spectral and Localization Properties for the One-Dimensional Bernoulli Discrete Dirac Operator
TL;DR: In this paper, a 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schr?odinger model.
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TL;DR: In this article, the singular spectrum produced by rank-one perturbations was studied from a spectral point of view, and it was shown that the spectrum is always of dimension zero, albeit sometimes pure point and sometimes singular continuous.