Ballistic transport in periodic and random media

2022/02/02 by de Monvel, Anne Boutet, Sabri, Mostafa · 3 citations
#47B80 #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 81Q10 #Secondary: 46N50

paper · doi:10.48550/arxiv.2202.00940

Abstract

We prove ballistic transport of all orders, that is, ‖ x^me^-itHψ‖\asymp t^m, for the following models: the adjacency matrix on ℤ^d, the Laplace operator on ℝ^d, periodic Schrödinger operators on ℝ^d, and discrete periodic Schrödinger operators on periodic graphs. In all cases we give the exact expression of the limit of ‖ x^me^-itHψ‖/t^m as t→+∞. We then move to universal covers of finite graphs (these are infinite trees) and prove ballistic transport in mean when the potential is lifted naturally, giving a periodic model, and when the tree is endowed with random i.i.d. potential, giving an Anderson model. The limiting distributions are then discussed, enriching the transport theory. Some general upper bounds are detailed in the appendix.

Cited by

• Measure of the spectra of periodic graph operators in the large-coupling limit
• Sharp Polynomial Velocity Decay Bounds for Multidimensional Periodic Schrödinger Operators
• Ballistic Transport for Discrete Multi-Dimensional Schrödinger Operators With Decaying Potential

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