Lower Transport Bounds for One-Dimensional Continuum Schr\"odinger Operators

TL;DR

This paper establishes lower bounds on quantum transport for one-dimensional continuum Schrödinger operators, especially at critical energies, with applications to models like the Bernoulli-Anderson model.

Contribution

It provides the first general quantum dynamical lower bounds for continuum Schrödinger operators with critical energies, extending previous discrete results.

Findings

Quantum dynamical lower bounds are proven at critical energies.

Results apply to models including Bernoulli-Anderson with constant potential.

Demonstrates slow growth of transfer matrix norms at critical energies.

Abstract

We prove quantum dynamical lower bounds for one-dimensional continuum Schr\"odinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including the Bernoulli-Anderson model with a constant single-site potential.