Power-law bounds on transfer matrices and quantum dynamics in one dimension II

TL;DR

This paper establishes lower bounds on quantum dynamics for one-dimensional Schrödinger operators by deriving power-law upper bounds on transfer matrix norms, extending previous methods to various models with complex spectral properties.

Contribution

It advances the approach from part I by providing quantum dynamical bounds for new classes of models with exceptional energies, including substitution, Sturmian, hierarchical, prime, and sparse potentials.

Findings

Quantum dynamical lower bounds are proven for multiple models.

Power-law bounds on transfer matrices are established for models with exceptional energies.

The approach is extended to a broader class of one-dimensional Schrödinger operators.

Abstract

We establish quantum dynamical lower bounds for a number of discrete one-dimensional Schr\"odinger operators. These dynamical bounds are derived from power-law upper bounds on the norms of transfer matrices. We develop further the approach from part I and study many examples. Particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power-law bounds on transfer matrices. The models discussed in this paper include substitution models, Sturmian models, a hierarchical model, the prime model, and a class of moderately sparse potentials.