Quantum quasiballistic dynamics and thick point spectrum

TL;DR

This paper establishes lower bounds on quantum dynamics for operators with pure point spectrum, demonstrating quasiballistic behavior in systems with thick point spectrum, with applications to Schrödinger operators.

Contribution

It introduces a method to relate eigenvalue spacing properties to dynamical bounds, revealing quasiballistic behavior in systems with thick point spectrum.

Findings

Systems with thick point spectrum exhibit quasiballistic dynamics.

Explicit applications to Schrödinger operators demonstrate the theory.

Dynamical lower bounds depend on eigenvalue spacing properties.

Abstract

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schr\"odinger operators.