Propagation estimates in the one-commutator theory

TL;DR

This paper extends propagation estimates in Mourre theory to cases with less regularity of the Hamiltonian, demonstrating new results for multi-dimensional Schrödinger operators without relying on the limiting absorption principle.

Contribution

It introduces novel propagation estimates for Hamiltonians with limited regularity, including a new estimate based on an improved RAGE formula applicable to multi-dimensional cases.

Findings

Spectral measure of H is a Rajchman measure under reduced regularity.

Derived propagation estimates using minimal escape velocities.

New propagation estimate for multi-dimensional Schrödinger operators.

Abstract

In the abstract framework of Mourre theory, the propagation of states is understood in terms of a conjugate operator $A$. A powerful estimate has long been known for Hamiltonians having a good regularity with respect to $A$ thanks to the limiting absorption principle (LAP). We study the case where $H$ has less regularity with respect to $A$, specifically in a situation where the LAP and the absence of singularly continuous spectrum have not yet been established. We show that in this case the spectral measure of $H$ is a Rajchman measure and we derive some propagation estimates. One estimate is an application of minimal escape velocities, while the other estimate relies on an improved version of the RAGE formula. Based on several examples, including continuous and discrete Schr\"odinger operators, it appears that the latter propagation estimate is a new result for multi-dimensional Hamiltonians.