Propagation Estimates for Two-cluster Scattering Channels of N-body Schr\"odinger Operators

TL;DR

This paper develops propagation estimates for two-cluster scattering channels in N-body Schrödinger operators, enhancing understanding of quantum scattering processes and providing tools for analyzing three-body problems.

Contribution

It introduces new propagation estimates based on Mourre's commutator and Skibsted's methods, with improved indices and applications to resolvent and microlocal estimates.

Findings

Propagation estimates for two-cluster channels established

Improved indices achieved with almost invariant subspace projections

Resolvent and microlocal estimates derived for three-body problems

Abstract

In this paper we prove propagation estimates for two-cluster scattering channels of N-body Schr\"odinger operators. These estimates are based on the estimate similar to Mourre's commutator estimate and the method of Skibsted. We also obtain propagation estimates with better indices using projections onto almost invariant subspaces close to two-cluster scattering channels. As an application of these estimates we obtain the resolvent estimate for two-cluster scattering channels and microlocal propagation estimates in three-body problems without projections. Our method clearly illustrates evolution of the solutions of the Schr\"odinger equation.