Low frequency asymptotics and local energy decay for the Schr{\"o}dinger equation

TL;DR

This paper establishes low frequency resolvent estimates and local energy decay for the Schrödinger equation in an asymptotically Euclidean setting, advancing understanding of its spectral and time decay properties.

Contribution

It provides the leading term for the resolvent near zero spectral parameter by comparing perturbed and free resolvents, extending the commutators method for generalized resolvents.

Findings

Derived the leading term of the resolvent near zero energy.

Proved local energy decay for the Schrödinger equation.

Applied results to large time asymptotics of the evolution.

Abstract

We prove low frequency resolvent estimates and local energy decay for the Schr{\"o}dinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schr{\"o}dinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Finally, we deduce similar results for the large time asymptotics of the corresponding evolution problem.