Asymptotics in all regimes for the Schr\"odinger equation with time-independent coefficients

TL;DR

This paper derives detailed asymptotic expansions for Schr"odinger equation solutions across all large-radii and large-time regimes on asymptotically conic manifolds, including Euclidean space.

Contribution

It provides comprehensive asymptotic analysis of Schr"odinger solutions in all regimes, extending previous results to a unified framework on asymptotically conic manifolds.

Findings

Asymptotic expansions valid in all joint large-radii and large-time regimes.

Unified analysis applicable to Euclidean space and similar manifolds.

Results depend on recent resolvent analysis of Schr"odinger operators.

Abstract

Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for the solutions of the initial-value problem for the Schr\"odinger equation (assuming Schwartz initial data). Asymptotics are calculated in all joint large-radii large-time regimes, these corresponding to the boundary hypersurfaces of a particular compactification of spacetime.