Weighted Strichartz estimates for radial Schr\"odinger equation on noncompact manifolds

TL;DR

This paper establishes weighted Strichartz estimates for radial solutions of the Schr"odinger equation on certain noncompact, rotationally symmetric manifolds, expanding the understanding of dispersive properties beyond Euclidean spaces.

Contribution

It generalizes known estimates to a broader class of noncompact manifolds with specific curvature conditions, not relying on algebraic structures like hyperbolic spaces.

Findings

Weighted Strichartz estimates hold on these manifolds.

Larger class of exponents than Euclidean case.

Improved scattering theory results.

Abstract

We prove global weighted Strichartz estimates for radial solutions of linear Schr\"odinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity.