Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''

TL;DR

This paper corrects a key step in the proof of local decay estimates for wave and Schrödinger equations on Schwarzschild manifolds, reaffirming some results while clarifying limitations for non-radial, large data, semilinear cases.

Contribution

It provides a corrected commutator argument that establishes local decay estimates for linear equations and radial semilinear equations on Schwarzschild manifolds.

Findings

Corrects a step in the commutator proof for decay estimates.

Reestablishes decay results for linear and radial semilinear equations.

Does not extend to non-radial, large data, semilinear equations.

Abstract

In ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'' (math-ph/0002030), ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'' (gr-qc/0310091), and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation'' (gr-qc/0310066), local decay estimates were proven for the (decoupled) Schrodinger, wave, and Regge-Wheeler equations on the Schwarzschild manifold, using commutator methods. Here, we correct a step in the commutator argument. The corrected argument works either for radial semilinear equations or general linear equations. This recovers the results in math-ph/0002030 and gr-qc/0310066, but does not recover the non radial, large data, semilinear result asserted in the gr-qc/0310091.