A nonlinear Poisson formula for the Schrodinger operator

TL;DR

This paper establishes a nonlinear Poisson formula for the Schrödinger group, extending previous results to include all short-range and energy-critical nonlinearities through a direct proof.

Contribution

It provides a direct proof of the nonlinear Poisson formula and broadens its applicability to all short-range and energy-critical nonlinearities.

Findings

Proved a nonlinear Poisson formula for the Schrödinger group.

Extended the formula to include all short-range nonlinearities.

Allowed for energy-critical nonlinearities in the formula.

Abstract

We prove a nonlinear Poisson type formula for the Schrodinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, energy-critical nonlinearities are allowed.