Nonlinear resonances with a potential: Multilinear estimates and an application to NLS

TL;DR

This paper develops multilinear estimates using the spectral theory of Schrödinger operators to analyze nonlinear dispersive equations with potentials, establishing global existence and scattering results for small-data solutions.

Contribution

It introduces a systematic approach combining space-time resonance analysis with spectral theory to handle low-degree nonlinearities in non-Euclidean settings.

Findings

Proved global existence and scattering for quadratic Schrödinger equations with potential.

Developed multilinear analysis tools in the distorted Fourier transform framework.

Extended the space-time resonance method to non-Euclidean spectral settings.

Abstract

This paper considers the question of global in time existence and asymptotic behavior of small-data solutions of nonlinear dispersive equations with a real potential $V$. The main concern is treating nonlinearities whose degree is low enough as to preclude the simple use of classical energy methods and decay estimates. In their place, we present a systematic approach that adapts the space-time resonance method to the non-Euclidean setting using the spectral theory of the Schroedinger operator $-\Delta+V$. We start by developing tools of independent interest, namely multilinear analysis (Coifman-Meyer type theorems) in the framework of the corresponding distorted Fourier transform. As a first application, this is then used to prove global existence and scattering for a quadratic Schroedinger equation.