  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.
