An inverse scattering problem for the nonlinear Schrödinger equation under partial harmonic confinement

We study an inverse problem for a nonlinear Schrödinger equation with a harmonic potential in some spatial directions and free propagation in the remaining directions. The nonlinearity has an unknown power and a real, spatially dependent coefficient. For small incoming data in a weighted energy space, we construct the wave operator and show that it determines both the power and the coefficient when the latter depends either on the confined variables or on the free variables. The proof uses the first nonlinear term of the wave operator and concentrated product data. We explain why dependence on the free variables requires a stronger lower bound on the power. We also discuss the outgoing wave operator, the small-data scattering operator, and coefficients depending on all spatial variables.