I would like to have some reference about the problem of resolution of a Poisson equation in a box, with Dirichlet or (better) Neumann boundary condition, by use of FFT.

The use of FFT to solve differential equations are not restricted to the Poisson equations (used a lot in Astrophysics), but also to solve all kind of PDEs. These are called the spectral methods, where the FFT is used just to compute the derivatives (or also the intergral) and the equations are solved either in the Physical space or in the spectral space.

See Canuto, Hussaini, Quarteroni and Zang, 1988, Spectral Methods in Fluid Dynamics (New York: Springer Verlag) - a very good book indeed.

Good luck,

Patrick