On the construction of almost periodic solutions for a nonlinear Schrödinger equation

We describe the construction of almost-periodic solutions for a particular Schrödinger equation on a finite x- interval, depending on some potential V. The construction proceeds by iterating a KAM theorem about the existence of quasi-periodic solutions for some partial differential equations. As a result, we find for 'almost all' potentials V uncountably many almost-periodic solutions accumulating at the zero solution.