  Under natural restrictions it is known that a nonlinear Schr\"odinger
equation is a Hamiltonian PDE which defines a symplectic flow on a symplectic
Hilbert space preserving the Hilbert norm. When the potential is one-periodic
in time and after passing to the projectivization, it makes sense to ask
whether the natural analogue of the Arnold conjecture holds. By employing
methods from non-standard model theory we show how Hamiltonian Floer theory can
be generalized from finite to infinite dimensions. While our proof entirely
builds on finite-dimensional results, we do not ask for any prior knowledge of
non-standard model theory.
