Detectability of critical points of smooth functionals from their finite-dimensional approximations

Given a critical point of a C²-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. ‘visible’) from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.