  We consider the Schr{\"o}dinger operator --$\Delta$ + V on the Euclidean
space with potential in the Lorentz space L^{n/2,1} and we find necessary and
sufficient conditions for zero to be a resonance or an eigenvalue. We consider
functions with gradient in L^2 and that verify the equation (--$\Delta$ +
V)$\psi$ = 0, namely the kernel of (--$\Delta$ + V) in the homogeneous Sobolev
space of order one. We prove that a function in this set is either in a weak
Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The
set of eigenfunctions is the hyperplane of functions that are orthogonal to V,
furthermore we show that under some classic orthogonality conditions a zero
eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We
study dimensions n $\ge$ 3 and in dimension three we generalize a result proved
by Beceanu.
