Hardy spaces related to Schr\"odinger operators with potentials which are sums of L^p-functions

We investigate the Hardy space H^1_L associated to the Schr"odinger operator L=-\Delta+V on R^n, where V=\sum_{j=1}^d V_j. We assume that each V_j depends on variables from a linear subspace VV_j of \Rn, dim VV_j \geq 3, and V_j belongs to L^q(VV_j) for certain q. We prove that there exist two distinct isomorphisms of H^1_L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H_L^1. We also prove that the space H_L^1 is described by means of the Riesz transforms R_{L,i} = \partial_i L^{-1/2}.