Let \(\mathcal {L}=-\Delta +V\) be a Schrödinger operator on \(\mathbb {R}^n\) with \(n\ge 3\) , where the non-trivial potential V is in certain reverse Hölder class. In this paper, we establish some atomic decompositions for \(H^1_\mathcal {L}(\omega )\) (the weighted Hardy space related to \(\mathcal {L}\) ), where the integral of an atom over its support merely admits the logarithmic/polynomial/exponential decay rather than the cancellation property of Goldberg’s local atom in Duke Math. J. (1979). This extends the work of Dafni–Lau–Picon–Vasconcelos in Nonlinear Anal. (2022) concerning the unweighted local Hardy space \(h^1\) of Goldberg, which involves with the operator \(-\Delta +1\) in essence, to the weighted Hardy space associated with the Schrödinger operator \(-\Delta +V\) . As applications, we further decompose a \(H^1_\mathcal {L}(\omega )\) -function into a sum of molecules, and the behaviour of Riesz potentials/Marcinkiewicz integrals/Hardy inequality on \(H^1_\mathcal {L}(\omega )\) are also considered.
