Let L=−Δ+V is a Schrödinger operator on \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{R}^{d}$\end{document}, d≥3, V≥0. Let \documentclass[12pt]{minimal}
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\begin{document}$H^{1}_{L}$\end{document} denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping \documentclass[12pt]{minimal}
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\begin{document}$$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$\end{document} is an isomorphism from the Hardy space \documentclass[12pt]{minimal}
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\begin{document}$H_{L}^{1}$\end{document} onto the classical Hardy space \documentclass[12pt]{minimal}
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\begin{document}$H^{1}(\mathbb{R}^{d})$\end{document} if and only if \documentclass[12pt]{minimal}
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\begin{document}$\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$\end{document} belongs to \documentclass[12pt]{minimal}
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\begin{document}$L^{\infty}(\mathbb{R}^{d})$\end{document}.
