Summary: We consider real potentials \(V\) such that the Schrödinger operator \(-\Delta+V\) maps the Sobolev space \(W^{2,p}(\mathbb{R}^N)\) continuously into \(L^p(\mathbb{R}^N)\) for a range of values of \(p\) which includes 2. Let \(\sigma_e\) denote the essential spectrum of \(-\Delta+V\) as a selfadjoint operator in \(L^2(\mathbb{R}^N)\). If \(\lambda\not\in \sigma_e\), we show that for all \(p\) in the range considered, \(-\Delta+ V-\lambda: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)\) is a Fredholm operator of index zero, that \(\text{ker}\{-\Delta+ V-\lambda\}\) is independent of \(p\) and that \(L^p(\mathbb{R}^N)= \text{ker}\{-\Delta_ V-\lambda\}\oplus \{-\Delta+ V-\lambda\} W^{2,p}(\mathbb{R}^N)\).