Fredholm properties of Schrödinger operators in $L^P(\mathbbR^N)$

Abstract

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 self-adjoint operator in $L^{2}(\mathbb{R}^{N}).$ If $\lambda\notin$ $\sigma_{e},$ we show that for all $p$ in the range considered, $-\Delta+V-\lambda:W^{2,p}% (\mathbb{R}^{N})\rightarrow L^{p}(\mathbb{R}^{N})$ is a Fredholm operator of index zero, that ker {$-\Delta+V-\lambda$\} is independent of $p$ and that $L^{p}(\mathbb{R}^{N})=$ker {$-\Delta+V-\lambda$\}$\oplus$\{$-\Delta +V-\lambda$\}$W^{2,p}(\mathbb{R}^{N}).$