  We consider the random Schr\"odinger operator on $\mathbb{R}$ obtained by
perturbing the Laplacian with a white noise. We prove that Anderson
localization holds for this operator: almost surely the spectral measure is
pure point and the eigenfunctions are exponentially localized. We give two
separate proofs of this result. We also present a detailed construction of the
operator and relate it to the parabolic Anderson model. Finally, we discuss the
case where the noise is smoothed out.
