We propose a simple construction of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau [KL81] that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is $\mathbf{R}$. To prove this result, we extend the method of Kotani [Kot85] to our setting of singular random operators.