  We construct the continuous Anderson hamiltonian on $(-L,L)^d$ driven by a
white noise and endowed with either Dirichlet or periodic boundary conditions.
Our construction holds in any dimension $d\le 3$ and relies on the theory of
regularity structures: it yields a self-adjoint operator in
$L^2\big((-L,L)^d\big)$ with pure point spectrum. In $d\ge 2$, a
renormalisation of the operator by means of infinite constants is required to
compensate for ill-defined products involving functionals of the white noise.
We also obtain left tail estimates on the distributions of the eigenvalues: in
particular, for $d=3$ these estimates show that the eigenvalues do not have
exponential moments.
