  We consider the continuous Anderson Hamiltonian with white noise potential on
$(-L/2,L/2)^d$ in dimension $d\le 3$, and derive the asymptotic of the smallest
eigenvalues when $L$ goes to infinity. We show that these eigenvalues go to
$-\infty$ at speed $(\log L)^{1/(2-d/2)}$ and identify the prefactor in terms
of the optimal constant of the Gagliardo-Nirenberg inequality. This result was
already known in dimensions $1$ and $2$, but appears to be new in dimension
$3$. We present some conjectures on the fluctuations of the eigenvalues and on
the asymptotic shape of the corresponding eigenfunctions near their
localisation centers.
