  We prove upper and lower bounds for the number of eigenvalues of semi-bounded
Schr\"odinger operators in all spatial dimensions. As a corollary, we obtain
two-sided estimates for the sum of the negative eigenvalues of atomic
Hamiltonians with Kato potentials. Instead of being in terms of the potential
itself, as in the usual Lieb-Thirring result, the bounds are in terms of the
landscape function, also known as the torsion function, which is a solution of
$(-\Delta + V +M)u_M =1$ in $\mathbb{R}^d$; here $M\in\mathbb{R}$ is chosen so
that the operator is positive. We further prove that the infimum of $(u_M^{-1}
- M)$ is a lower bound for the ground state energy $E_0$ and derive a simple
iteration scheme converging to $E_0$.
