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We prove an upper and a lower bound on the rank of the spectral projectionsof the Schr\"odinger operator $-\Delta + V$ in terms of the volume of thesublevel sets of an effective potential $\frac{1}{u}$. Here, $u$ is the`landscape function' of [(David, G., Filoche, M., & Mayboroda, S. (2021)Advances in Mathematics, 390, 107946)], namely a solution of $(-\Delta + V)u =1$ in $\bbR^d$. We prove the result for non-negative potentials satisfying aKato-type and a doubling condition, in all spatial dimensions, in infinitevolume, and show that no coarse graining is required. Our result yields inparticular a necessary and sufficient condition for discreteness of thespectrum. In the case of polynomial potentials, we prove that the spectrum isdiscrete if and only if no directional derivative vanishes identically.

Counting eigenvalues of Schr\"odinger operators using the landscape function