Counting eigenvalues of Schrödinger operators using the landscape function

Sven Bachmann, Richard Froese, Severin Schraven

Published 2023-06-06Version 1

We prove an upper and a lower bound on the rank of the spectral projections of the Schr\"odinger operator $-\Delta + V$ in terms of the volume of the sublevel 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 the solution of $(-\Delta + V)u = 1$ in $\mathbb{R}^d$. We prove the result for non-negative potentials satisfying a Kato and doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum.