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

TL;DR

This paper establishes bounds on the spectral projection rank of Schrödinger operators using the landscape function, linking eigenvalue counting to the geometry of sublevel sets of an effective potential.

Contribution

It provides the first bounds on spectral projections based on the landscape function for non-negative potentials in all dimensions, without coarse graining.

Findings

Bounds on spectral projections in terms of sublevel set volume

Necessary and sufficient conditions for spectrum discreteness

Characterization of spectrum discreteness for polynomial potentials

Abstract

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 a solution of $(-\Delta + V)u = 1$ in $\bbR^d$. We prove the result for non-negative potentials satisfying a Kato-type and a 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. In the case of polynomial potentials, we prove that the spectrum is discrete if and only if no directional derivative vanishes identically.