Weekly Bulletin

We discuss upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. For atomic Hamiltonians with Kato potentials one can strengthen the result to obtain two-sided estimates for the sum of the negative eigenvalues. 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. This talk is based on the preprint \href{https://arxiv.org/abs/2403.19023}{arXiv:2403.19023} which is joint work with S. Bachmann and R. Froese.