Bounds for eigenvalue sums of Schrödinger operators with complex radial potentials

Abstract

We consider eigenvalue sums of Schrödinger operators $-\Delta +V$ on $L^2({\mathbb{R}}^d)$ with complex radial potentials $V\in L^q({\mathbb{R}}^d)$, $q<d$. We prove quantitative bounds on the distribution of the eigenvalues in terms of the $L^q$ norm of $V$. A consequence of our bounds is that, if the eigenvalues $(z_j)$ accumulate to a point in $(0,\infty )$, then $(\mathrm{Im}{z}_j)$ is summable. The key technical tools are resolvent estimates in Schatten spaces. We show that these resolvent estimates follow from spectral measure estimates by an epsilon removal argument.