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 eigenvlaues in terms of the $L^q$ norm of $V$. A consequence of our bounds is that, if the eigenvlaues $(z_j)$ accumulates to a point in $(0,\infty )$, then $(\im z_j)$ 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.

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