Bounds for Eigenvalue Sums of Schr\"odinger Operators with Complex Radial Potentials

TL;DR

This paper establishes bounds on the distribution of eigenvalues for Schrödinger operators with complex radial potentials, revealing that eigenvalues accumulating in the positive real axis have summable imaginary parts, using resolvent and spectral measure estimates.

Contribution

It provides new quantitative bounds on eigenvalue distributions for Schrödinger operators with complex radial potentials, utilizing resolvent estimates derived from spectral measure bounds.

Findings

Eigenvalues accumulating in (0,∞) have summable imaginary parts.

Quantitative bounds relate eigenvalue sums to the $L^q$ norm of the potential.

Resolvent estimates are obtained via spectral measure bounds and epsilon removal.

Abstract

We consider eigenvalue sums of Schr\"odinger operators $-\Delta+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q<d$. 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.