Schr\"odinger operator with non-zero accumulation points of complex eigenvalues

TL;DR

This paper constructs non-selfadjoint Schr"odinger operators with complex potentials that have infinitely many eigenvalues accumulating at every point of the essential spectrum, challenging existing Lieb-Thirring inequalities.

Contribution

It provides explicit examples of non-selfadjoint Schr"odinger operators with complex potentials exhibiting accumulation of eigenvalues at the essential spectrum.

Findings

Eigenvalues accumulate at all points of the essential spectrum.

Lieb-Thirring inequalities do not hold for these non-selfadjoint operators.

Constructed potentials decay at infinity and are in $L^p$ for $p>d$.

Abstract

We study Schr\"odinger operators $H=-\Delta+V$ in $L^2(\Omega)$ where $\Omega$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential $V\in L^p(\Omega)\cap L^{\infty}(\Omega)$ that decays at infinity so that $H$ has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum $\sigma_{\rm ess}(H)=[0,\infty)$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schr\"odinger operators are no longer true in the non-selfadjoint case.