Schrödinger operator with non-zero accumulation points of complex eigenvalues

Abstract

We study Schrödinger operators $H=-Δ+V$ in $L^2(Ω)$ where $Ω$ 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 $V\in L^p(Ω)\cap L^{\infty}(Ω)$ that decays at infinity so that $H$ has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum $σ_{\rm ess}(H)=[0,\infty)$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.