Schrödinger operator with non-zero accumulation points of complex eigenvalues

Abstract

We study Schrödinger 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 ${\sigma }_{\mathrm{ess}}(H)=[0,\infty )$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.

10 pages