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.

arXiv comment10 pages

References in corpus (2)

• Eigenvalue bounds for Schrödinger operators with complex potentials
• Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials

Cited by in corpus (6)

• Counterexample to the Laptev--Safronov conjecture
• Keller-type bounds for Dirac operators perturbed by rigid potentials
• Eigenvalue bounds for non-self-adjoint Schrödinger operators with non-trapping metrics
• Schrödinger operators with complex sparse potentials
• $L_p$-spectrum and Lieb-Thirring inequalities for Schrödinger operators on the hyperbolic plane
• On Lieb-Thirring inequalities for multidimensional Schrödinger operators with complex potentials