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Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues

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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 \({\sigma_{\rm ess}(H)=[0,\infty)}\). This demonstrates that the Lieb–Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.

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  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333, München, Germany

    Sabine Bögli

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  1. Sabine Bögli
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Correspondence to Sabine Bögli.

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Communicated by R. Seiringer

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Bögli, S. Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues. Commun. Math. Phys. 352, 629–639 (2017). https://doi.org/10.1007/s00220-016-2806-5

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  • DOI: https://doi.org/10.1007/s00220-016-2806-5

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