| Summary: | We discuss the eigenvalues E_j of Schrödinger operators −Δ+V in L^2(R^d) with complex potentials V ∈ L^p, p < ∞. We show that (A) Re E_j → ∞ implies Im E_j → 0, and (B) Re E_j → E ∈ [0,∞) implies (Im E_j) ∈ ℓ^q for some q depending on p. We prove quantitative versions of (A) and (B) in terms of the L^p-norm of V. © 2017 by the Author. Received by the editors October 12, 2015 and, in revised form, March 14, 2016. Published electronically: July 13, 2017. The author was supported by NSF grant DMS-1363432. Fundamental for several of the new theorems here are results from [13], which were obtained jointly with J. Sabin to whom the author is most grateful. He would also like to thank M. Demuth and M. Hansmann for fruitful discussions and M. Demuth, L. Golinskii and F. Hanauska for helpful remarks on a previous version of this manuscript. This paper has its origin at the conference "Mathematical aspects of physics with non-self-adjoint operators" in June 2015 and the author is grateful to the organizers and the American Institute of Mathematics for the invitation. This paper was finished at the Mittag–Leffler Institute and the author is grateful to A. Laptev for the hospitality. Submitted - 1510.03411.pdf
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