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Eigenvalue bounds for Schrödinger operators with complex potentials. III


Author: Rupert L. Frank
Journal: Trans. Amer. Math. Soc. 370 (2018), 219-240
MSC (2010): Primary 35P15, 31Q12
DOI: https://doi.org/10.1090/tran/6936
Published electronically: July 13, 2017
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Abstract: We discuss the eigenvalues $ E_j$ of Schrödinger operators $ -\Delta +V$ in $ L^2(\mathbb{R}^d)$ with complex potentials $ V\in L^p$, $ p<\infty $. We show that (A) $ \operatorname {Re} E_j\to \infty $ implies $ \operatorname {Im} E_j\to 0$, and (B) $ \operatorname {Re} E_j\to E\in [0,\infty )$ implies $ (\operatorname {Im} E_j)\in \ell ^q$ for some $ q$ depending on $ p$. We prove quantitative versions of (A) and (B) in terms of the $ L^p$-norm of $ V$.


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Additional Information

Rupert L. Frank
Affiliation: Deparment of Mathematics 253-37, Caltech, Pasadena, California 91125
Email: rlfrank@caltech.edu

DOI: https://doi.org/10.1090/tran/6936
Received by editor(s): October 12, 2015
Received by editor(s) in revised form: March 14, 2016
Published electronically: July 13, 2017
Additional Notes: The author was supported by NSF grant DMS–1363432.
Article copyright: © Copyright 2017 by the author

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