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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

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
MathSciNet review: 3717979
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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
MR Author ID: 728268
ORCID: 0000-0001-7973-4688
Email: rlfrank@caltech.edu

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