Eigenvalue bounds for Schrödinger operators with complex potentials. III
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- by Rupert L. Frank PDF
- Trans. Amer. Math. Soc. 370 (2018), 219-240
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 $V$.References
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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.
- © Copyright 2017 by the author
- Journal: Trans. Amer. Math. Soc. 370 (2018), 219-240
- MSC (2010): Primary 35P15, 31Q12
- DOI: https://doi.org/10.1090/tran/6936
- MathSciNet review: 3717979