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Eigenvalue bounds for Schrödinger operators with complex potentials. II

Frank, Rupert L. and Simon, Barry (2017) Eigenvalue bounds for Schrödinger operators with complex potentials. II. Journal of Spectral Theory, 7 (3). pp. 633-658. ISSN 1664-039X. https://resolver.caltech.edu/CaltechAUTHORS:20170502-082840587

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Abstract

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator -Δ+V in L^2 (R^ν) with complex potential has absolute value at most a constant times ||V||^(γ+ν/2)/γ)_(γ+ν/2) for 0 < γ ≤ ν/2 in dimension ν ≥ 2. We prove this conjecture for radial potentials if 0 < γ < ν/2 and we ‘almost disprove’ it for general potentials if 1/2 < γ < ν/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.4171/JST/173DOIArticle
http://www.ems-ph.org/journals/show_abstract.php?issn=1664-039X&vol=7&iss=3&rank=1PublisherArticle
https://arxiv.org/abs/1504.01144arXivDiscussion Paper
ORCID:
AuthorORCID
Simon, Barry0000-0003-2561-8539
Additional Information:© 2017 European Mathematical Society. Received April 5, 2015; revised May 26, 2015. Published online: 2017-09-28. Work partially supported by U.S. National Science Foundation grants PHY-1347399, DMS-1363432 (R. L. Frank), and DMS-1265592 (B. Simon).
Funders:
Funding AgencyGrant Number
NSFPHY-1347399
NSFDMS-1363432
NSFDMS-1265592
Subject Keywords:Schrödinger operator, complex-valued potential, eigenvalue bounds, embedded eigenvalue
Issue or Number:3
Classification Code:Mathematics Subject Classification (2010): Primary 35P15; Secondary 81Q12
Record Number:CaltechAUTHORS:20170502-082840587
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170502-082840587
Official Citation:Frank Rupert, Simon Barry: Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory 7 (2017), 633-658. doi: 10.4171/JST/173
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77120
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:02 May 2017 17:58
Last Modified:03 Oct 2019 17:53

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