Counterexample to the Laptev-Safronov Conjecture
Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009) conjectured an inequality between the magnitude of eigenvalues of a non-self-adjoint Schrödinger operator on Rd, d≥2, and an Lq norm of the potential, for any q∈[d/2,d]. Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the conjectur...
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ftunivdurham:oai:dro.dur.ac.uk.OAI2:37316 2023-05-15T17:07:13+02:00 Counterexample to the Laptev-Safronov Conjecture Boegli, Sabine Cuenin, Jean-Claude 2022 application/pdf http://dro.dur.ac.uk/37316/ http://dro.dur.ac.uk/37316/1/37316.pdf http://dro.dur.ac.uk/37316/2/37316.pdf https://doi.org/10.1007/s00220-022-04546-z unknown Springer dro:37316 issn:0010-3616 issn: 1432-0916 doi:10.1007/s00220-022-04546-z http://dro.dur.ac.uk/37316/ https://doi.org/10.1007/s00220-022-04546-z http://dro.dur.ac.uk/37316/1/37316.pdf http://dro.dur.ac.uk/37316/2/37316.pdf This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. CC-BY Communications in Mathematical Physics, 2022 [Peer Reviewed Journal] Article PeerReviewed 2022 ftunivdurham https://doi.org/10.1007/s00220-022-04546-z 2022-12-08T23:25:51Z Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009) conjectured an inequality between the magnitude of eigenvalues of a non-self-adjoint Schrödinger operator on Rd, d≥2, and an Lq norm of the potential, for any q∈[d/2,d]. Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the conjecture is true for q∈[d/2,(d+1)/2]. We construct a counterexample that disproves the conjecture in the remaining range q∈((d+1)/2,d]. As a corollary of our main result we show that, for any q>(d+1)/2, there is a complex potential in Lq∩L∞ such that the discrete eigenvalues of the corresponding Schrödinger operator accumulate at every point in [0,∞). In some sense, our counterexample is the Schrödinger operator analogue of the ubiquitous Knapp example in Harmonic Analysis. We also show that it is adaptable to a larger class of Schrödinger type (pseudodifferential) operators, and we prove corresponding sharp upper bounds. Article in Journal/Newspaper laptev Durham University: Durham Research Online Communications in Mathematical Physics |
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Durham University: Durham Research Online |
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Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009) conjectured an inequality between the magnitude of eigenvalues of a non-self-adjoint Schrödinger operator on Rd, d≥2, and an Lq norm of the potential, for any q∈[d/2,d]. Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the conjecture is true for q∈[d/2,(d+1)/2]. We construct a counterexample that disproves the conjecture in the remaining range q∈((d+1)/2,d]. As a corollary of our main result we show that, for any q>(d+1)/2, there is a complex potential in Lq∩L∞ such that the discrete eigenvalues of the corresponding Schrödinger operator accumulate at every point in [0,∞). In some sense, our counterexample is the Schrödinger operator analogue of the ubiquitous Knapp example in Harmonic Analysis. We also show that it is adaptable to a larger class of Schrödinger type (pseudodifferential) operators, and we prove corresponding sharp upper bounds. |
format |
Article in Journal/Newspaper |
author |
Boegli, Sabine Cuenin, Jean-Claude |
spellingShingle |
Boegli, Sabine Cuenin, Jean-Claude Counterexample to the Laptev-Safronov Conjecture |
author_facet |
Boegli, Sabine Cuenin, Jean-Claude |
author_sort |
Boegli, Sabine |
title |
Counterexample to the Laptev-Safronov Conjecture |
title_short |
Counterexample to the Laptev-Safronov Conjecture |
title_full |
Counterexample to the Laptev-Safronov Conjecture |
title_fullStr |
Counterexample to the Laptev-Safronov Conjecture |
title_full_unstemmed |
Counterexample to the Laptev-Safronov Conjecture |
title_sort |
counterexample to the laptev-safronov conjecture |
publisher |
Springer |
publishDate |
2022 |
url |
http://dro.dur.ac.uk/37316/ http://dro.dur.ac.uk/37316/1/37316.pdf http://dro.dur.ac.uk/37316/2/37316.pdf https://doi.org/10.1007/s00220-022-04546-z |
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laptev |
genre_facet |
laptev |
op_source |
Communications in Mathematical Physics, 2022 [Peer Reviewed Journal] |
op_relation |
dro:37316 issn:0010-3616 issn: 1432-0916 doi:10.1007/s00220-022-04546-z http://dro.dur.ac.uk/37316/ https://doi.org/10.1007/s00220-022-04546-z http://dro.dur.ac.uk/37316/1/37316.pdf http://dro.dur.ac.uk/37316/2/37316.pdf |
op_rights |
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
op_rightsnorm |
CC-BY |
op_doi |
https://doi.org/10.1007/s00220-022-04546-z |
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Communications in Mathematical Physics |
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