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 R d , d≥ 2 , and an L q norm of the potential, for any q∈ [d/ 2 , d]. Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the...
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ftloughboroughun:oai:figshare.com:article/22094606 2023-05-15T17:07:14+02:00 Counterexample to the Laptev–Safronov conjecture Sabine Bögli Jean-Claude Cuenin 2022-11-17T00:00:00Z https://figshare.com/articles/journal_contribution/Counterexample_to_the_Laptev_Safronov_conjecture/22094606 unknown 2134/22094606.v1 https://figshare.com/articles/journal_contribution/Counterexample_to_the_Laptev_Safronov_conjecture/22094606 CC BY 4.0 Mathematical physics Pure mathematics Quantum physics Science & Technology Physical Sciences Physics Mathematical Text Journal contribution 2022 ftloughboroughun 2023-03-23T00:09:11Z 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 R d , d≥ 2 , and an L q 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 L q ∩ 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. Other Non-Article Part of Journal/Newspaper laptev Loughborough University: Figshare |
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Mathematical physics Pure mathematics Quantum physics Science & Technology Physical Sciences Physics Mathematical |
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Mathematical physics Pure mathematics Quantum physics Science & Technology Physical Sciences Physics Mathematical Sabine Bögli Jean-Claude Cuenin Counterexample to the Laptev–Safronov conjecture |
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Mathematical physics Pure mathematics Quantum physics Science & Technology Physical Sciences Physics Mathematical |
description |
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 R d , d≥ 2 , and an L q 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 L q ∩ 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 |
Other Non-Article Part of Journal/Newspaper |
author |
Sabine Bögli Jean-Claude Cuenin |
author_facet |
Sabine Bögli Jean-Claude Cuenin |
author_sort |
Sabine Bögli |
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 |
publishDate |
2022 |
url |
https://figshare.com/articles/journal_contribution/Counterexample_to_the_Laptev_Safronov_conjecture/22094606 |
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laptev |
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laptev |
op_relation |
2134/22094606.v1 https://figshare.com/articles/journal_contribution/Counterexample_to_the_Laptev_Safronov_conjecture/22094606 |
op_rights |
CC BY 4.0 |
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1766062564035264512 |