The periodic Lieb-Thirring inequality

We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D L...

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Bibliographic Details
Main Authors: Frank, Rupert L., Gontier, David, Lewin, Mathieu
Format: Article in Journal/Newspaper
Language:unknown
Published: arXiv 2020
Subjects:
Mathematical Physics math-ph
Spectral Theory math.SP
FOS Physical sciences
FOS Mathematics
Ari
laptev
Online Access:https://dx.doi.org/10.48550/arxiv.2010.02981
https://arxiv.org/abs/2010.02981
id ftdatacite:10.48550/arxiv.2010.02981
record_format openpolar
spelling ftdatacite:10.48550/arxiv.2010.02981 2023-05-15T17:07:16+02:00 The periodic Lieb-Thirring inequality Frank, Rupert L. Gontier, David Lewin, Mathieu 2020 https://dx.doi.org/10.48550/arxiv.2010.02981 https://arxiv.org/abs/2010.02981 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Mathematical Physics math-ph Spectral Theory math.SP FOS Physical sciences FOS Mathematics Article CreativeWork article Preprint 2020 ftdatacite https://doi.org/10.48550/arxiv.2010.02981 2022-03-10T15:15:57Z We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb-Thirring inequality at the special exponent $γ=3/2$ admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic. : To Ari Laptev on the occasion of his 70th birthday. Final version Article in Journal/Newspaper laptev DataCite Metadata Store (German National Library of Science and Technology) Ari ENVELOPE(147.813,147.813,59.810,59.810)
institution Open Polar
collection DataCite Metadata Store (German National Library of Science and Technology)
op_collection_id ftdatacite
language unknown
topic Mathematical Physics math-ph
Spectral Theory math.SP
FOS Physical sciences
FOS Mathematics
spellingShingle Mathematical Physics math-ph
Spectral Theory math.SP
FOS Physical sciences
FOS Mathematics
Frank, Rupert L.
Gontier, David
Lewin, Mathieu
The periodic Lieb-Thirring inequality
topic_facet Mathematical Physics math-ph
Spectral Theory math.SP
FOS Physical sciences
FOS Mathematics
description We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb-Thirring inequality at the special exponent $γ=3/2$ admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic. : To Ari Laptev on the occasion of his 70th birthday. Final version
format Article in Journal/Newspaper
author Frank, Rupert L.
Gontier, David
Lewin, Mathieu
author_facet Frank, Rupert L.
Gontier, David
Lewin, Mathieu
author_sort Frank, Rupert L.
title The periodic Lieb-Thirring inequality
title_short The periodic Lieb-Thirring inequality
title_full The periodic Lieb-Thirring inequality
title_fullStr The periodic Lieb-Thirring inequality
title_full_unstemmed The periodic Lieb-Thirring inequality
title_sort periodic lieb-thirring inequality
publisher arXiv
publishDate 2020
url https://dx.doi.org/10.48550/arxiv.2010.02981
https://arxiv.org/abs/2010.02981
long_lat ENVELOPE(147.813,147.813,59.810,59.810)
geographic Ari
geographic_facet Ari
genre laptev
genre_facet laptev
op_rights arXiv.org perpetual, non-exclusive license
http://arxiv.org/licenses/nonexclusive-distrib/1.0/
op_doi https://doi.org/10.48550/arxiv.2010.02981
_version_ 1766062638166441984

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