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...
Main Authors: | , , |
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Format: | Article in Journal/Newspaper |
Language: | unknown |
Published: |
arXiv
2020
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Subjects: | |
Online Access: | https://dx.doi.org/10.48550/arxiv.2010.02981 https://arxiv.org/abs/2010.02981 |
Summary: | 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 |
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