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...

Full description

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
Description
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

Similar Items