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arXiv · 2010.02981

The periodic Lieb-Thirring inequality

Rupert L. Frank·David Gontier·Mathieu Lewin
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Abstract

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.

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Rupert L. Frank, David Gontier, Mathieu Lewin. 2020-11-23. The periodic Lieb-Thirring inequality. https://arxiv.org/abs/2010.02981

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