# The periodic Lieb–Thirring inequality

• ### Rupert L. Frank

Ludwig-Maximilians-Universität München, Germany; California Institute of Technology Pasadena, USA
• ### David Gontier

Université Paris-Dauphine, France
• ### Mathieu Lewin

Université Paris-Dauphine, France

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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 $\gamma=\frac{3}{2}$ 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.