The periodic Lieb-Thirring inequality

Rupert L. Frank (1, 2) , David Gontier (3, 4) , Mathieu Lewin (3, 4)

1 CALTECH - California Institute of Technology
2 LMU - Ludwig Maximilian University [Munich] = Ludwig Maximilians Universität München
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 PSL - Université Paris Sciences et Lettres

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 $\gamma=3/2$ 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.

Domaines: Physique mathématique [math-ph]

Rupert L. Frank, David Gontier, Mathieu Lewin. The periodic Lieb-Thirring inequality. Pavel Exner, Rupert Frank, Fritz Gesztesy, Helge Holden, Timo Weidl. Partial Differential Equations, Spectral Theory, and Mathematical Physics. The Ari Laptev Anniversary Volume, 18, EMS Publishing House, pp.135--154, 2021, EMS Series of Congress Reports, 978-3-98547-007-5. ⟨10.4171/ECR/18⟩. ⟨hal-02964212⟩