The periodic Lieb-Thirring inequality

TL;DR

This paper explores the Lieb-Thirring inequality in periodic systems, proposing a new conjecture on its optimal constant, and demonstrates the existence of periodic optimizers in 1D with numerical support in 2D.

Contribution

It extends the Lieb-Thirring inequality to periodic systems, introduces a new conjecture on its best constant, and shows the existence of periodic optimizers in 1D and numerical evidence in 2D.

Findings

Periodic systems share the same optimal constant as finite systems.

Existence of a family of periodic optimizers in 1D at $oldsymbol{ ext{γ=3/2}}$.

Numerical simulations in 2D support the conjecture of periodic optimizers.

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 $\gamma=3/2$ 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.