The periodic Lieb–Thirring inequality

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.