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We discuss the Lieb-Thirring inequality for periodic systems, which has thesame optimal constant as the original inequality for finite systems. Thisallows us to formulate a new conjecture about the value of its best constant.To demonstrate the importance of periodic states, we prove that the 1DLieb-Thirring inequality at the special exponent $\gamma=3/2$ admits aone-parameter family of periodic optimizers, interpolating between theone-bound state and the uniform potential. Finally, we provide numericalsimulations in 2D which support our conjecture that optimizers could beperiodic.

The periodic Lieb-Thirring inequality