Lieb–Thirring inequality

Remarks

The right side is, up to a constant, the semiclassical value: counting phase-space volume gives $\sum_j \abs{E_j}^\gamma \approx L^{\mathrm{cl}}_{\gamma,n}\int V_-^{\gamma + n/2} \, dx$ for large potentials. Lieb and Thirring proved the inequality for $\gamma > 1/2$ when $n = 1$ and $\gamma > 0$ when $n \ge 2$. The case $\gamma = 0$, $n \ge 3$ is the Cwikel–Lieb–Rozenblum bound, and the endpoint $\gamma = 1/2$, $n = 1$ is due to Weidl. The restrictions on $\gamma$ are needed. For $n = 1, 2$, every nonzero $V \le 0$ produces a negative eigenvalue, so no bound with $\gamma = 0$ is possible there. For $n = 1$ the bound also fails for $0 < \gamma < 1/2$.

For $\gamma = 1$, a duality argument turns the inequality into a lower bound on kinetic energy. For an $N$-fermion wave function with one-particle density $\rho$, the kinetic energy is at least $K_n\int \rho(x)^{1 + 2/n} \, dx$ for a constant $K_n > 0$.

Going deeper

For a student

Lieb and Thirring found the inequality to give a short proof of the stability of matter. This says that the energy of $N$ electrons and $N$ nuclei is at least $-CN$ for a constant $C$ independent of $N$. The case $\gamma = 1$ is the key, through the kinetic energy bound, which resembles the Thomas–Fermi model. The sharp constants are an active topic; for $n = 1$ and $\gamma = 3/2$ the sharp constant equals the semiclassical one. The inequality is a quantitative version of the fact that $-\Delta + V$ has finitely many negative eigenvalues when $V_-$ is small at infinity.