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  "body": "\n## Statement\n\nLet $n \\ge 1$ be an integer and let $\\gamma$ be a real number with\n$$\\gamma \\ge \\tfrac{1}{2} \\text{ if } n = 1, \\qquad \\gamma \u003e 0 \\text{ if } n = 2, \\qquad \\gamma \\ge 0 \\text{ if } n \\ge 3.$$\nThen there is a constant $L_{\\gamma,n}$, depending only on $\\gamma$ and $n$, with the following property.\nLet $V \\colon \\R^n \\to \\R$ be measurable with positive part $V_+ = \\max(V, 0)$ locally integrable and negative part $V_- = \\max(-V, 0)$ in the Lebesgue space $L^{\\gamma + n/2}(\\R^n)$.\nLet $E_1 \\le E_2 \\le \\cdots \u003c 0$ be the negative [[eigenvalues|0BQAA2BK3Q]], counted with multiplicity, of the [[Schrödinger operator|89AA6XJY5K]] $-\\Delta + V$ on $L^2(\\R^n)$, defined through its quadratic form.\nThen\n$$\\sum_j \\abs{E_j}^\\gamma \\le L_{\\gamma,n}\\int_{\\R^n} V_-(x)^{\\gamma + n/2} \\, dx,$$\nwhere for $\\gamma = 0$ the left side is the number of negative eigenvalues.\n\n## Remarks\n\nThe 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.\nLieb and Thirring proved the inequality for $\\gamma \u003e 1/2$ when $n = 1$ and $\\gamma \u003e 0$ when $n \\ge 2$.\nThe case $\\gamma = 0$, $n \\ge 3$ is the Cwikel–Lieb–Rozenblum bound, and the endpoint $\\gamma = 1/2$, $n = 1$ is due to Weidl.\nThe restrictions on $\\gamma$ are needed.\nFor $n = 1, 2$, every nonzero $V \\le 0$ produces a negative eigenvalue, so no bound with $\\gamma = 0$ is possible there.\nFor $n = 1$ the bound also fails for $0 \u003c \\gamma \u003c 1/2$.\n\nFor $\\gamma = 1$, a duality argument turns the inequality into a lower bound on kinetic energy.\nFor 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 $K_n \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nThe negative eigenvalues of $-\\Delta + V$ are the bound states of a quantum particle in a potential well $V$.\nThe inequality says that a shallow or small well cannot hold many deep bound states.\nThe total, measured by the sum of powers of the binding energies, is controlled by a single integral of the well depth.\n\n### L2 human\n\nLieb and Thirring found the inequality to give a short proof of the stability of matter.\nThis says that the energy of $N$ electrons and $N$ nuclei is at least $-CN$ for a constant $C$ independent of $N$.\nThe case $\\gamma = 1$ is the key, through the kinetic energy bound, which resembles the Thomas–Fermi model.\nThe sharp constants are an active topic; for $n = 1$ and $\\gamma = 3/2$ the sharp constant equals the semiclassical one.\nThe inequality is a quantitative version of the fact that $-\\Delta + V$ has finitely many negative eigenvalues when $V_-$ is small at infinity.\n",
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