### Metadata

### Abstract

We propose a general strategy to derive Lieb–Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb–Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.

### References

[Car61] A polynomial in each variable separately is a polynomial, Am. Math. Mon., Volume 68 (1961), pp. 42-44 | Article | MR 125183 | Zbl 0098.01502

[Dau83] An uncertainty principle for fermions with generalized kinetic energy, Commun. Math. Phys., Volume 90 (1983) no. 4, pp. 511-520 | Article | MR 719431 | Zbl 0946.81521

[DL67] Stability of matter. I, J. Math. Phys., Volume 8 (1967) no. 3, pp. 423-434 | Article | MR 2408896 | Zbl 0948.81665

[DNPV12] Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521-573 | Article | MR 2944369 | Zbl 1252.46023

[Dys67] Ground-state energy of a finite system of charged particles, J. Math. Phys., Volume 8 (1967) no. 8, pp. 1538-1545 | Article | MR 2408895

[FHJN18] The Lieb–Thirring inequality revisited (2018) (https://arxiv.org/abs/1808.09017, to appear in Journal of the European Mathematical Society)

[FS12] Lieb–Thirring inequality for a model of particles with point interactions, J. Math. Phys., Volume 53 (2012) no. 9, 095201, 11 pages | Article | MR 2905783 | Zbl 1278.81061

[Gir60] Relationship between systems of impenetrable bosons and fermions in one dimension, J. Mathematical Phys., Volume 1 (1960), pp. 516-523 | Article | MR 128913 | Zbl 0098.21704

[HSV13] On fractional Poincaré inequalities, J. Anal. Math., Volume 120 (2013), pp. 85-104 | Article | Zbl 1288.26019

[Lap12] Spectral inequalities for Partial Differential Equations and their applications, Fifth International Congress of Chinese Mathematicians. Part 1, 2 (Studies in Advanced Mathematics) Volume 51 (2012), pp. 629-643 | MR 2908096 | Zbl 1269.35050

[LL01] Analysis, Graduate Studies in Mathematics, Volume 14, American Mathematical Society, 2001 | Zbl 0966.26002

[LL18] Exclusion bounds for extended anyons, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 1, pp. 309-365 | Article | MR 3740376 | Zbl 1391.81217

[LNP16] Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 3, pp. 1343-1382 | Article | MR 3448930 | Zbl 1332.81292

[LPS15] Lieb–Thirring bounds for interacting Bose gases, Commun. Math. Phys., Volume 335 (2015) no. 2, pp. 1019-1056 | Article | MR 3316649 | Zbl 1310.81170

[LS09] The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2009 | Zbl 1179.81004

[LS13a] Hardy and Lieb–Thirring inequalities for anyons, Commun. Math. Phys., Volume 322 (2013) no. 3, pp. 883-908 | Article | MR 3079335 | Zbl 1270.81248

[LS13b] Local exclusion principle for identical particles obeying intermediate and fractional statistics, Phys. Rev. A, Volume 88 (2013) no. 6, 062106, 9 pages | Article

[LS14] Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics, Ann. Henri Poincaré, Volume 15 (2014) no. 6, pp. 1061-1107 | Article | MR 3205745 | Zbl 1294.81385

[LS18] Fermionic behavior of ideal anyons, Lett. Math. Phys., Volume 108 (2018) no. 11, pp. 2523-2541 | Article | MR 3861388 | Zbl 1402.81269

[LT75] Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett., Volume 35 (1975) no. 11, pp. 687-689 | Article

[LT76] Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics: Essaus in Honor (Lieb, Elliott H., ed.) (Princeton Series in Physics), Princeton University Press, 1976, pp. 269-303 | Zbl 0342.35044

[Lun18] Methods of modern mathematical physics: Uncertainty and exclusion principles in quantum mechanics (2018) (https://arxiv.org/abs/1805.03063)

[LY01] The ground state energy of a dilute two-dimensional Bose gas, J. Stat. Phys., Volume 103 (2001) no. 3-4, pp. 509-526 | Article | MR 1827922 | Zbl 1115.82306

[Nam18] Lieb–Thirring inequality with semiclassical constant and gradient error term, J. Funct. Anal., Volume 274 (2018) no. 6, pp. 1739-1746 | MR 3758547 | Zbl 1414.35185

[Rum11] Balanced distribution-energy inequalities and related entropy bounds, Duke Math. J., Volume 160 (2011) no. 3, pp. 567-597 | Article | MR 2852369 | Zbl 1239.47019

[Sve81] The effect of submanifolds upon essential self-adjointness and deficiency indices, J. Math. Anal. Appl., Volume 80 (1981) no. 2, pp. 551-565 | Article | Zbl 0473.47039

[Yaf99] Sharp Constants in the Hardy–Rellich Inequalities, J. Funct. Anal., Volume 168 (1999) no. 1, pp. 121-144 | Article | MR 1717839 | Zbl 0981.26016