Volume 4 (2021)
Larson, Simon; Lundholm, Douglas; Nam, Phan Thành
Lieb–Thirring inequalities for wave functions vanishing on the diagonal set
Annales Henri Lebesgue, Volume 4 (2021), pp. 251-282.

Metadata

DOI10.5802/ahl.72
Keywords Lieb–Thirring inequalities, uncertainty principle, exclusion principle, Poincaré inequality

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] Carroll, Francis W. A polynomial in each variable separately is a polynomial, Am. Math. Mon., Volume 68 (1961), pp. 42-44 | DOI | MR | Zbl

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

[DL67] Dyson, Freeman J.; Lenard, Andrew Stability of matter. I, J. Math. Phys., Volume 8 (1967) no. 3, pp. 423-434 | DOI | MR | Zbl

[DNPV12] Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521-573 | DOI | MR | Zbl

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

[FHJN18] Frank, Rupert L.; Hundertmark, Dirk; Jex, Michael; Nam, Phan Thành The Lieb–Thirring inequality revisited (2018) (https://arxiv.org/abs/1808.09017, to appear in Journal of the European Mathematical Society)

[FS12] Frank, Rupert L.; Seiringer, Robert Lieb–Thirring inequality for a model of particles with point interactions, J. Math. Phys., Volume 53 (2012) no. 9, 095201, 11 pages | DOI | MR | Zbl

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

[HSV13] Hurri-Syrjänen, Ritva; Vähäkangas, Antti V. On fractional Poincaré inequalities, J. Anal. Math., Volume 120 (2013), pp. 85-104 | DOI | Zbl

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

[LL01] Lieb, Elliott H.; Loss, Michael Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001 | Zbl

[LL18] Larson, Simon; Lundholm, Douglas Exclusion bounds for extended anyons, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 1, pp. 309-365 | DOI | MR | Zbl

[LNP16] Lundholm, Douglas; Nam, Phan Thành; Portmann, Fabian Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 3, pp. 1343-1382 | DOI | MR | Zbl

[LPS15] Lundholm, Douglas; Portmann, Fabian; Solovej, Jan Philip Lieb–Thirring bounds for interacting Bose gases, Commun. Math. Phys., Volume 335 (2015) no. 2, pp. 1019-1056 | DOI | MR | Zbl

[LS09] Lieb, Elliott H.; Seiringer, Robert The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2009 | Zbl

[LS13a] Lundholm, Douglas; Solovej, Jan Philip Hardy and Lieb–Thirring inequalities for anyons, Commun. Math. Phys., Volume 322 (2013) no. 3, pp. 883-908 | DOI | MR | Zbl

[LS13b] Lundholm, Douglas; Solovej, Jan Philip Local exclusion principle for identical particles obeying intermediate and fractional statistics, Phys. Rev. A, Volume 88 (2013) no. 6, 062106, 9 pages | DOI

[LS14] Lundholm, Douglas; Solovej, Jan Philip Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics, Ann. Henri Poincaré, Volume 15 (2014) no. 6, pp. 1061-1107 | DOI | MR | Zbl

[LS18] Lundholm, Douglas; Seiringer, Robert Fermionic behavior of ideal anyons, Lett. Math. Phys., Volume 108 (2018) no. 11, pp. 2523-2541 | DOI | MR | Zbl

[LT75] Lieb, Elliott H.; Thirring, Walter E. Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett., Volume 35 (1975) no. 11, pp. 687-689 | DOI

[LT76] Lieb, Elliott H.; Thirring, Walter E. 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

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

[LY01] Lieb, Elliott H.; Yngvason, Jakob The ground state energy of a dilute two-dimensional Bose gas, J. Stat. Phys., Volume 103 (2001) no. 3-4, pp. 509-526 | DOI | MR | Zbl

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

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

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

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