Hardy inequalities for large fermionic systems. (English) Zbl 07902930
Summary: Given \(0<s< \frac{d}{2}\) with \(s \leq 1\), we are interested in the large \(N\)-behavior of the optimal constant \(\kappa N\) in the Hardy inequality \(\sum_{n=1}^N (- \Delta_n)^s \geq \kappa_N \sum_{n<m} | X_n- X_m|^{-2s}\), when restricted to antisymmetric functions. We show that \(N^{1- \frac{2s}{d}} \kappa_N\) has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
References:
| [1] | M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, DC, 1964 Zbl 0171.38503 MR 167642 · Zbl 0171.38503 |
| [2] | V. Calvez, J. A. Carrillo, and F. Hoffmann, The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime. In Nonlocal and nonlinear diffu-sions and interactions: new methods and directions, pp. 1-71, Lecture Notes in Math. 2186, Springer, Cham, 2017 Zbl 1384.35134 MR 3588121 · Zbl 1384.35134 · doi:10.1007/978-3-319-61494-6_1 |
| [3] | V. Calvez, J. A. Carrillo, and F. Hoffmann, Uniqueness of stationary states for singular Keller-Segel type models. Nonlinear Anal. 205 (2021), article no. 112222 Zbl 1458.35004 MR 4190637 · Zbl 1458.35004 · doi:10.1016/j.na.2020.112222 |
| [4] | A. J. Coleman, Structure of fermion density matrices. Rev. Modern Phys. 35 (1963), 668-689 MR 0155637 · doi:10.1103/RevModPhys.35.668 |
| [5] | J. G. Conlon, The ground state energy of a classical gas. Comm. Math. Phys. 94 (1984), no. 4, 439-458 Zbl 0946.82501 MR 0763746 · Zbl 0946.82501 · doi:10.1007/BF01403881 |
| [6] | E. B. Davies, A review of Hardy inequalities. In The Maz’ya anniversary collection, Vol. 2 (Rostock, 1998), pp. 55-67, Oper. Theory Adv. Appl. 110, Birkhäuser, Basel, 1999 Zbl 0936.35121 MR 1747888 · Zbl 0936.35121 · doi:10.1007/978-3-0348-8672-7_5 |
| [7] | C. Fefferman and R. de la Llave, Relativistic stability of matter. I. Rev. Mat. Iberoameri-cana 2 (1986), no. 1-2, 119-213 Zbl 0602.58015 MR 0864658 · Zbl 0602.58015 · doi:10.4171/RMI/30 |
| [8] | S. Fournais, M. Lewin, and J. P. Solovej, The semi-classical limit of large fermionic sys-tems. Calc. Var. Partial Differential Equations 57 (2018), no. 4, article no. 105 Zbl 1395.81313 MR 3814648 · Zbl 1395.81313 · doi:10.1007/s00526-018-1374-2 |
| [9] | R. L. Frank, The Lieb-Thirring inequalities: Recent results and open problems. In Nine mathematical challenges-an elucidation, pp. 45-86, Proc. Sympos. Pure Math. 104, American Mathematical Society, Providence, RI, 2021 Zbl 1518.35002 MR 4337417 · Zbl 1518.35002 · doi:10.1090/pspum/104/01877 |
| [10] | R. L. Frank, Lieb-Thirring inequalities and other functional inequalities for orthonormal systems. In ICM-International Congress of Mathematicians. Vol. 5. Sections 9-11, pp. 3756-3774, EMS Press, Berlin, 2023 Zbl 07823084 MR 4680381 · Zbl 07823084 · doi:10.4171/icm2022/105 |
| [11] | R. L. Frank, D. Hundertmark, M. Jex, and P. T. Nam, The Lieb-Thirring inequality revis-ited. J. Eur. Math. Soc. (JEMS) 23 (2021), no. 8, 2583-2600 Zbl 1467.35014 MR 4269422 · Zbl 1467.35014 · doi:10.4171/jems/1062 |
| [12] | R. L. Frank, A. Laptev, and T. Weidl, Schrödinger operators: eigenvalues and Lieb-Thirring inequalities. Cambridge Stud. Adv. Math. 200, Cambridge University Press, Cambridge, 2023 Zbl 7595814 MR 4496335 · Zbl 1536.35001 · doi:10.1017/9781009218436 |
| [13] | R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), no. 12, 3407-3430 Zbl 1189.26031 MR 2469027 · Zbl 1189.26031 · doi:10.1016/j.jfa.2008.05.015 |
| [14] | D. Gontier, M. Lewin, and F. Q. Nazar, The nonlinear Schrödinger equation for orthonor-mal functions: existence of ground states. Arch. Ration. Mech. Anal. 240 (2021), no. 3, 1203-1254 Zbl 1470.35328 MR 4264945 · Zbl 1470.35328 · doi:10.1007/s00205-021-01634-7 |
| [15] | C. Hainzl and R. Seiringer, General decomposition of radial functions on R n and applica-tions to N -body quantum systems. Lett. Math. Phys. 61 (2002), no. 1, 75-84 Zbl 1016.81059 MR 1930084 · Zbl 1016.81059 · doi:10.1023/A:1020204818938 |
| [16] | I. W. Herbst, Spectral theory of the operator .p 2 C m 2 / 1=2 Ze 2 =r. Comm. Math. Phys. 53 (1977), no. 3, 285-294 Zbl 0375.35047 MR 0436854 · Zbl 0375.35047 · doi:10.1007/BF01609852 |
| [17] | M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, “Schrödinger inequalities” and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A (3) 16 (1977), no. 5, 1782-1785 MR 0471726 · doi:10.1103/PhysRevA.16.1782 |
| [18] | M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom, Many-particle Hardy inequalities. J. Lond. Math. Soc. (2) 77 (2008), no. 1, 99-114 Zbl 1135.35012 MR 2389919 · Zbl 1135.35012 · doi:10.1112/jlms/jdm091 |
| [19] | V. F. Kovalenko, M. A. Perel’muter, and Y. A. Semenov, Schrödinger operators with L l=2 W .R l /-potentials. J. Math. Phys. 22 (1981), no. 5, 1033-1044 Zbl 0463.47027 MR 0622855 · Zbl 0463.47027 · doi:10.1063/1.525009 |
| [20] | G. Leoni, A first course in Sobolev spaces. 2nd edn., Grad. Stud. Math. 181, American Mathematical Society, Providence, RI, 2017 Zbl 1382.46001 MR 3726909 · Zbl 1382.46001 · doi:10.1090/gsm/181 |
| [21] | J.-M. Lévy-Leblond, Nonsaturation of gravitational forces. J. Math. Phys. 10 (1969), 806-812 Zbl 0181.56602 · doi:10.1063/1.1664909 |
| [22] | E. H. Lieb, A lower bound for Coulomb energies. Phys. Lett. A 70 (1979), no. 5-6, 444-446 MR 0588128 · doi:10.1016/0375-9601(79)90358-X |
| [23] | E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys. 53 (1981), no. 4, 603-641 Zbl 1049.81679 MR 0629207 · Zbl 1114.81336 · doi:10.1103/RevModPhys.53.603 |
| [24] | E. H. Lieb and M. Loss, Analysis. 2nd edn., Grad. Stud. Math. 14, American Mathematical Society, Providence, RI, 2001 Zbl 0966.26002 MR 1817225 · Zbl 0966.26002 · doi:10.1090/gsm/014 |
| [25] | E. H. Lieb and S. Oxford, Improved lower bound on the indirect Coulomb energy. Int. J. Quantum Chem. 19 (1981), 427-439 · doi:10.1002/qua.560190306 |
| [26] | E. H. Lieb and R. Seiringer, The stability of matter in quantum mechanics. Cambridge University Press, Cambridge, 2010 Zbl 1179.81004 MR 2583992 · Zbl 1179.81004 · doi:10.1017/CBO9780511819681 |
| [27] | E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids. Advances in Math. 23 (1977), no. 1, 22-116 Zbl 0938.81568 MR 0428944 · Zbl 0938.81568 · doi:10.1016/0001-8708(77)90108-6 |
| [28] | E. H. Lieb, J. P. Solovej, and J. Yngvason, Ground states of large quantum dots in magnetic fields. Phys. Rev. B 51 (1995), 10646-10665 · doi:10.1103/PhysRevB.51.10646 |
| [29] | E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Stud. Math. Phys. (1976), 269-303 Zbl 0342.35044 · Zbl 0342.35044 |
| [30] | E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativis-tic kinetic energy. Ann. Physics 155 (1984), no. 2, 494-512 MR 0753345 · doi:10.1016/0003-4916(84)90010-1 |
| [31] | E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys. 112 (1987), no. 1, 147-174 Zbl 0641.35065 MR 0904142 · Zbl 0641.35065 · doi:10.1007/BF01217684 |
| [32] | E. H. Lieb and H.-T. Yau, The stability and instability of relativistic matter. Comm. Math. Phys. 118 (1988), no. 2, 177-213 Zbl 0686.35099 MR 0956165 · Zbl 0686.35099 · doi:10.1007/BF01218577 |
| [33] | D. Lundholm, P. T. Nam, and F. Portmann, Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems. Arch. Ration. Mech. Anal. 219 (2016), no. 3, 1343-1382 Zbl 1332.81292 MR 3448930 · Zbl 1332.81292 · doi:10.1007/s00205-015-0923-5 |
| [34] | V. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations. Grundlehren Math. Wiss. 342, Springer, Heidelberg, 2011 Zbl 1217.46002 MR 2777530 · Zbl 1217.46002 · doi:10.1007/978-3-642-15564-2 |
| [35] | B. Opic and A. Kufner, Hardy-type inequalities. Pitman Res. Notes Math. Ser. 219, Long-man Scientific & Technical, Harlow, 1990 Zbl 0698.26007 MR 1069756 · Zbl 0698.26007 |
| [36] | J. Peetre, Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279-317 Zbl 0151.17903 MR 0221282 · Zbl 0151.17903 · doi:10.5802/aif.232 |
| [37] | W. Thirring, A lower bound with the best possible constant for Coulomb Hamiltonians. Comm. Math. Phys. 79 (1981), no. 1, 1-7 MR 0609223 · doi:10.1007/BF01208281 |
| [38] | D. Yafaev, Sharp constants in the Hardy-Rellich inequalities. J. Funct. Anal. 168 (1999), no. 1, 121-144 Zbl 0981.26016 MR 1717839 · Zbl 0981.26016 · doi:10.1006/jfan.1999.3462 |
| [39] | . |
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