## Duke Mathematical Journal

### A positive density analogue of the Lieb–Thirring inequality

#### Abstract

The Lieb–Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger operator in terms of an $L^{p}$-norm of the potential. These are dual to bounds on the $H^{1}$-norms of a system of orthonormal functions. Here we extend these bounds to analogous inequalities for perturbations of the Fermi sea of noninteracting particles (i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials).

#### Article information

Source
Duke Math. J. Volume 162, Number 3 (2013), 435-495.

Dates
First available in Project Euclid: 14 February 2013

https://projecteuclid.org/euclid.dmj/1360874852

Digital Object Identifier
doi:10.1215/00127094-2019477

Mathematical Reviews number (MathSciNet)
MR3024090

Zentralblatt MATH identifier
1260.35088

#### Citation

Frank, Rupert L.; Lewin, Mathieu; Lieb, Elliott H.; Seiringer, Robert. A positive density analogue of the Lieb–Thirring inequality. Duke Math. J. 162 (2013), no. 3, 435--495. doi:10.1215/00127094-2019477. https://projecteuclid.org/euclid.dmj/1360874852.

#### References

• [1] R. D. Benguria and M. Loss, “Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane” in Partial Differential Equations and Inverse Problems, Contemp. Math. 362, Amer. Math. Soc., Providence, 2004, 53–61.
• [2] M. S. Birman and V. A. Sloushch, Discrete spectrum of the periodic Schrödinger operator with a variable metric perturbed by a nonnegative potential, Math. Model. Nat. Phenom. 5 (2010), 32–53.
• [3] M. S. Birman and D. R. Yafaev, The scattering matrix for a perturbation of a periodic Schrödinger operator by decreasing potential (in Russian), Algebra i Analiz 6, no. 3 (1994), 17–39; English Translation in St. Petersburg Math. J. 6, no. 3 (1995), 453–474.
• [4] É. Cancès, A. Deleurence, and M. Lewin, A new approach to the modeling of local defects in crystals: the reduced Hartree-Fock case, Comm. Math. Phys. 281 (2008), 129–177.
• [5] A. J. Coleman, Structure of fermion density matrices, Rev. Modern Phys. 35 (1963), 668–689.
• [6] J. Dolbeault, A. Laptev, and M. Loss, Lieb–Thirring inequalities with improved constants, J. Eur. Math. Soc. (JEMS) 10 (2008), 1121–1126.
• [7] R. L. Frank, M. Lewin, E. H. Lieb, and R. Seiringer, Energy cost to make a hole in the Fermi sea, Phys. Rev. Lett. 106 (2011), 150402.
• [8] R. L. Frank and B. Simon, Critical Lieb–Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices, Duke Math. J. 157 (2011), 461–493.
• [9] C. Hainzl, M. Lewin, and É. Séré, Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation, Comm. Math. Phys. 257 (2005), 515–562.
• [10] C. Hainzl, M. Lewin, and É. Séré, Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics, Arch. Ration. Mech. Anal. 192 (2009), 453–499.
• [11] D. Hundertmark, “Some bound state problems in quantum mechanics” in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math. 76, Part I, Amer. Math. Soc., Providence, 2007, 463–496.
• [12] D. Hundertmark, A. Laptev, and T. Weidl, New bounds on the Lieb–Thirring constants, Invent. Math. 140 (2000), 693–704.
• [13] A. D. Ionescu and D. Jerison, On the absence of positive eigenvalues of Schrödinger operators with rough potentials, Geom. Funct. Anal. 13 (2003), 1029–1081.
• [14] T. Kennedy and E. H. Lieb, Proof of the Peierls instability in one dimension, Phys. Rev. Lett. 59 (1987), 1309–1312.
• [15] H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues, Comm. Math. Phys. 267 (2006), 419–449.
• [16] W. Kohn, Image of the Fermi surface in the vibration spectrum of a metal, Phys. Rev. Lett. 2 (1959), 393–394.
• [17] A. Laptev and T. Weidl, “Recent results on Lieb–Thirring inequalities” in Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), Univ. Nantes, Nantes, 2000, Exp. No. XX.
• [18] A. Laptev and T. Weidl, Sharp Lieb–Thirring inequalities in high dimensions, Acta Math. 184 (2000), 87–111.
• [19] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), 309–318.
• [20] E. H. Lieb, Density functionals for coulomb systems, Int. J. Quantum Chem. 24 (1983), 243–277.
• [21] E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, Amer. Math. Soc., Providence, 2001.
• [22] E. H. Lieb and B. Nachtergaele, “Dimerization in ring-shaped molecules: The stability of the Peierls instability” in XIth International Congress of Mathematical Physics (Paris, 1994), International Press, Cambridge, Mass., 1995, 423–431.
• [23] E. H. Lieb and B. Nachtergaele, Stability of the Peierls instability for ring-shaped molecules, Phys. Rev. B 51 (1995), 4777–4791.
• [24] E. H. Lieb and B. Nachtergaele, Bond alternation in ring-shaped molecules: The stability of the Peierls instability, Int. J. Quantum Chem. 58 (1996), 699–706.
• [25] E. H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge Univ. Press, Cambridge, 2010.
• [26] E. H. Lieb and W. E. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett. 35 (1975), 687–689.
• [27] 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., Princeton Univ. Press, 1976, 269–303.
• [28] A. Migdal, Interactions between electrons and lattice vibrations in a normal metal (in Russian), Zh. Eksp. Teor. Fiz. 34 (1958), 1438–1446; English translation in Sov. Phys. JETP 7, 996 (1958), 996–1001.
• [29] R. E. Peierls, Quantum Theory of Solids, Oxford Univ. Press, London, 1955.
• [30] A. Pushnitski, The scattering matrix and the differences of spectral projections, Bull. Lond. Math. Soc. 40 (2008), 227–238.
• [31] A. Pushnitski and D. Yafaev, Spectral theory of discontinuous functions of self-adjoint operators and scattering theory, J. Funct. Anal. 259 (2010), 1950–1973.
• [32] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
• [33] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.
• [34] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978.
• [35] M. Rumin, Balanced distribution-energy inequalities and related entropy bounds, Duke Math. J. 160 (2011), 567–597.
• [36] B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35, Cambridge Univ. Press, Cambridge, 1979.
• [37] A. V. Sobolev, “Weyl asymptotics for the discrete spectrum of the perturbed Hill operator” in Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90), Adv. Soviet Math. 7, Amer. Math. Soc., Providence, 1991, 159–178.
• [38] J. Voit, One-dimensional Fermi liquids, Rep. Progr. Phys. 58 (1995), 977–1116.
• [39] J. von Neumann and E. Wigner, Über merkwürdige diskrete Eigenwerte, Phys. Z. 30 (1929), 465–467.
• [40] D. R. Yafaev, Mathematical Scattering Theory, Math. Surveys Monogr. 158, Amer. Math. Soc., Providence, 2010.