Abstract
We provide lower bounds for the sum of the negative eigenvalues of the operator \({|\sigma\cdot p_A|^{2s} - C_s/|x|^{2s} + V}\) in three dimensions, where \({s\in (0, 1]}\), covering the interesting physical cases s = 1 and s = 1/2. Here \({\sigma}\) is the vector of Pauli matrices, \({p_A = p - A}\), with \({p = -i\nabla}\) the three-dimensional momentum operator and A a given magnetic vector potential, and Cs is the critical Hardy constant, that is, the optimal constant in the Hardy inequality \({|p|^{2s} \geq C_s/|x|^{2s}}\). If spin is neglected, results of this type are known in the literature as Hardy–Lieb–Thirring inequalities, which bound the sum of negative eigenvalues from below by \({-M_s\int V_{-}^{1 + 3/(2s)}}\), for a positive constant Ms. The inclusion of magnetic fields in this case follows from the non-magnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for \({1/2 \leq s \leq 1}\) we are able to express the bound purely in terms of the magnetic field energy \({\|B\|_2^2}\) and integrals of powers of the negative part of V.
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Benguria R., Loss M.: A simple proof of a theorem by Laptev and Weidl. Math. Res. Lett. 7, 195 (2000)
Birman, M.S., Koplienko, L.S., Solomyak, M.Z.: Estimates for the spectrum of the difference between fractional powers of two self-adjoint operators, Izv. Vysš. Učebn. Zaved. Matematika, no. 3 (154), 3–10 (1975). (Russian) Translation to English in: Soviet Mathematics, 19 (3), 1–6 (1975)
Bugliaro L., Fefferman C., Fröhlich J., Graf G.M., Stubbe J.: A Lieb–Thirring bound for a magnetic Pauli Hamiltonian. Commun. Math. Phys. 187, 567 (1997)
Cwikel M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93 (1977)
Chen, S., Frank, R.L., Weth, T.: Remainder terms in the fractional sobolev inequality. Indiana Univ. Math. J. 62, 1381 (2013)
Daubechies I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511 (1983)
Dolbeault J., Laptev A., Loss M.: Lieb–Thirring inequalities with improved constants. J. Eur. Math. Soc. 10, 1121 (2008)
Ekholm T., Frank R.L.: On Lieb–Thirring inequalities for Schrödinger operators with virtual level. Commun. Math. Phys. 264, 725 (2006)
Erdös L.: Magnetic Lieb–Thirring inequalities. Commun. Math. Phys. 170, 629 (1995)
Erdös L., Fournais S., Solovej J.P.: Relativistic Scott correction in self-generated magnetic fields. J. Math. Phys. 53, 095202 (2012)
Erdös L., Solovej J.P.: Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields, I: nonasymptotic Lieb–Thirring-type estimate. Duke Math. J 96, 127 (1999)
Erdös L., Solovej J.P.: Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields, II: leading order asymptotic estimates. Commun. Math. Phys. 188, 599 (1997)
Erdös L., Solovej J.P.: Uniform Lieb–Thirring inequality for the three-dimensional Pauli operator with a strong non-homogeneous magnetic field. Ann. Henri Poincaré 5, 671 (2004)
Frank R.L.: A simple proof of Hardy–Lieb–Thirring inequalities. Commun. Math. Phys. 290, 789 (2009)
Frank, R.L.: Eigenvalue bounds for the fractional laplacian: a review (2017). arXiv:1603.09736
Frank R.L., Lieb E.H., Seiringer R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21, 925 (2008)
Frank R.L., Lieb E.H., Seiringer R.: Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value. Commun. Math. Phys. 275, 479 (2007)
Fröhlich J., Lieb E.H., Loss M.: Stability of Coulomb systems with magnetic fields I. The one-electron atom. Commun. Math. Phys. 104, 251 (1986)
Hansen F., Pedersen G.: Jensen’s operator inequality. Bull. Lond. Math. Soc. 35, 553 (2003)
Herbst I.W.: Spectral theory of the operator \({\left(p^2 + m^2\right)^{1/2} - Ze^2/r}\). Commun. Math. Phys. 53, 285–294 (1977)
Hundertmark D., Lieb E.H., Thomas L.E.: A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. Adv. Theor. Math. Phys. 2, 719 (1998)
Hundertmark D., Laptev A., Weidl T.: New bounds on the Lieb–Thirring constants. Invent. Math. 140, 693 (2000)
Laptev, A., Weidl, T.: Sharp Lieb–Thirring inequalities in high dimensions. Acta Math. 184, 87 (2000)
Lenzmann E., Lewin M.: Minimizers for the Hartree–Fock–Bogoliubov theory of neutron stars and white dwarfs. Duke Math J. 152, 257 (2010)
Lieb, E.H.: Lieb–Thirring Inequalities, Kluwer Encyclopedia of Mathematics, Supplement Vol. II, 311 (2000). arXiv:math-ph/0003039
Lieb E.H.: Bounds on the eigenvalues of the Laplace and Schroedinger operators. Bull. Am. Math. Soc. 82, 751 (1976)
Lieb, E.H.: The number of bound states of one-body Schrödinger operators and the Weyl Problem, Geometry of the Laplace Operator. In: Proceedings of Symposia in Pure Mathematics, vol. 36, p. 250. American Mathematical Society (1980)
Lieb E.H., Aizenman M.: On semi-classical bounds for eigenvalues of Schrödinger operators. Phys. Lett. A 66, 427 (1978)
Lieb E.H., Loss M.: Stability of Coulomb systems with magnetic fields II. The many-electron atom and the one-electron molecule. Commun. Math. Phys. 104, 271 (1986)
Lieb E.H., Loss M., Solovej J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985 (1995)
Lieb E.H., Siedentop H., Solovej J.P.: Stability and instability of relativistic electrons in classical electromagnetic fields. J. Stat. Phys. 89, 37 (1997)
Lieb, E.H., Siedentop, H., Solovej, J.P.: Stability of relativistic matter with magnetic fields. Phys. Rev. Lett. 79, 1785 (1997)
Lieb E.H., Solovej J.P., Yngvason J.: Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band region. Commun. Pure Appl. Math. 47, 513 (1994)
Lieb E.H., Solovej J.P., Yngvason J.: Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys. 161, 77 (1994)
Lieb E.H., Solovej J.P., Yngvason J.: Ground states of large quantum dots in magnetic fields. Phys. Rev. B 51, 10646 (1995)
Lieb, E.H., Thirring, W.E.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687. Erratum: Phys. Rev. Lett. 35, 1116 (1975)
Lieb, E.H., Thirring, W.E.: Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton University Press (1976)
Lieb E.H., Yau H-.T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118, 177 (1988)
Loss M., Yau H-.T.: Stability of Coulomb systems with magnetic fields. III. Zero energy bound states of the Pauli operator. Commun. Math. Phys. 104, 283 (1986)
Rosenbljum, G.V.: The distribution of the discrete spectrum for singular differential operators. Dokl. Akad. Nauk SSSR 202, 1012 (1972). See also Sov. Math. Dokl. 13, 245 (1972) (English), Izv. Vyss. Ucebn. Zaved. Matem. 164, 75 (1976), and Sov. Math. (Iz VUZ) 20, 63 (1976) (English)
Schwinger J.: On the bound states of a given potential. Proc. Natl. Acad. Sci. 47, 122 (1961)
Sobolev A.V.: Lieb–Thirring inequalities for the Pauli operator in three dimensions. IMA Vol. Math. Appl. 95, 155–188 (1997)
Sobolev A.V.: On the Lieb–Thirring estimates for the Pauli operator. Duke Math. J. 82, 607 (1996)
Solovej J.P., Østergaard Sørensen T., Spitzer W.L.: Relativistic Scott correction for atoms and molecules. Commun. Pure Appl. Math. 63, 39 (2010)
Weidl, T.: Remarks on virtual bound states for semi-bounded operators. Commun. Part. Differ. Equ. 24(1& 2), 25 (1999)
Acknowledgements
The authors were partially supported by a Sapere Aude grant from the Independent Research Fund Denmark, Grant number DFF–4181-00221. They would also like to thank the anonymous referee for precise and useful comments that helped make the article better, in particular for pointing out the article by Hansen and Pedersen cited by Eq. (1.28), of which the authors were unaware.
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Bley, G.A., Fournais, S. Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators. Commun. Math. Phys. 365, 651–683 (2019). https://doi.org/10.1007/s00220-018-3204-y
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DOI: https://doi.org/10.1007/s00220-018-3204-y