Abstract
We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namely p 2 /2m is replaced by (p 2 c 2 + m 2 c 4 ) 1/2 — mc 2 . The electrons are allowed to have q spin states (q = 2 in nature). For one electron and one nucleus instability occurs if Zα>2/π, where z is the nuclear charge and a is the fine structure constant. We prove that stability occurs in the many-body case if z α ≦ 2/π and α l/(47q). For small z, a better bound on a is also given. In the other direction we show that there is a critical αc (no greater than 128/15π) such that if α>αc then instability always occurs for all positive z (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.
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Work partially supported by U.S. National Science Foundation grant PHY-85–15288-A02
The author thanks the Institute for Advanced Study for its hospitality and the U.S. National Science Foundation for support under grant DMS-8601978
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References
Baxter, J.R.: Inequalities for potentials of particle systems. 111. J. Math. 24, 645–652 (1980)
Chandrasekhar, S.: Phil. Mag. 11, 592 (1931)
Chandrasekhar, S.: Astro. J. 74, 81 (1931)
Chandrasekhar, S.: Monthly Notices Roy. Astron. Soc. 91, 456 (1931)
Chandrasekhar, S.: Rev. Mod. Phys. 56, 137 (1984)
Conlon, J.G.: The ground state energy of a classical gas. Commun. Math. Phys. 94, 439–458 (1984)
Conlon, J.G., Lieb, E.H., Yau, H.-T.: The N7/5 law for charged bosons. Commun. Math. Phys. 116,417–448 (1988)
Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987
Daubechies, I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511–520 (1983)
Daubechies, I.: One electron molecules with relativistic kinetic energy: properties of the discrete spectrum. Commun. Math. Phys. 94, 523–535 (1984)
Daubechies, I., Lieb, E.H.: One-electron relativistic molecules with Coulomb interaction. Commun. Math. Phys. 90, 497–510 (1983)
Dyson, F.J.: Ground state energy of a finite system of charged particles. J. Math. Phys. 8, 1538–1545(1967)
Dyson, F.J., Lenard, A.: Stability of matter I and II. J. Math. Phys. 8, 423–434 (1967); ibid 9, 698–711 (1968). See also Lenard’s Battelle lecture. In: Lecture Notes in Physics, vol. 23. Berlin, Heidelberg, New York: Springer 1973
Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of integral transforms, Vol. 1. New York, Toronto, London: McGraw-Hill 1954, p. 75, 2.4 (35)
Federbush, P.: A new approach to the stability of matter problem. II. J. Math. Phys. 16, 706–709 (1975)
Fefferman, C.: The N-body problem in quantum mechanics. Commun. Pure Appl. Math. Suppl. 39, S67–S109 (1986)
Fefferman, C., de la Llave, R.: Relativistic stability of matter. I. Rev. Math. Iberoamericana 2, 119–215(1986)
Fröhlich, J., Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields. I. The one-electron atom. Commun. Math. Phys. 104, 251–270 (1986)
Herbst, I.: Spectral theory of the operator (p 2 + m 2 ) 1/2 -ze 2 /rCommun. Math. Phys. H, 285–294 (1977); Errata ibid 55, 316 (1977)
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966. See remark 5.12, p. 307
Kovalenko, V., Perelmuter, M., Semenov, Ya.: Schrödinger operators with (Math) potentials. J. Math. Phys. 22, 1033–1044 (1981)
Lieb, E.H.: Stability of matter. Rev. Mod. Phys. 48, 553–569 (1976)
Lieb, E.H.: The N5/3 law for bosons. Phys. Lett. 70A, 71–73 (1979)
Lieb, E.H. : Density functionals for Coulomb systems. Int. J. Quant. Chem. 24,243–277 (1983)
Lieb, E.H.: On characteristic exponents in turbulence. Commun. Math. Phys. 92, 473–480 (1984)
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, 271–282 (1986)
Lieb, E., Simon, B.: Thomas Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116(1977)
Lieb, E.H., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975). Errata, ibid 35,1116 (1975); see also their article: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, Essays in honor of Valentine Bargmann. Lieb, E.H., Simon, B., Wightman, A.S. (eds.). Princeton, NJ: Princeton University Press 1976
Lieb, E.H., Thirring, W.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. (NY) 155, 494–512 (1984)
Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112,147–174(1987).
See also Lieb, E.H. and Yau, H.-T.: A rigorous examination of the Chandrasekhar theory of stellar collapse. Astro. J. 323,140–144 (1987)
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, 283–290 (1986)
Weder, R.: Spectral analysis of pseudodifferential operators. J. Funct. Anal. 20,319–337 (1975)
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Lieb, E.H., Yau, HT. (2001). The Stability and Instability of Relativistic Matter. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_34
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DOI: https://doi.org/10.1007/978-3-662-04360-8_34
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