The N7/5 Law for Charged Bosons
Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfies E 0 ≦ −AN 7/5 We prove that 7/5 is the correct power by proving that E o ≧ −BN 7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardeau showed that E 0 ≦ − CN p 1/4 This 1/4 law is also validated here by showing that E 0 ≧ −DN p 1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.
KeywordsKinetic Energy Thermodynamic Limit Coulomb Potential Virial Theorem Yukawa Potential
Unable to display preview. Download preview PDF.
- 1.Bogoliubov, N. N.: On the theory of superfluidity. J. Phys. (USSR) 11, 23–32 (1947)Google Scholar
- 2.Conlon, J. G.: The ground state energy of a Bose gas with Coulomb interaction II. Commun. Math. Phys. 108, 363–374 (1987). See also part I, ibid 100, 355–397 (1985)Google Scholar
- 4.Dyson, F. J.: Ground-state energy of a finite system of charged particles. J. Math. Phys. 8, 1538–1545 (1967)Google Scholar
- 5.Dyson, F. J., Lenard, A.: Stability of matter I and II. J. Math. Phys. 8, 423–434 (1967); ibid 9, 698–711 (1968)Google Scholar
- 7.Foldy, L. L.: Charged boson gas. Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962)Google Scholar
- 10.Lieb, E. H.: The Bose fluid. In: Lectures in Theoretical Physics, Vol. VII C, pp. 175–224. Brittin, W. E. (ed.). Boulder: University of Colorado Press 1965Google Scholar
- 11.Lieb, E. H.: On characteristic exponents in turbulence. Commun. Math. Phys. 92, 473–480 (1984)Google Scholar
- 12.Lieb, E. H., Narnhofer, H.: The thermodynamic limit for jellium. J. Stat. Phys. 12, 291–310 (1975). Errata, ibid 14, 465 (1976)Google Scholar
- 13.Lieb, E. H., Sakakura, A. Y.: Simplified approach to the ground-state energy of an imperfect Bose gas II. Charged Bose gas at high density. Phys. Rev. 133, A899–A906 (1964)Google Scholar
- 14.Lieb, E. H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977). See also, Lieb, E. H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981); Errata ibid 54, 311 (1982)Google Scholar
- 15.Lieb, E. H., Thirring, W. E.: 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).Google Scholar
- 16.Thirring, W. E.: A course in mathematical physics, Vol. 4. Quantum mechanics of large systems. New York, Wien: Springer 1983Google Scholar