Abstract
We present a generalization of the Fefferman–de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions V on Rn going to zero at infinity. This generalized decomposition can be used to extend previous results on N-body quantum systems with Coulomb interaction to a more general class of interactions. As an example of such an application, we derive the high density asymptotics of the ground state energy of jellium with Yukawa interaction in the thermodynamic limit, using a correlation estimate by Graf and Solovej.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Askey, R.: Refinements of Abel summability for Jacobi series, In: Harmonic Analysis on Homogeneous Spaces, Proc. Sympos. Pure Math. 26, Amer. Math. Soc. Providence, 1973, pp. 335-338.
Askey, R.: Summability of Jacobi Series, Trans. Amer. Math. Soc. 179 (1973), 71-84.
Bach, V.: Error bound for the Hartree-Fock energy of atoms and molecules, Comm. Math. Phys. 147 (1992), 527-548.
Dirac, P. A. M.: Note on exchange phenomena in the Thomas-Fermi atom, Proc. Cambridge Philos. Soc. 26 (1931), 376-385.
Debye, P. and Hückel, E.: Zur Theorie der Elektrolyte, Phys. Z. 24 (1923), 185-206.
Fields, J. L. and Ismail, M. E.: On the positivity of some 1 F 2 's, SIAM J. Math. Anal. 6 (1975), 551-559.
Fefferman, C. L. and de la Llave, R.: Relativistic stability of matter, I, Rev. Mat. Iber. 2 (1986), 119-161.
Gneiting, T.: Radial positive definite functions generated by Euclid's hat, J. Multivariate Anal. 69 (1999), 88-119.
Gneiting, T.: Criteria of Pó lya type for radial positive definite functions, Proc. Amer. Math. Soc. 129 (2001), 2309-2318.
Grosse, H. and Martin, A.: Particle Physics and the Schro¨dinger Equation, Cambridge Univ. Press, 1997, p. 137.
Graf, G. M. and Solevej, J. P.: A correlation estimate with applications to quantum systems with coulomb interaction, Rev. Math. Phys. 6 (1994), 977-997.
Hammersley, J. M. and Nelder, J. A.: Sampling from an isotropic Gaussian process, Proc. Cambridge Philos. Soc. 51 (1955), 652-662.
Hainzl, C. and Seiringer, R.: Bounds on one-dimensional exchange energies with application to lowest Landau band quantum mechanics, Lett. Math. Phys. 55 (2001), 133-142.
Lieb, E. H. and Narnhofer, H.: The thermodynamic limit for Jellium, J. Statist. Phys. 12 (1975), 291-310. Errata J. Statist. Phys. 14 (1976), 465.
Lieb, E. H. and Oxford, S.: Improved lower bound on the indirect Coulomb energy, Internat. J. Quantum Chem. 19 (1981), 427-439.
Mahan, G. D.: Many-Particle Physics, 2nd edn, Plenum Press, New York, 1990.
Pó lya, G.: Remarks on characteristic functions, In: Proc. Berkeley Sympos. Mathematical Statistics and Probability, Univ. California Press, 1949, pp. 115-123.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hainzl, C., Seiringer, R. General Decomposition of Radial Functions on Rn and Applications to N-Body Quantum Systems. Letters in Mathematical Physics 61, 75–84 (2002). https://doi.org/10.1023/A:1020204818938
Issue Date:
DOI: https://doi.org/10.1023/A:1020204818938