Abstract
Consider N bosons in a finite box Λ=[0,L]3⊂R 3 interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle
where a is the scattering length of the potential. Previously, an upper bound of the form C16/15π 2 for some constant C>1 was obtained in (Erdös et al. in Phys. Rev. A 78:053627, 2008). Our result proves the upper bound of the prediction by Lee and Yang (Phys. Rev. 105(3):1119–1120, 1957) and Lee et al. (Phys. Rev. 106(6):1135–1145, 1957).
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References
Bogoliubov, N.N.: On the theory of superfluidity. Izv. Akad. Nauk USSR 11(1), 77 (1947) (in Russian)
Bogoliubov, N.N.: On the theory of superfluidity. J. Phys. 11(1), 23–32 (1947) (in English)
Dyson, F.J.: Ground-state energy of a hard-sphere gas. Phys. Rev. 106, 20–26 (1957)
Erdős, L., Schlein, B., Yau, H.-T.: Ground-state energy of a low-density Bose gas: A second-order upper bound. Phys. Rev. A 78, 053627 (2008)
Girardeau, M., Arnowitt, R.: Theory of many-boson systems: Pair theory. Phys. Rev. 113(3), 755–761 (1959)
Giuliani, A., Seiringer, R.: The ground state energy of the weakly interacting Bose gas at high density, arXiv:0811.1166v1 (2008)
Huang, K., Yang, C.N.: Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev. 105(3), 767–775 (1957)
Landau, L., Lifshitz, E.: Quantum Mechanics, Non-relativistic Theory, 3rd. edn. Pergamon, Elmsford (1991)
Lee, T.D., Huang, K., Yang, C.N.: Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106(6), 1135–1145 (1957)
Lee, T.D., Yang, C.N.: Many body problem in quantum mechanics and quantum statistical mechanics. Phys. Rev. 105(3), 1119–1120 (1957)
Lieb, E.H.: Simplified approach to the ground state energy of an imperfect Bose gas. Phys. Rev. 130, 2518–2528 (1963). See also Phys. Rev. 133, A899–A906 (1964) (with A.Y. Sakakura) and Phys. Rev. 134, A312–A315 (1964) (with W. Liniger)
Lieb, E.H., Solovej, J.P.: Ground state energy of the one-component charged Bose gas. Commun. Math. Phys. 217, 127–163 (2001)
Lieb, E.H., Solovej, J.P.: Ground state energy of the two-component charged Bose gas. Commun. Math. Phys. 252, 448–534 (2004)
Lieb, E.H., Yngvason, J.: Ground state energy of the low density Bose gas. Phys. Rev. Lett. 80, 2504–2507 (1998)
Solovej, J.P.: Upper bounds to the ground state energies of the one- and two-component charged Bose gases. Comm. Math. Phys. 266(3), 797–818 (2006)
Yang, C.N.: Dilute hard “sphere” Bose gas in dimensions 2, 4 and 5, arXiv:0807.0938
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Partially supported by NSF grants DMS-0757425, 0804279.
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Yau, HT., Yin, J. The Second Order Upper Bound for the Ground Energy of a Bose Gas. J Stat Phys 136, 453–503 (2009). https://doi.org/10.1007/s10955-009-9792-3
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DOI: https://doi.org/10.1007/s10955-009-9792-3