2008/06/30 by László Erdős, Laszlo Erdos, Benjamin Schlein +2 · 2 citations
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Quantum many-body systems #Quantum, superfluid, helium dynamics #math-ph #math.MP #msc:82B10
paper · pdf · doi:10.1103/physreva.78.053627
10 pages, no figures
openalex publication_date 2008/11/19 · arxiv created 2009/02/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider N bosons in a finite box \ensuremathΛ=[0,L]3\ensuremath⊂ℝ3 interacting via a two-body non-negative soft potential V=\ensuremathλ\stackrel\ifmmode \else \~\fiV with \stackrel\ifmmode \else \~\fiV fixed and \ensuremathλ>0 small. We will take the limit L,N\ensuremath→\ensuremath∞ by keeping the density \ensuremath\varrho=N∕L3 fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle \ensuremathε, \ensuremathε\ensuremath\leqslant4\ensuremathπ\ensuremath\varrhoa[1+(128∕15√\ensuremathπ)(\ensuremath\varrhoa3)1∕2S_\ensuremathλ]+O(\ensuremath\varrho2\ensuremath|ln\phantom\rule0.2em0ex\ensuremath\varrho\ensuremath|), as \ensuremath\varrho\ensuremath→0, with a constant satisfying 1\ensuremath\leqslantS_\ensuremathλ\ensuremath\leqslant1+C\ensuremathλ. Here a is the scattering length of V and thus depends on \ensuremathλ. In comparison, the prediction by Lee and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that S_\ensuremathλ=1 independent of \ensuremathλ.