Ground State Energy of the Low Density Bose Gas

Abstract

Now that the properties of low temperature Bose gases at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. One of these is that the leading term in the energy/particle is 2πh 2 ρa/m, where a is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.PACS numbers: 05.30. Jp, 03.75.Fi, With the renewed experimental interest in low density, low temperature Bose gases, some of the formulas posited four and five decades ago have been dusted off and reexamined. One of these is the leading term in the ground state energy. In the limit of small particle density, ρ,where e 0 (ρ) is the ground state energy (g.s.e.) per particle in the thermodynamic limit, a is the scattering length (assumed positive) of the two-body potential v for bosons of mass m, and µ ≡h 2 /2m. Is (1) correct? In particular, is it true for the hardsphere gas? While there have been many attempts at a rigorous proof of (1) in the past forty years, none has been found so far. Our aim here is to supply that proof for finite range, positive potentials. As remarked below, (1) cannot hold unrestrictedly; more than a > 0 is needed.An upper bound for e 0 (ρ) agreeing with (1) is not easy to derive, but it was achieved for hard spheres by a variational calculation [1], which can be extended to include general, positive potentials of finite range. What remained unknown was a good lower bound. The only one available is Dyson's [1], and that is about fourteen times smaller than (1). In this paper we shall provide a lower bound of the desired form, and thus prove (1). We can also give explicit error bounds for small enough values of the dimensionless parameter Y ≡ 4πρa 3 /3:for some fixed C (which is not evaluated explicitly because C and the exponent 1/17 are only of academic interest). The bound (2) holds for all non-negative, finite range, spherical, two-body potentials. A typical experimental value [2] is Y ≈ 10 −5 . Dyson's upper bound is µ 4πρa(1 + 2Y 1/3 )(1 − Y 1/3 ) −2 . We conjecture that (1) requires only a positive scattering length and the absence of any many-body, negative energy bound state. If there are such bound states then (1) is certainly wrong, but this obvious caveat does not seem to have been clearly emphasized before. There is a 'nice' potential with positive scattering length, no 2-body bound state, but with a 3-body bound state [3].Our method also obviously applies to the positive temperature free energy (because Neumann boundary conditions give an upper bound to the solution to the heat (or Bloch) equation).We also give some explicit bounds for finite systems, which might be useful for experiments with traps, but we concentrate here on the thermodynamic limit for simplicity. For traps with slowly varying...

A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas

“…This fact has been 'known' since the early 50's but a rigorous proof is fairly recent [3]. In two dimensions the corresponding formula is 4πρ 2 | ln(ρa 2 )| −1 as proved in [4] by extension of the c 2000 by the authors. Reproduction of this work, in its entirety, by any means, is permitted for non-commercial purposes.…”

“…The coefficients I, J and K are given by Eqs. (2.4)-(2.10) in [4]. They depend on the scattering length and a parameter b.…”

“…For the proof see [4], where generalizations to other dimensions and potentials with a negative part are also discussed. Note that the factor 1 2 in (A.1) and (A.2) is due to the reduced mass of the two body problem.…”

“…In fact, one may then simply cut off the potential at some point R 0 > R 1 (i.e., set v(x) = 0 for |x| > R 0 ) and consider the limit of the scattering lengths of the cut off potentials as R 0 → ∞. See [4] for details.…”

The kinetic and interaction energies of a trapped Bose gas: beyond the mean field

“…Indeed, in the paper [3] the third term of the GP functional is treated as the energy of the pairwise boson interaction. While according to another point of view [4], in the homogeneous case V ext = 0 this term makes contribution only to the kinetic energy if the boson-boson interaction potential V (r) is of the hardsphere form: V (r) = +∞ at r < a, and V (r) = 0 at r a. This problem is usually considered to be trivial [2].…”