 02183 Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason
 The Ground State of the Bose Gas
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Apr 11, 02

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Abstract. Now that the low temperature properties of quantummechanical manybody
systems (bosons) at low density, $\rho$, can be examined experimentally
it is appropriate to revisit some of the formulas deduced by many
authors 45 decades ago. For systems with repulsive (i.e. positive)
interaction potentials the experimental low temperature state and the
ground state are effectively synonymous  and this fact is used in all
modeling. In such cases, the leading term in the energy/particle is
$2\pi\hbar^2 a \rho/m$ where $a$ is the scattering length of the
twobody potential. Owing to the delicate and peculiar nature of
bosonic correlations (such as the strange $N^{7/5}$ law for charged
bosons), four decades of research failed to establish this plausible
formula rigorously. The only previous lower bound for the energy was
found by Dyson in 1957, but it was 14 times too small. The correct
asymptotic formula has recently been obtained by us and this work will
be presented. The reason behind the mathematical difficulties will be
emphasized. A different formula, postulated as late as 1971 by Schick,
holds in twodimensions and this, too, will be shown to be correct.
With the aid of the methodology developed to prove the lower bound for
the homogeneous gas, two other problems have been successfully
addressed. One is the proof by us that the GrossPitaevskii equation
correctly describes the ground state in the `traps' actually used in
the experiments. For this system it is also possible to prove complete
Bose condensation, as we have shown. Another topic is a proof that
Foldy's 1961 theory of a high density Bose gas of charged particles
correctly describes its ground state energy. All of this is quite
recent work and it is hoped that the mathematical methodology might be
useful, ultimately, to solve more complex problems connected with
these interesting systems.
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