Summary
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) 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, and to explore new regimes not treated before. 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 2πħ2 aρ/m where a is the scattering length of the two-body 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 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 two dimensions 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, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii 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 and superfluidity as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. 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; using this we can also prove the N 7/5 formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. 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.
Keywords
- Ground State Energy
- Thermodynamic Limit
- Neumann Boundary Condition
- Einstein Condensation
- Trace Class Norm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
On leave from Dept. of Math., University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
The work was supported in part by US NSF grants PHY 0139984-A01 (EHL), PHY 0353181 (RS) and DMS-0111298 (JPS); by EU grant HPRN-CT-2002-00277 (JPS and JY); by FWF grant P17176-N02 (JY); by MaPhySto — A Network in Mathematical Physics and Stochastics funded by The Danish National Research Foundation (JPS), and by grants from the Danish research council (JPS).
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Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J. (2005). The Quantum-Mechanical Many-Body Problem: The Bose Gas. In: Benedicks, M., Jones, P.W., Smirnov, S., Winckler, B. (eds) Perspectives in Analysis. Mathematical Physics Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-30434-7_9
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