# The quantum-mechanical many-body problem: The Bose gas

Lieb É, Seiringer R, Solovej J, Yngvason J. 2005.The quantum-mechanical many-body problem: The Bose gas. In: Perspectives in Analysis. Mathematical Physics Studies, vol. 27, 97–183.

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Author

Lieb, Élliott H;
Seiringer, Robert

^{ISTA}^{}; Solovej, Jan P; Yngvason, JakobBook Editor

Benedicks, Michael;
Jones, Peter W;
Smirnov, Stanislav;
Winckler, Björn

Series Title

Mathematical Physics Studies

Abstract

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.

Publishing Year

Date Published

2005-01-01

Book Title

Perspectives in Analysis

Volume

27

Page

97 - 183

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### Cite this

Lieb É, Seiringer R, Solovej J, Yngvason J. The quantum-mechanical many-body problem: The Bose gas. In: Benedicks M, Jones P, Smirnov S, Winckler B, eds.

*Perspectives in Analysis*. Vol 27. Springer; 2005:97-183. doi:10.1007/3-540-30434-7_9Lieb, É., Seiringer, R., Solovej, J., & Yngvason, J. (2005). The quantum-mechanical many-body problem: The Bose gas. In M. Benedicks, P. Jones, S. Smirnov, & B. Winckler (Eds.),

*Perspectives in Analysis*(Vol. 27, pp. 97–183). Springer. https://doi.org/10.1007/3-540-30434-7_9Lieb, Élliott, Robert Seiringer, Jan Solovej, and Jakob Yngvason. “The Quantum-Mechanical Many-Body Problem: The Bose Gas.” In

*Perspectives in Analysis*, edited by Michael Benedicks, Peter Jones, Stanislav Smirnov, and Björn Winckler, 27:97–183. Springer, 2005. https://doi.org/10.1007/3-540-30434-7_9.É. Lieb, R. Seiringer, J. Solovej, and J. Yngvason, “The quantum-mechanical many-body problem: The Bose gas,” in

*Perspectives in Analysis*, vol. 27, M. Benedicks, P. Jones, S. Smirnov, and B. Winckler, Eds. Springer, 2005, pp. 97–183.Lieb, Élliott, et al. “The Quantum-Mechanical Many-Body Problem: The Bose Gas.”

*Perspectives in Analysis*, edited by Michael Benedicks et al., vol. 27, Springer, 2005, pp. 97–183, doi:10.1007/3-540-30434-7_9.**All files available under the following license(s):**

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