Vol. 2, No. 3, 2020
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Analysis of a simple equation for the ground state energy of the Bose gas

Eric A. Carlen, Ian Jauslin and Elliott H. Lieb
Vol. 2 (2020), No. 3, 659–684
DOI: 10.2140/paa.2020.2.659
Abstract

In 1963 a partial differential equation with a convolution nonlinearity was introduced in connection with a quantum mechanical many-body problem, namely the gas of bosonic particles. This equation is mathematically interesting for several reasons. Although the equation was expected to be valid only for small values of the parameters, further investigation showed that predictions based on the equation agree well over the entire range of parameters with what is expected to be true for the solution of the true many-body problem. Additionally, the novel nonlinearity is easy to state but seems to have almost no literature up to now. Finally, the earlier work did not prove existence and uniqueness of a solution, which we provide here along with properties of the solution such as decay at infinity.

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Keywords
Bose gas, partial differential equations
Mathematical Subject Classification 2010
Primary: 35Q40
Secondary: 82B10
Milestones
Received: 6 January 2020
Revised: 26 February 2020
Accepted: 25 March 2020
Published: 17 November 2020
Authors
Eric A. Carlen
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Ian Jauslin
Department of Physics
Princeton University
Princeton, NJ
United States
Elliott H. Lieb
Departments of Mathematics and Physics
Princeton University
Princeton, NJ
United States