Analysis of a simple equation for the ground state energy of the Bose gas

TL;DR

This paper investigates a 1963 PDE with convolution non-linearity linked to the quantum Bose gas, demonstrating its validity across all parameters, establishing existence and uniqueness of solutions, and analyzing their properties.

Contribution

The paper proves existence and uniqueness of solutions for the PDE and explores their properties, extending understanding of the equation's applicability to the Bose gas problem.

Findings

Equation predictions align with the true many-body problem across all parameters.

Existence and uniqueness of solutions are established.

Solutions exhibit decay at infinity.

Abstract

In 1963 a partial differential equation with a convolution non-linearity 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. (1) 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 {\it entire range} of parameters with what is expected to be true for the solution of the true many-body problem. (2) The novel nonlinearity is easy to state but seems to have almost no literature up to now. (3) 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.