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

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.