Great Lakes Mathematical Physics Meeting 2019, Oberlin, Ohio, USA
In 1963, Lieb introduced an effective theory to approximate the ground state energy of a system of Bosons interacting with each other via a repulsive pair potential, in the thermodynamic limit. Lieb showed that in one dimension, this effective theory predicts a ground state energy that differs at most by 20% from its exact value, for any density. The main idea is that instead of considering marginals of the square of the wave function, as in Hartree theory, we consider marginals of the wave function itself, which is positive in the ground state. The effective theory Lieb obtained is a non-linear integro-differential equation. In this talk, I will discuss some recent work about this effective equation, in which we proved that the ground state energy obtained from it agrees exactly with the asymptotic prediction of Lee, Huang and Yang for the full N-body system at low densities, and that, for potentials whose Fourier transform is positive, the effective theory produces the correct asymptotics for large densities as well. This is joint work with E. Carlen and E.H. Lieb.
This presentation is based on