The Many-Body Problem

Cécil Dewitt, Ph. Nozières, H. Mendlowitz

Daiichiro Sugimoto

Physics is full of many-body problems. But many-problems cannot be solved exactly in terms of known mathematical functions. Even the three-body problem can only be solved exactly in a few special cases, and most many-body systems consist of many more than three bodies. Enter supercomputers.

Christian Frønsdal

John W. Norbury

A solution to quantum mechanical problems of two, three, and four bodies is discussed from the point of view of two-body harmonic oscillator forces. An exact solution to the three- and four-body problems is found for harmonic oscillator forces, assuming equal masses. The aim of the discussion is to make students aware of the real difficulties of the few-body problem.

Jerome K. Percus

Manuel Valiente

Here, I focus on the use of microscopic, few-body techniques that are relevant in the many-body problem. These methods can be divided into indirect and direct. In particular, indirect methods are concerned with the simplification of the many-body problem by substituting the full, microscopic interactions by pseudopotentials which are designed to reproduce collisional information at specified energies, or binding energies in the few-body sector. These simplified interactions yield more tractable theories of the many-body problem, and are equivalent to effective field theory of interactions. Direct methods, which so far are most useful in one spatial dimension, have the goal of attacking the many-body problem at once by using few-body information only. Here, I will present non-perturbative direct methods to study one-dimensional fermionic and bosonic gases in one dimension.