Adrian Escobar, CRM

Title:Four-body problem in d-dimensional space: ground state.

Abstract: In this talk, we will consider aspects of the quantum and classical dynamics of a 4-body system in $d$-dimensional space. The study is restricted to solutions which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For $d geq 3$, this describes a six-dimensional quantum particle moving in a curved space while for $d=1$ it corresponds to a three-dimensional particle and coincides with the $A_3$ (4-body) rational Calogero model. The kinetic energy of the system has a hidden $sl(7,mathbb{R})$ Lie (Poisson) algebra structure, but for the special case $d=1$ it becomes degenerate with hidden algebra $sl(4,R)$. Based on the geometrical properties of the tetrahedron whose vertices correspond to the positions of the particles, exactly-solvable potentials will be presented as well.