Beyond Mean Field: on the Role of Pair Excitations in the Evolution of Condensates

This paper is in part a summary of our earlier work [17, 18, 19], and in part an announcement introducing a refinement of the equations for the pair excitation function used in our previous work with D. Margetis. The new equations are Euler-Lagrange equations, and the solutions conserve energy and the number of particles. 2. Introduction The problem, which has received a lot of attention in recent years, is concerned with the evolution of theN -body linear Schrödinger equation 1 i ∂ ∂t ψN(t, ·) = HNψN(t, ·) with ψN(0, x1, · · · , xN) = φ0(x1)φ0(x2) · · ·φ0(xN) ‖ψN(t, ·)‖L2(R3N ) = 1 The Hamiltonian is an operator of the form