Dynamics of N-spins Quantum Ferromagnet with Uniform Exchange: Exact Solution

The transverse spin autocorrelation function at an arbitrary temperature are calculated rigorously for the system of $N$ uniformly exchanged quantum spins. At large $N$ the correlator in the para-phase is found to have a Gaussian bump at small times and a non vanishing tail at large times. A possible application of those exact results as forming a starting dynamical mean-field approximation for long-range quantum magnets and an accompanied physical picture are discussed.