Exact Nonperturbative Unitary Amplitudes for 1->8 Transitions in a Field Theoretic Model

We present a quantum mechanical model with an infinite number of (discrete) degrees of freedom, which can serve as a laboratory for multiparticle production in a collision. There is a cubic coupling between modes without, however, any problems associated with unstable ground states. The model is amenable to precise numerical calculations of nonperturbative 1->N transition amplitudes. On an ordinary workstation, time and memory limitations effectively restrict N to be $\le\ 8,$ and we present results for this case. We find (1) that there is reasonable period of time for which there is a constant rate for the 1->8 transition; (2) at the end of the linear period, the eight particle amplitude attains a maximum value $\aemax$ which is about $3-4$ orders of magnitude larger than the comparable amplitude for excitation of the $N=8$ state in the anharmonic oscillator; (3) for values of the coupling in the region where the Born approximation fails, the amplitude is much larger than the naive estimates $A_8\simeq \exp{(-1/\g2)}\$ or $\ \exp{(-8)};$ it is more like $A_8\sim\exp{(-0.20/\g2)}.$