  We investigate the emergence of chaotic dynamics in a quantum Fermi - Pasta -
Ulam problem for anharmonic vibrations in atomic chains applying
semi-quantitative analysis of resonant interactions complemented by exact
diagonalization numerical studies. The crossover energy separating chaotic high
energy phase and localized (integrable) low energy phase is estimated. It
decreases inversely proportionally to the number of atoms until approaching the
quantum regime where this dependence saturates. The chaotic behavior appears at
lower energies in systems with free or fixed ends boundary conditions compared
to periodic systems. The applications of the theory to realistic molecules are
discussed.
