The Quantum Fermi-Pasta-Ulam Problem

In this work we consider the quantum version of the classical Fermi-Pasta-Ulam problem, i.e. we study the quantum dynamics of a one-dimensional chain of particles interacting through nonlinear forces. Using the quantum analogue of the classical Hamiltonian perturbation theory, in the Heisenberg picture, we eliminate through a canonical transformation the nonresonant anharmonic terms, computing the quantum version of the Birkhoff normal form to second order. Such a normal form is shown to display small divisors for large size systems, being thus useless to describe anharmonic lattice vibrations. We then show that, for the initial excitation of long wavelength modes (acoustic modes), which is the case of low temperature lattices in thermal equilibrium, the dynamics of the system is close to that of the quantum Korteweg-de Vries equation.