On the quantization of the three-particle Toda lattice

The authors compare the Einstein-Brillouin-Keller quantization procedure and the canonical quantization of a three-particle Toda chain with periodic boundary conditions. In particular, the transition from very low energies, at which the system may be approximated by harmonic oscillators, to intermediate energies is investigated. This is the regime of a general integrable nonlinear system, for which they find a Poissonian statistics for the energy levels. In the limit of very high energies they exploit the fact that the system may be described essentially by a triangular billiard and thus can derive some exact results.