SEMICLASSICAL QUANTIZATION OF THE PERIODIC TODA CHAIN

Gutzwiller's semiclassical quantization scheme for the trace of Green's function is applied to the periodic Toda chain. We obtain a set of algebraic equations that determine the energy levels arising from a special periodic orbit, namely the single cnoidal wave solution. Our formulae show a simple dependence on the number of particles N in the chain. N merely occurs as a parameter. We perform the soliton limit of our equations and get a semiclassical correction to first order in h to the dispersion relation E = E(p) of a soliton on the infinite chain which is in remarkable agreement with the Bethe ansatz result. The classical data which enter into the semiclassical quantization formula are of interest in their own right. We give a complete treatment of the linear stability analysis of a single cnoidal wave and also some new expressions for its dispersion relation which expresses the frequency v as a function of the wavenumber k.