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 a lgebraic equations that determine the energy levels arising from a spe cial periodic orbit, namely the single cnoidal wave solution. Our form ulae show a simple dependence on the number of particles N in the chai n. 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 t he dispersion relation E = E(p) of a soliton on the infinite chain whi ch is in remarkable agreement with the Bethe ansatz result. The classi cal data which enter into the semiclassical quantization formula are o f interest in their own right. We give a complete treatment of the lin ear stability analysis of a single cnoidal wave and also some new expr essions for its dispersion relation which expresses the frequency v as a function of the wavenumber k.