Dispersionless Toda and Toeplitz Operators

In this paper we present some results on the dispersionless limit of the Toda lattice equations viewed as the semiclassical limit of an equation involving certain Toeplitz operators. We consider both nonperiodic and periodic boundary conditions. For the nonperiodic case the phase space is the Riemann sphere, while in the periodic case it is the torus C/Z2. In both cases we prove precise estimates on the dispersionless limit. In addition, we show that the Toda equations, although they are nonlinear, propagate a Toeplitz operator into an operator arbitrarily close to a Toeplitz operator as long as the Toda partial differential equation (PDE) (dispersionless limit) admits smooth solutions.