From Toda to KdV

For periodic Toda chains with a large number N of particles we consider states which are N−2-close to the equilibrium and constructed by discretizing arbitrary given C2−functions with mesh size N−1. Our aim is to describe the spectrum of the Jacobi matrices LN appearing in the Lax pair formulation of the dynamics of these states as N → ∞. To this end we construct two Hill operators H± – such operators come up in the Lax pair formulation of the Korteweg-de Vries equation – and prove by methods of semiclassical analysis that the asymptotics as N → ∞ of the eigenvalues at the edges of the spectrum of LN are of the form ±(2− (2N)λn + · · · ) where (λn )n≥0 are the eigenvalues of H±. In the bulk of the spectrum, the eigenvalues are o(N−2)-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to LN .