From Toda to KdV

For periodic Toda chains with a large number N of particles we consider states which areN(-2)-close to the equilibrium and constructed by discretizing arbitrary given C-2 -functions with mesh sizeN(-1). Our aim is to describe the spectrum of the Jacobi matrices L-N appearing in the Lax pair formulation of the dynamics of these states as N -> infinity. 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 -> infinity of the eigenvalues at the edges of the spectrum of L-N are of the form +/-(2 - (2N)(-2) lambda(+/-)(n) + ...) where (lambda(+/-)(n))(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 L-N.