From Toda to KdV. (22nd June 2015) - Bambusi, D; Kappeler, T; Paul, T
Abstract: 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 C 2 −functions with mesh size N −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  → ∞. 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 L N are of the form where 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 .