From Toda to KdV

Abstract

For periodic Toda chains with a large number $\mathit{N}$ of particles we consider states which are $\mathit{N}^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given C$^2$−functions with mesh size $\mathit{N}^{-1}$. 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 $\mathit{N} \rightarrow \infty$. To this end we construct two Hill operators $\mathit{H}_\pm$—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 $\mathit{N} \rightarrow \infty$ of the eigenvalues at the edges of the spectrum of $\mathit{L}_N$ are of the form $\pm$(2-(2$\mathit{N})^{-2}\lambda^\pm_n$ where ($\lambda^\pm_n$)$_n\geqslant0$ are the eigenvalues of $\mathit{H}_\pm$. In the bulk of the spectrum, the eigenvalues are o($\mathit{N}^{-2}$)-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $\mathit{L}_\mathit{N}$.