Global well-posedness of the Toda lattice on an exact spectral phase space

Abstract

We identify an exact spectral phase space for the two-sided Toda lattice. Let $q=\{a_n,b_n\}_{n\in\mathbb Z}$ be coefficients of the right and left half-line Jacobi operators and denote their spectral measures by $\sigma_{\pm}^{q}$. Define a phase space \[ \mathcal Q=\left\{ \begin{array} [c]{c}% q=\{a_n,b_n\}_{n\in\mathbb Z}: a_n>0,\ b_{n} \in \mathbb{R} \text{ and} \int_{\mathbb R}e^{c|\lambda|}\sigma^q_\pm(d\lambda)<\infty \text{ for every }c>0 \end{array} \right\} . \] The integrability condition makes the representing measures unique. We prove that $q\in\mathcal Q$ if and only if the Toda lattice with initial datum $q$ admits a classical solution for all positive and negative times. Moreover, the solution remains in $\mathcal Q$, is unique, and depends continuously on the initial datum, uniformly on compact time intervals.