Continued fractions and integrable systems

Stieltjes’ solution of the classical moment problem is the forerunner of inverse spectral theory. In the modern theory of completely integrable systems, the method of inverse scattering is used to obtain explicit solutions of a class of Hamiltonian systems which arise as isospectral deformations of a linear operator. In this lecture we explain how Stieltjesʹs formulas are used to obtain explicit solutions to a number of completely integrable systems, including discrete reductions of some nonlinear partial differential equations, and the isospectral deformations of Jacobi matrices a.k.a. Toda flows.