Integrable Systems and Random Matrix Theory

In this lecture, we consider the famed doubly-infinite Toda lattice which is completely-integrable. We present the inverse scattering transform method for the solution of the Cauchy initial value problem for sufficiently decaying initial data. As is well known, the Toda lattice equations can be recast as an isospectral flow on Jacobi matrices and this gives rise to the existence of a Lax pair. Thus we move on to cover scattering theory for Jacobi matrices, introduce the scattering transform and scattering data associated with a Jacobi matrix. Then we cover the time evolution of the scattering data under the dynamics induced by the Toda lattice equations and present the Riemann-Hilbert formulation of the inverse scattering transform. We review some results on long-time asymptotics of the solutions of the Cauchy initial problem for sufficiently decaying initial data. Time permitting, we plan to consider Hamiltonian perturbations of the Toda lattice. Speaker(s): Deniz Bilman (University of Michigan)