Description
We begin by introducing central examples of Schr¨odinger operators and
Jacobi matrices with ergodic coefficients. As we will see, such operators open a Pandora’s
box of spectral theory: The spectrum can be a Cantor set and the spectral type can be
anything, absolutely continuous, singular continuous, or even a dense set of eigenvalues.
Our model classes of operators are singled out for their relevance to the Korteweg–de
Vries and Toda evolutions. After a brief review of the analysis of the periodic problem,
we will demonstrate how one goes about solving these evolutionary PDEs using the
spectral theory of ergodic operators.