Spectral Theory, Jacobi Matrices, Continued Fractions and Difference Operators

Abstract

The aim of this paper is to describe some connections between spectral theory in infinitedimensional Lie algebras, deformation theory and linearization of nonlinear dynamicalsystems. We explain how results from isospectral deformations, cohomology groups andalgebraic geometry can be used to obtain insight into integrable systems. Another part willbe dedicated to the study of infinite continued fractions and isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical pointof view. Also, the notion of algebraically completely integrable systems is explained andtechniques to solve such systems are presented. Several nonlinear problems in mathematicalphysics illustrate these results.