Spectral Theory, Jacobi Matrices, Continued Fractions and Difference Operators

Abstract

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

Keywords

• difference operators
• Jacobian varieties
• Lax equations
• Cohomology group
• Continued fraction
• Deformation
• Integrable systems
• Periodic Jacobi matrices
• Spectral curves

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