# Isospectral deformations, the spectrum of Jacobi matrices, infinite continued fraction and difference operators. Application to dynamics on infinite dimensional systems

## TL;DR

This paper explores the connections between isospectral deformations, Jacobi matrices, continued fractions, and difference operators, applying algebraic geometry to analyze integrable systems and their dynamics in infinite-dimensional settings.

## Contribution

It introduces new algebraic geometric methods to study isospectral deformations and integrable systems, linking spectral theory with infinite continued fractions and difference operators.

## Key findings

- Connections established between coadjoint orbits and integrable dynamics
- Algebraic geometric techniques applied to infinite continued fractions and Jacobi matrices
- Examples provided from mathematical physics demonstrating the theory

## Abstract

This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory, cohomology groups and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. Another part will be dedicated to the study of infinite continued fraction, orthogonal polynomials, the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Some connections with Cauchy-Stieltjes transform of a suitable measure and Abelian integrals are given. Finally the notion of algebraically completely integrable systems is explained, techniques to solve such systems are presented and some interesting cases appear as coverings of such dynamical systems. These results are exemplified by several problems of dynamical systems of relevance in mathematical physics.