Isochronous systems and orthogonal polynomials

Abstract

The theory of orthogonal polynomials has seen many remarkable developments during the last two decades, due to its connections with integrable systems, spectral theory and random matrices. Indeed, in recent years the interest for this theory has often arisen from outside the orthogonal polynomial community after their connection with integrable systems was found. In this thesis the study has been restricted to classical integrable, and isochronous, dynamical systems and a new connection with orthogonal polynomials is presented. The main results in this thesis are related to Diophantine findings obtained from an important class of integrable systems: Isochronous systems. The most famous of these systems is the harmonic oscillator. We explain in detail the properties of these systems and how we can construct isochronous systems from a large class of integrable systems.We give the proof of one Diophantine conjecture and we will see the connection of the Diophantine properties with orthogonal polynomials and their complete factorization. We identify classes of orthogonal polynomials defined by three term recursion relations depending on a parameter ν , which satisfy also a second recursion involving that parameter, and some of which feature zeros given by formulas involving integers. After, we apply the machinery developed previously to all the polynomials of the Askey scheme. For these polynomials we identify other, new, additional recursion relations involving a shift of some parameters that they feature. For several of these polynomials we obtain factorization formulas for special values of their parameters. We show the connection of our machinery with the discrete integrability, comparing the three term recursion relation with a spectral problem involving a discrete Schr¨dinger operator, o and the second recursion with a discrete time evolution for the eigenfunctions. Following the Lax technique developed in the last three decades we will construct an entire hierarchy of equations, and we will see the relation of this hierarchy with the hierarchy of the discrete time Toda lattice. Finally, we present another approach of the our machinery applied to integrable ODE. We consider the stationary KdV’s hierarchy, but this general procedure could be extended to various soliton hierarchies.