New families of exactly solvable many-body models and superintegrable systems

Abstract

In this thesis, we study various extensions of Calogero models and superintegrable systems. We construct a kN-body one-dimensional model which reduces to the familiar Calogero model when k = 1. We present a class of many-body systems that are equivalent to harmonic oscillators. We study interesting extensions of the D-dimensional Coulomb-Kepler system and show that when the extension satisfies certain conditions, then some components of the Laplace-Runge-Lenz vector can be extendedto conserved quantity of the new models. By introducing block separation of variables, we construct the Kepler-singular oscillator type models which are a new family of superintegrable systems. We also use separation of variables to obtain the energy spectrum, eigenfunctions and corresponding quadratic algebraic structures. Separation of variables is not only used for solving eigenvalue problems but also provides us with new tools for generalizing superintegrable systems. We generalize the double harmonic-singular oscillators and Kepler-singular oscillator systems. The integrals of new models now can involve angles from block spherical coordinates. A derivation of more general quadratic algebraic structures is presented as well. We also give examples to show they can be solved in terms of so-called X1 Jacobi polynomials.