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On the status of DELL systems

A. Mironov, A. Morozov

Abstract

A detailed review of the $p,q$-duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e. trigonometric-elliptic ones), and explain their relations to the bi-elliptic Koroteev-Shakirov (KS) model. The most interesting self-dual double-elliptic (DELL) system remains a mystery, but we provide a clearer formulation of the problem and describe the steps that are still to be done.

On the status of DELL systems

Abstract

A detailed review of the -duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e. trigonometric-elliptic ones), and explain their relations to the bi-elliptic Koroteev-Shakirov (KS) model. The most interesting self-dual double-elliptic (DELL) system remains a mystery, but we provide a clearer formulation of the problem and describe the steps that are still to be done.
Paper Structure (35 sections, 158 equations)

This paper contains 35 sections, 158 equations.

Table of Contents

  1. Introduction
  2. The notation. Symmetric functions.
  3. The notation. Elliptic functions.
  4. Duality map
  5. Duality map in a quantum system
  6. Quasiclassical limit
  7. Classical duality map
  8. Classical duality at $N=2$
  9. A set of simple examples
  10. Harmonic oscillator.
  11. Non-periodic Toda chain (Liouville equation).
  12. Linear potential.
  13. Coulomb potential.
  14. Trigonometric RS model
  15. Periodic Toda chain
  16. ...and 20 more sections