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Ruijsenaars-van Diejen-Takemura Hamiltonians as rational Heun operators

Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov

Abstract

The most general Ruijsenaars-van Diejen-Takemura Hamiltonians are characterized as Heun operators defined as second order $q$-difference operators with a raising action on elementary rational functions with poles on the Askey-Wilson grid.

Ruijsenaars-van Diejen-Takemura Hamiltonians as rational Heun operators

Abstract

The most general Ruijsenaars-van Diejen-Takemura Hamiltonians are characterized as Heun operators defined as second order -difference operators with a raising action on elementary rational functions with poles on the Askey-Wilson grid.
Paper Structure (15 sections, 53 equations)

This paper contains 15 sections, 53 equations.

Table of Contents

  1. Introduction
  2. Raising properties
  3. An intermediary step: the operator $W_{\!\text{AW}}$
  4. General properties
  5. Restricting $W$ to the Askey-Wilson grid
  6. The Heun operator $W^{(1)}_{\!\text{AW}}$ associated to one series of poles on the Askey-Wilson grid
  7. The Heun operator $W^{(2)}_{\!\text{AW}}$ associated to two series of poles on the Askey-Wilson grid
  8. Two additional conditions
  9. Construction of $W_{\!\text{\mdseries AW}}^{(2)}$
  10. Sufficiency
  11. Identification with the Takemura operator ${\mathcal{A}}^{\langle 1\rangle}$
  12. $W_{\!\text{\mdseries AW}}^{(2)}$ and ${\mathcal{A}}^{\langle 1\rangle}$
  13. $W_{\!\text{\mdseries AW}}^{(1)}$ and ${\mathcal{A}}^{\langle 1\rangle}$
  14. Concluding remarks
  15. Derivation of $\hat{W}_{\!\text{AW}}^{(2)}$ and $\hat{{\mathcal{W}}}$ from the Ruijsenaars-van Diejen Operator

Theorems & Definitions (3)

  • Definition 2.1
  • Remark 5.1
  • Remark 5.2