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Baker-Akhiezer specialisation of joint eigenfunctions for hyperbolic relativistic Calogero-Moser Hamiltonians

Martin Hallnäs

Abstract

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_N$ for quantum Hamiltonians of the hyperbolic relativistic $N$-particle Calogero--Moser system. For generic coupling values, they are non-elementary functions that in the $N=2$ case essentially amount to a `relativistic' generalisation of the conical function specialisation of the Gauss hypergeometric function ${}_2F_1$. In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions $ψ_N$ of Baker--Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that $J_N$ essentially amounts to the antisymmetrisation of $ψ_N$ and, as a byproduct, we obtain a recursive construction of $ψ_N$ in terms of an iterated residue formula.

Baker-Akhiezer specialisation of joint eigenfunctions for hyperbolic relativistic Calogero-Moser Hamiltonians

Abstract

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions for quantum Hamiltonians of the hyperbolic relativistic -particle Calogero--Moser system. For generic coupling values, they are non-elementary functions that in the case essentially amount to a `relativistic' generalisation of the conical function specialisation of the Gauss hypergeometric function . In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions of Baker--Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that essentially amounts to the antisymmetrisation of and, as a byproduct, we obtain a recursive construction of in terms of an iterated residue formula.

Paper Structure

This paper contains 7 sections, 10 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Joint eigenfunctions
  4. Baker--Akhiezer functions
  5. Baker--Akhiezer specialisation
  6. The $N=2$ case
  7. The inductive step

Key Result

Theorem 3.1

Let $m\in\mathbb{Z}_+$ and let $a>m-1$ be such that $\exp(-i\pi/a)$ is not a root of unity. Then, we have with the constant

Theorems & Definitions (15)

  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • proof
  • ...and 5 more