Double Inozemtsev Limits of the Quantum DELL System

TL;DR

This paper explores complex limits of the quantum double elliptic system, revealing connections to elliptic Ruijsenaars-Schneider Hamiltonians and providing eigenvalue expressions in terms of eigenfunctions.

Contribution

It introduces novel double Inozemtsev limits of the quantum DELL system, linking it to eRS Hamiltonians and deriving eigenvalue formulas based on eigenfunctions.

Findings

Double Inozemtsev limits produce elliptic Ruijsenaars-Schneider Hamiltonians.

Unconventional normalization of Hamiltonians in certain regimes.

Formal eigenvalue expressions in terms of eigenfunctions.

Abstract

In this letter we study various Inozemtsev-type limits of the quantum double elliptic (DELL) system when both elliptic parameters are sent to zero at different rates, while the coupling constant is sent to infinity, such that a certain combination of the three parameters is kept fixed. We find a regime in which such double Inozemtsev limit of DELL produces the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians albeit in an unconventional normalization. We discuss other double scaling limits and anisotropic scaling of coordinates and momenta. In addition, we provide a formal expression for the eigenvalues of the eRS Hamiltonians solely in terms of their eigenfunctions.