The $BC_{1}$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions

TL;DR

This paper analyzes the algebraic structure and polynomial eigenfunctions of the $BC_1$ elliptic quantum model, revealing hidden $sl(2)$ symmetry and special coupling constants leading to finite polynomial solutions.

Contribution

It demonstrates the hidden $sl(2)$ algebra in the $BC_1$ elliptic model and identifies conditions for polynomial eigenfunctions, advancing understanding of its algebraic and spectral properties.

Findings

Hidden $sl(2)$ algebra exists for all coupling constants.

Three families of coupling constants yield polynomial eigenfunctions.

Potential simplifies to a rational function in invariant variables.

Abstract

The potential of the $BC_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the Lam\'e Hamiltonian. It is shown that in the space of $BC_1$ elliptic invariant, the potential becomes a rational function, while the flat space metric becomes a polynomial. The model possesses the hidden $sl(2)$ algebra for arbitrary coupling constants: it is equivalent to $sl(2)$-quantum top in three different magnetic fields. It is shown that there exist three one-parametric families of coupling constants for which a finite number of polynomial eigenfunctions (up to a factor) occur.