The $B{C}_1$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions

Abstract

The potential of the $B{C}_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $B{C}_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the Lamé Hamiltonian. It is shown that in the space of $B{C}_1$ 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.

10 pages, some references added, introduction extended

References in corpus (1)

• Particular Integrability and (Quasi)-exact-solvability

Cited by in corpus (3)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Quasi-exact-solvability of the $A_2/G_2$ Elliptic model: algebraic forms, $sl(3)/g^{(2)}$ hidden algebra, polynomial eigenfunctions
• A quasi-exactly solvable model: two charges in a magnetic field, subject to a non-Coulomb mutual interaction