Quasi-exact-solvability of the A₂/G₂ Elliptic model: algebraic forms, sl(3)/g⁽²⁾ hidden algebra, polynomial eigenfunctions

The potential of the A₂ quantum elliptic model (3-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through Weierstrass wp-function and has a single coupling constant. A change of variables has been found, which are A₂ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown that the model possesses the hidden sl(3) algebra - the Hamiltonian is an element of the universal enveloping algebra U_sl(3) for arbitrary coupling constant - thus, it is equivalent to sl(3)-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial, is constructed explicitly, being also an element of U_sl(3). It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the G₂ quantum elliptic model (3-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through Weierstrass wp-function and has two coupling constants. A change of variables has been found, which are G₂ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown the model possesses the hidden g⁽²⁾ algebra. It is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a (non-singular) gauge factor occur.