Quasi-exact-solvability of the $A_{2}/G_2$ Elliptic model: algebraic forms, $sl(3)/g^{(2)}$ hidden algebra, polynomial eigenfunctions

摘要

The potential of the $A_2$ quantum elliptic model (3-body Calogero-Moserelliptic model) is defined by the pairwise three-body interaction throughWeierstrass $\wp$-function and has a single coupling constant. A change ofvariables has been found, which are $A_2$ elliptic invariants, such that thepotential becomes a rational function, while the flat space metric as well asits associated vector are polynomials in two variables. It is shown that themodel possesses the hidden $sl(3)$ algebra - the Hamiltonian is an element ofthe universal enveloping algebra $U_{sl(3)}$ for arbitrary coupling constant -thus, it is equivalent to $sl(3)$-quantum Euler-Arnold top. The integral, in aform of the third order differential operator with polynomial, is constructedexplicitly, being also an element of $U_{sl(3)}$. It is shown that there existsa discrete sequence of the coupling constants for which a finite number ofpolynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the $G_2$ quantum elliptic model (3-body Wolfes ellipticmodel) is defined by the pairwise and three-body interactions throughWeierstrass $\wp$-function and has two coupling constants. A change ofvariables has been found, which are $G_2$ elliptic invariants, such that thepotential becomes a rational function, while the flat space metric as well asits associated vector are polynomials in two variables. It is shown the modelpossesses the hidden $g^{(2)}$ algebra. It is shown that there exists adiscrete family of the coupling constants for which a finite number ofpolynomial eigenfunctions up to a (non-singular) gauge factor occur.