Quasi-exact-solvability of the elliptic model: algebraic forms, hidden algebra, and polynomial eigenfunctions

Abstract

The potential of the A2 quantum elliptic model (three-body Calogero–Moser elliptic model) is defined by the pairwise three-body interaction through the Weierstrass ℘-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 algebra—the Hamiltonian is an element of the universal enveloping algebra for the arbitrary coupling constant—thus, it is equivalent to the -quantum Euler–Arnold top. The integral, in a form of the third order differential operator with polynomial coefficients, is constructed explicitly, being also an element of . 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, occurs. For these values of the coupling constants there exists a particular integral: it commutes with the Hamiltonian in action on the space of polynomial eigenfunctions, and the Hamiltonian is invariant with respect to two-dimensional projective transformations. It is shown that the A2 model has another hidden algebra introduced in Rosenbaum et al (1998 Int. J. Mod. Phys. A 13 3885). The potential of the G2 quantum elliptic model (three-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through the Weierstrass ℘-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 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 occurs. For these values of the coupling constants, there exists a particular integral.