Matrix Jacobi Elliptic Hamiltonian and Quasi-Exact Solvability

Résumé

We construct a new example of 2 × 2 -matrix quasi-exactly solvable (QES )Hamiltonian which is associated to a potential depending on the Jacobi elliptic functions. We apply the QES analytic method in order to establish three necessary and sufficient algebraic conditions for the previous Hamiltonian to have an invariant vector space whose generic elements are polynomials. This Hamiltonian is called quasi-exactly solvable.