Study about Matrix Quasi-Exactly Solvable Jacobi Elliptic Hamiltonian

Abstract

In the last few years, a new class of operators which is intermediate to exactly solvable and non-solvable operators has been discovered: the quasi-exactly solvable (QES) operators, for which a finite part of the spectrum can computed algebraically. A new example of a 2 × 2 -matrix quasi-exactly solvable (QES) Hamiltonian was constructed which is associated with a potential depending on the Jacobi elliptic functions. The QES analytic method was applied in order to establish three necessary and sufficient algebraic conditions for the 2 × 2 -matrix Hamiltonian to have an invariant vector space whose generic elements are polynomials. This Hamiltonian is called quasi-exactly solvable.