Quasi-Exactly Solvable Jacobi Elliptic Potential

A new example of 2 2 × -matrix quasi-exactly solvable (QES) Hamiltonian which is associated to a Jacobi elliptic potential is constructed. We compute algebraically three necessary and sufficient conditions with the QES analytic method for the Jacobi Hamiltonian to have a finite dimensional invariant vector space. The matrix Jacobi Hamiltonian is called quasi-exactly solvable.


Introduction
In quantum mechanics, the goal consists in computing the eigenvalues of linear Hamiltonian. In most cases, the spectrum of the Hamiltonian cannot be calculated algebraically. However, in few cases, some of which have the eigenvalues found explicitly. This type of Hamiltonian is called exactly solvable.
In the last few years, a new class of Hamiltonians which is intermediate to exactly solvable and non-solvable Hamiltonians has been discovered: the quasi-exactly solvable operators, for which a finite part of the eigenvalues can be computed algebraically. Many examples of QES Hamiltonians are studied in [1]- [13].
In the Refs. [10] [11] [12] [13], the QES analytic method is applied in order to establish a set of three necessary and sufficient conditions for Hamiltonians to have finite dimensional invariant vector spaces.
In this paper, we apply the same QES analytic method established in the Refs.
This paper is organized as follows: in Section 2, based on [10] [11] [12] [13], we briefly recall the QES analytic method used to investigate the quasi-exact solvability of 2 2 × -matrix operators. In Section 3, along the same lines as in the 1 δ = and the case 2 δ = .
The interest results will be computed.

QES Analytic Method
Taking account to the same lines as in [10] [11] [12] [13], we recall a general method to check whether a 2 2 × -matrix differential operator H (in a variable x) preserves a vector space whose components are polynomials.
Consider the 2 2 × -matrix Hamiltonian of the following form [10] [11] [12] [13]: which can be written in his components as follows More precisely, the diagonal components of 1 H  are differential operators and the off-diagonal components ( ) In order to obtain the QES conditions for H  , the generic vector of the above vector space is of the form The three necessary and sufficient QES conditions for H  to have an invariant vector space are 1) In the next step, we will apply in a same lines of this QES analytic method in order to prove the quasi-exact solvability of the 2 2 × -matrix QES Hamiltonian associated to Jacobi Elliptic Potential.

Case δ = 1
In this section, we apply the QES analytic method established in previous section to check whether the 2 × 2-matrix operator is quasi-exactly solvable. We consid- where 2 1 is the matrix identity, 1 2 1 2 , , , , Note that the sum D I V V + is the Jacobi elliptic potential associated to the previous Hamiltonian H(z).
Using the following the gauge transformation, the gauge Hamiltonian is written as follows The relevant change of variable consists in posing Taking account to the reference [9] [11], the differential symbol We recall that for generic values of k, the Jacobi functions obey the following The following identities are used to establish the gauge Hamiltonian (11)   Referring to the above relations (16), for 1 g cn = , the second term and the third term of the operator 11 H  of the Equation (12) are written as follows: Referring to the same identities given by the Equation (16) The next step is to establish the QES conditions of the gauge Hamiltonian. In other words, we put out the expressions of the real parameters 1 , a b and θ .
Let us express the gauge Hamiltonian H  given by the above relations (24) in its components according to As 1 δ = , the generic wave function ψ of the gauge Hamiltonian given by the Equation (11) is written as follows Note that the action of these above three gauge components of H  given by the relations (26) on the wave function ψ given by the relation (27) leads to the following expressions: After some algebraic manipulations, one can easily obtain the 2 2 × -matrices Taking account to these above expressions given by the Equation (29)   1) The first QES condition is 2) The second QES condition is as follows 3) The final and the third QES condition is computed as follows N k n k n k n k n k n k n k n a a n a n a k n a a k k k k k

Conclusion
In this paper, we have applied the QES analytic method in order to construct a 2 2 × -matrix QES Hamiltonian which is associated to a Jacobi elliptic potential.
For both two cases considered, 1 δ = and 2 δ = , more precisely, we have computed the three necessary and sufficient algebraic QES conditions for the Jacobi elliptic Hamiltonian to have an invariant vector space.