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

In quantum physics, one of the main mathematical problems consists in constructing the spectrum of a linear operator defined on a suitable domain of Hilbert space. In most cases, this type of problem cannot be explicitly solved, in other words the eigenvalues of the Hamiltonian cannot be computed algebraically. However, in few cases, some of which turn out to be physically fundamental, the spectrum can indeed be found explicitly. The two major examples of this kind are the celebrated harmonic quantum oscillator and the hydrogen atom (i.e. 3- dimensional Schrödinger equation coupled to an external Coulomb potential). These examples are called exactly solvable in the sense that the full spectrum of the Hamiltonian is found explicitly.

In the last few years, a new class of operators which is intermediate to exactly solvable and non solvable operators has been discovered [1-4]: the quasi-exactly solvable (QES) operators, for which a finite part of the spectrum can computed algebraically.

Although scalar QES operators have been classified in one variable [

More recently, interesting tools for classification of 2 × 2-matrix QES operators in one spatial dimensional [7- 9] and in creation and annihilation operators [

In the Ref. [

A set of necessary and sufficient conditions (i.e. QES conditions) for 2 × 2-matrix operators to preserve a vector space of polynomials have been proposed. These QES conditions constitute the so-called QES analytic method.

This paper is organized as follows: In the Section 2, based on the Ref. [

A general test to check whether a -matrix differential operator H (in a variable) preserves a vector space whose components are polynomials is proposed [

with

.

Here denote homogeneous differential operators, C_{s}, D_{s} are arbitrary constant and are integers. More precisely, the diagonal components of are differential operators and the off-diagonal components

and are respectively proportional to and with and. The operators and have lower degrees in all their components than the corresponding components in.

In order to obtain QES conditions for, the generic vector of the vector space is

Where are complex parameters. As a consequence the 2 × 2-matrices are defined by

The three QES conditions for to have an invariant vector space are as follows [

In the next step, we will apply in a systematic way the previous QES analytic method in order to construct a 2 × 2-matrix QES Hamiltonian associated to a potential depending on the Jacobi elliptic functions [11,12].

In this section we apply the QES analytic method established previously to check whether a particular 2 × 2-matrix operator is QES. We consider Schrödinger N × N-matrix operator with potential depending on the Jacobi elliptic functions of the form [

with

where a_{j}, b_{j} denote real constants (without loss of generality we assume) and V_{I} is symmetric off-diagonal matrix of the form

Note that the above Hamiltonian is to be considered on the Hilbert space of periodic functions on [0, 4 K(k)].

The properties of the Jacobi functions that are useful to make calculations are listed in the relations (12) and (13).

The case corresponds to the Lamé equation [

with

is the matrix identity and denote real constants. Note that the sum is the potential associated to the Hamiltonian H(z).

Using the following change of function (i.e. the gauge transformation), the gauge Hamiltonian is written as follows

where

and

Notice that two operators H and are called equivalent based on the Equation (8).

The relevant change of variable consists in posing

. In particular the differential symbol

is transformed into the following expression

We recall that for generic values of, the Jacobi functions obey the following relations [

These identities as well as the following ones are useful to establish the gauge Hamiltonian (8) in the variable after the prefactor including the Jacobi functions has been extracted [

Referring to the above relations (12) and (13), for, the second term and the third term of the operator of the Equation (9) are written as follows:

Referring to the same relations used previously, for, the second term and the third term of the operator of the Equation (9) are of the following form:

Replacing the terms of the components of the Hamiltonian given by the Equation (9) by the expressions (11), (14)-(17) and considering the change of variable, one can easily check the following components in variable:

The next step is to establish the conditions such that the gauge operator becomes quasi-exactly solvable. The so called QES conditions help to give the values of the real parameters and in terms of and. Indeed remain free parameters and is an integer.

Let us decompose the operator given by its components (18) according to

where

The generic vector of the invariant vector space under the action of the Hamiltonian has the following form as it is given by the Equation (2)

as, the above wave function is written as follows

acting on the wave function, he increases the degree by one unit,

doesn’t change the degree of the wave function, and reduces the degree of the wave function by one unit.

Let the operator acts on the above vector, the components of the vector are then polynomials in whose components are linear in the constants.

As a consequence the vector can be decomposed uniquely according to [

This above vector defines in particular the constant 2 × 2-matrices and which are found as follows

One can easily find The three necessary QES conditions for the operator to have a finite dimensional invariant vector space are successively obtained [

1) The first QES condition is

2) the second QES condition is as follows

In this above equation replacing by its value (25) and after some algebraic manipulations, the second QES condition is obtained

3) finally the third QES condition for the operator to have a finite dimensional invariant vector space (i.e. the operator is said quasi-exactly solvable) is obtained by the condition involving the matrix as

where is a constant and

The above expression is given by the first QES condition

After some algebraic manipulations, the Equations (27) and (28) lead to the third QES condition

Now, referring to the QES conditions given by the Equations (25), (26) and (29), we are allowed to conclude that the operator (therefore) is quasi-exactly solvable [

Along the same lines applied for the previous case, i.e. for the case δ = 1, one has to perform a gauge transformation according to

after some algebraic manipulations, the components of the above Hamiltonian are of the following form

with

and the operator is given by the Equation (7).

Referring to the relations (12) and to the table of identities given by the Equation (13), the second term and the third term of the operator (31) are of the following form

with.

For, the same relations (12) and the same table of identities (13) lead to the following second term and the third term of the operator (31):

Referring to the relations (11), (32), (33), (34), (35) and after performing the change of variable, the different components of the Hamiltonian given by the Equation (31) take the following form Decomposing now the above operator according the Equation (11), we obtain

The generic element of the invariant vector space under the action of the operator is given by the Equation (2) as in the QES analytic method

the case leads to

Notice that the above operators and given by the Equations (37) are respectively the matrix operators which increases, preserves and reduces the degree of the above generic vector given by the Equation (38). As a consequence the vector can be decomposed as follows

where the constant 2 × 2-matrices and can be computed explicitly after a straightforward calculation

where

One can deduce the matrix from the following expression

where

finally the matrix is easily found by

where

Along the same lines used in the QES analytic method, the three necessary conditions (4) for the operator whose components are given by the Equation (36) to be quasi-exactly solvable are successively obtained:

1) the first QES condition is as follows

2) the second QES condition is easily checked

3) Finally the third QES condition is found

In this paper, we have applied the QES analytic method established in the Ref. [

I thank Pr. Yves Brihaye for useful discussions.