PT-Symmetric Matrix Quasi-Exactly Solvable Razhavi Potential

A PT-symmetric Hamiltonian associated with a trigonometric Razhavi potential is analyzed. Along the same lines of the general quasi-exactly solvable analytic method considered in the [1] [2] [3], three necessary and sufficient algebraic conditions for this Hamiltonian to have a finite-dimensional invariant vector space are established. This PT-symmetric 2 2 × -matrix Hamiltonian is called quasi-exactly solvable (QES).


Introduction
In quantum physics, one of the main mathematical problems consists in constructing the set of eigenvalues of a linear operator defined on a suitable domain of a Hilbert space. In most cases, this type of problem cannot be explicitly solved, that is to say that the spectrum of the Hamiltonian cannot be found algebraically. However, in few cases, some of which turn out to be physically fundamental, the spectrum can indeed be computed explicitly. These cases are so called completely solvable (or exactly solvable). The example of this kind is the harmonic quantum oscillator.
In the last few years, the intermediate class between exactly solvable operators and non solvable operators has been discovered. This class is called quasi-exactly solvable. It means that a finite part of eigenvalues associated to this type of operators can be found algebraically [1]- [10].
Another concept that we consider throughout this paper is the PT-symmetric operator. It means that this operator is invariant under the combination of the parity operator P and the time-reversal operator T. Note that the Hamiltonian we analyze is both QES and PT-symmetric. This paper is organized as follows: In Section 2, based on the Ref. [1] [2] [3], we briefly recall the general QES analytic method used to investigate the quasi-exact solvability of 2 2 × -matrix Hamiltonians.
In Section 3, we show in details that the non hermitian Razhavi Hamiltonian is invariant under the combined PT-symmetry.
In Section 4, along the same lines as in the Refs. [

QES Analytic Method
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 systematic lines of this QES analytic method in order to construct a 2 2 × -matrix QES Hamiltonian associated to a PT-symmetric and trigonometric potential.

PT-Symmetric and Trigonometric QES Potential
In this section, we consider the PT-symmetric and trigonometric 2 2 × -matrix cos 2 cos cos cos 2 In order to prove that this above potential V is invariant under the combined action of the parity operator P and the time-reversal operator T (i.e. the combined PT-symmetry), the potential V has to satisfy the following relation [1] [2] [4]: Note that the time-reversal operator T reverses the sign of the complex number i: cos cos 2 sin sin 2 , After the similar algebraic manipulations used in the previous equation, one can easily check that the other three components elements of the potential PT V given by the Equation (12) satisfy the following relations: Referring to the previous relations (13) and (14), we are allowed to write that

PT-Symmetric Hamiltonian and His Quasi-Exact Solvability
In this section, we apply the QES analytic method established in the section 2 to prove that the PT-symmetric Hamiltonian given by the Equation (8) is quasi-exactly solvable.
In order to reveal the quasi-exact solvability of the previous Hamiltonian H, it is therefore necessary to transform H with the gauge function as follows [1] [2] where the gauge function is written as follows The next step is to perform the change in the variable z ( ) in order to find the final form of the above gauge Hamiltonian.
In the following, we will consider in details the cases 12 cos H C x = and   (21) Taking account to the change in variable z given by the Equation (19), one can easily find that The function ψ can be written as follows: Referring to the Equation (23), one can find that Considering the relation (23) and replacing the parameters , , , ε φ φ ε

H H H H
Note that the action of these above three gauge components of H  given by the Equation (27) on the wave function ψ given by the relation (20) leads to the following expressions: After some algebraic manipulations, one can easily obtain the 2 2 × -matrices ( ) Replacing in the relations (29) respectively  1) The first QES condition is obtained as follows 2) The second QES condition is of the following the form where Λ is a constant.

Conclusion
In this paper, we have applied the general QES analytic method established in the Ref. [1] in order to prove that the PT-symmetric 2 2