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. [1] [2] [3], we apply the QES analytic method in order to prove the quasi-exact solvability of the 2 × 2 -matrix Hamiltonian depending on the PT-symmetric potential. Three necessary and sufficient QES conditions for the 2 × 2 -matrix trigonometric Hamiltonian to have a finite dimensional invariant vector space will be established for two cases: ψ = ( p n , cos x q n − 1 ) t and ψ = ( p n − 1 cos x , q n ) t .

Taking account to the same lines as in the Refs. [1] [2] [3], 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 [1] [2] [3]:

H = ( − d 2 d x 2 + V 11 ( x ) x δ x δ ′ − d 2 d x 2 + V 22 ( x ) ) (1)

where

V ( x ) = ( V 11 ( x ) x δ x δ ′ V 22 ( x ) ) , δ = 0 , 1 , 2 , δ ′ = 1 − δ , V 12 ( x ) = x δ , V 21 ( x ) = x δ ′

V ( x ) is the potential associated to the Hamiltonian H given by this above relation (1).

A gauge transformation and a change of variable on the Hamiltonian H lead to the following Hamiltonian called the gauge one

H ˜ = ϕ − 1 H ϕ (2)

which can be written in his components as follows

H ˜ = H ˜ 1 + H ˜ 0 + H ˜ − 1 . (3)

More precisely, the diagonal components of H ˜ 1 are differential operators and the off-diagonal components ( H ˜ 1 ) 12 and ( H ˜ 1 ) 21 are respectively proportional to x δ and x δ ′ . The operators H ˜ 0 and H ˜ − 1 have lower degrees in all their components than the corresponding components in H ˜ 1 .

Note that the invariant vector space of the Hamiltonian H ˜ has the following form [1] [2] [3]:

W = { ( p n q n ) } , m = n − δ + 1 and n , m ∈ Ν . (4)

In order to obtain the QES conditions for H ˜ , the generic vector of the above vector space is of the form

φ = ( α 0 x n + α 1 x n − 1 β 0 x n − δ + 1 + β 1 x n − δ ) , (5)

where α i , β i ( i = 0 , 1 ) are complex parameters. As a consequence the 2 × 2 -matrices M 1 , M ˜ 1 , M 0 are defined by

H ˜ 1 ( α 0 x n β 0 x n − δ + 1 ) = d i a g ( x n + 1 , x n − δ + 2 ) M 1 ( α 0 β 0 ) ,

H ˜ 1 ( α 1 x n − 1 β 1 x n − δ ) = d i a g ( x n , x n − δ + 1 ) M ˜ 1 ( α 1 β 1 ) ,

H ˜ 0 ( α 0 x n β 0 x n − δ + 1 ) = d i a g ( x n , x n − δ + 1 ) M 0 ( α 0 β 0 ) . (6)

The three necessary and sufficient QES conditions for H ˜ to have an invariant vector space are

1) M 1 ( α 0 β 0 ) = ( 0 0 ) ,

2) M ˜ 1 t ( − β 0 α 0 ) = ( 0 0 ) ,

3) M 0 ( 1 β 0 α 0 ) = Λ ( 1 β 0 α 0 ) . (7)

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.

In this section, we consider the PT-symmetric and trigonometric 2 × 2 -matrix Hamiltonian [1] [2]:

H = ( − d 2 d x 2 + ( ρ cos 2 x − i M ) 2 + A H 12 H 21 − d 2 d x 2 + ( ρ cos 2 x − i M ) 2 + A ˜ ) , (8)

where A , A ˜ , M , M ˜ , ρ are free real parameters.

The non diagonal components of the Hamiltonian H (i.e. H 12 , H 21 ) are written as follows

H 12 = C cos x ,

H 12 = C ˜ cos x , (9)

with C and C ˜ are the coupling constants.

Now, we can reveal the PT-symmetry property of the trigonometric non hermitian 2 × 2 -matrix potential associated to the above Hamiltonian given by the Equation (8)

V = ( V 11 V 12 V 21 V 22 ) ,

V = ( ( ρ cos 2 x − i M ) 2 + A C cos x C ˜ cos x ( ρ cos 2 x − i M ) 2 + A ˜ ) (10)

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]:

V P T = V (11)

Note that the time-reversal operator T reverses the sign of the complex number i:

T i T − 1 = − i

while the parity operator replaces x by π 2 − x leading to

V P T = ( [ ρ cos ( π − 2 x ) + i M ] 2 + A C cos ( π 2 − x ) C ˜ cos ( π 2 − x ) [ ρ cos ( π − 2 x ) + i M ˜ ] 2 + A ˜ ) (12)

with

V 11 P T ( x ) = [ ρ cos ( π − 2 x ) + i M ] 2 + A ,

V 11 P T ( x ) = ρ 2 ( cos π cos 2 x + sin π sin 2 x ) 2     + 2 i ρ M ( cos π cos 2 x + sin π sin 2 x ) − M 2 + A ,

V 11 P T ( x ) = ρ 2 cos 2 2 x − 2 i ρ M cos 2 x − M 2 + A ,

V 11 P T ( x ) = ( ρ cos 2 x − i M ) 2 + A ,

V 11 P T ( x ) = V 11 ( x ) . (13)

After the similar algebraic manipulations used in the previous equation, one can easily check that the other three components elements of the potential V P T given by the Equation (12) satisfy the following relations:

V 12 P T ( x ) = V 12 ( x ) ,

V 21 P T ( x ) = V 21 ( x ) ,

V 22 P T ( x ) = V 22 ( x ) . (14)

Referring to the previous relations (13) and (14), we are allowed to write that

V P T ( x ) = V ( x ) . (15)

Thus, taking account to this Equation (15), we can conclude that the potential V ( x ) is invariant under the combined action of the parity operator P and the time-reversal operator T. In other words, the potential V ( x ) given by the Equation (10) is PT-symmetric. As a consequence, it is this PT-invariant symmetry property which explains the reality of some of the eigenvalues associated to the above trigonometric potential V ( x ) [1] [2] [3].

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] [3]:

H ˜ = R − 1 H R ,

H ˜ = e − θ cos 2 x ( z − ε ( 1 − z ) − ϕ 0 0 z − ε ˜ ( 1 − z ) − ϕ ˜ ) H e θ cos 2 x ( z ε ( 1 − z ) ϕ 0 0 z ε ˜ ( 1 − z ) ϕ ˜ ) , (16)

where the gauge function is written as follows

R ( x ) = e θ cos 2 x ( z ε ( 1 − z ) ϕ 0 0 z ε ˜ ( 1 − z ) ϕ ˜ ) (17)

and the gauge Hamiltonian is of the following form

H ˜ = ( H ˜ 11 H ˜ 12 H ˜ 21 H ˜ 22 ) . (18)

The next step is to perform the change in the variable z

z = cos ( 2 x + 1 ) 2 (19)

in order to find the final form of the above gauge Hamiltonian.

In the following, we will consider in details the cases H 12 = C cos x and H 12 = C ˜ cos x which are the non diagonal elements of the matrix PT-symmetric Hamiltonian H given by the Equation (8).

In this case, the possible values for the parameters ε , ε ˜ , ϕ , ϕ ˜ allow two cases for the wave function, ψ = ( ψ 1 , ψ 2 ) t which have the following forms:

Case 1:

ψ = ( p n , cos x q n − 1 ) t , (20)

Case 2:

ψ = ( p n − 1 cos x , q n ) t . (21)

If the parameter θ of the gauge function of the Equation (17) is chosen as θ = i ρ 2 , the components of the gauge Hamiltonian H ˜ ( z ) are written as follows:

H ˜ 11 = − 4 z ( 1 − z ) d 2 d z 2 + 2 [ 2 z − 1 − 4 ( 1 − z ) ε + 4 ϕ z ] d d z + ρ 2 − M 2     + 8 ϕ ε + 2 ε + 2 ϕ + A − 8 i ρ [ z ( 1 − z ) d d z + ε ( 1 − z ) − ϕ z + M − 1 4 ( 2 z − 1 ) ]

H ˜ 12 = z ε ˜ − ε ( 1 − z ) ϕ ˜ − ϕ C cos x ,

H ˜ 21 = z ε − ε ˜ ( 1 − z ) ϕ − ϕ ˜ C ˜ cos x ,

H ˜ 22 = − 4 z ( 1 − z ) d 2 d z 2 + 2 [ 2 z − 1 − 4 ( 1 − z ) ε ˜ + 4 ϕ ˜ z ] d d z + ρ 2 − M ˜ 2     + 8 ϕ ˜ ε ˜ + 2 ε ˜ + 2 ϕ ˜ + A ˜ − 8 i ρ [ z ( 1 − z ) d d z + ε ˜ ( 1 − z ) − ϕ ˜ z + M ˜ − 1 4 ( 2 z − 1 ) ] (22)

Taking account to the change in variable z given by the Equation (19), one can easily find that

cos x = z 1 2 . (23)

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 -matrix Razhavi Hamiltonian is quasi-exactly solvable. It means that a finite part of eigenvalues associated to this Razhavi Hamiltonian can be computed algebraically. We have considered two cases for the wave function: ψ = ( p n , cos x q n − 1 ) t and ψ = ( p n − 1 cos x , q n ) t . The three necessary and sufficient QES conditions for the PT-symmetric 2 × 2 -matrix trigonometric Hamiltonian to have a finite vector space invariant under its action are found algebraically.

I thank Pr. Yves Brihaye of useful discussions.

The author declares no conflicts of interest regarding the publication of this paper.

Nininahazwe, A. (2020) PT-Symmetric Matrix Quasi-Exactly Solvable Razhavi Potential. Open Journal of Microphysics, 10, 9-20. https://doi.org/10.4236/ojm.2020.102002