The Generalized PT-Symmetric Sinh-Gordon Potential Solvable 
within Quantum Hamilton-Jacobi Formalism
Özlem Yeşiltaşa, S. Bilge Ocakb,*
a Gazi University, Faculty of Arts and Sciences, Department of Physics, 06500, Teknikokullar, 
Ankara, Turkiye
b Sarayköy Nuclear Research and Training Center
Kazan, Ankara, Turkiye.
Abstract
The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework
of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh-
Gordon potential  are found for 0, 1n   states. 
Keywords: Quantum Hamilton Jacobi,  Sinh-Gordon Potential.
* Corresponding Author: semamuzo@yahoo.com
1. Introduction
The discovery of new class of physically significant spectral problems, called  quasi-exactly solvable 
(QES) models, has attracted much attention [1-3]. Several methods [4-5] for the generation of QES 
model have been worked out. These are the models for which a part of the bound state energy 
spectrum and corresponding wave functions can be obtained exactly. These models have been 
constructed and studied extensively by means of Lie algebraic approach. In order that a part of the 
spectrum can be obtained exactly, the parameters appearing in the relation must satisfy a condition 
known as the condition for quasi-exact solvability. 
Within the Quantum Hamiltonian Jacobi approach (QHJ) [6-7], it has been found to be an elegant and 
the simple method to determine the energy spectrum of exactly solvable models in quantum 
mechanics. The advantage of this method is that it is possible to determine the energy eigen-values 
without having to solve for the eigen-functions. In this formalism, a quantum analog of classical action 
angle variables is introduced [6-7] . The quantization condition represents well known results on the 
number nodes of the   wave function, translated in terms of logarithmic derivative, also is called 
quantum momentum function (QMF) [3, 6-7]. The equation satisfied by the QMF is a non-linear 
differential equation, called quantum Hamilton-jacobi equation leads to two solutions. The application 
of QHJ to eigen-values has been explored in great detail in ref. [3, 6-9].
In recent years, PT -symmetric Hamiltonians have generated much interest in quantum mechanics
[10-11]. Physical motivation for the PT - symmetric but non-Hermitian Hamiltonians have been 
emphasized by many authors [12-16]. Recently, Mostafazadeh [16, 17] has introduced a different 
concept  which is known as the class of pseudo-Hermitian Hamiltonians, and argued that the basic 
structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-
Hermiticity [17]. Following these detailed works, non-Hermitian Hamiltonians with real or complex 
spectra have been  analyzed by using various numerical  and   analytical techniques [18-22]. In the 
applications, Ranjani and her collaborators applied the QHJ formalism, to Hamiltonians with Khare-
Mandal potential and Scarf potential, characterized by discrete parity and time reversal PT
symmetries [3].
The aim of the present work is to calculate the energy eigenvalues and the corresponding 
eigenfunctions of the Sinh-Gordon potential field which is used in coherent spin states [23]. Originally 
the form of the potential is a hyperbolic one given as 
                                       
( ) sinh ( ) cosh
V x x B S x
         (1)        
We  study  general form of this hyperbolic effective potential which is called as Sinh-Gordon potential 
and is also in PT-symmetric form where the  potential parameters 1V and 2V  are real [23],
               xCoshVxSinhVxV  2
1)(                  (2)
in a Hamiltonian system generating quasi-exactly-solvable problems.
In this letter, we first give a brief description of the QHJ formalism. The Sinh-Gordon potential is 
studied in one dimension. In each of these cases the condition of quasi-exactly solvability is derived 
within the QHJ approach. The energy and eigenfunctions of the generalized hyperbolic effective 
potential   are obtained by using the QES condition.  
2. Quantum Hamiltonian-Jacobi Formalism
In the quantum Hamilton-Jacobi (QHJ) formalism, the logarithmic derivative of the wave function 
)(x is given  by [3, 6-7] :
          )(ln
ip                                            (3)
which is called as quantum momentum function QMF.  QMF plays an important role since it is stated 
analogous to the classical momentum function as, 
p   where S is the Hamilton’s characteristic
function and it is related to the wave function by 
x exp)( . Substituting )(x  in terms of S
in the Schrödinger equation  EH   and using the potential relation in Eq.(2), the following 
expression is obtained:
                                 )(2'2 xVEmpip   .                                    (4)
Leacock and Padgett [6-7] proposed the quantization condition for the bound states in order to find the 
eigenvalues. Quantum action variable is defined as QMF and exact quantization condition for the 
bound states of a real potential is given as [3, 6-7]
npdx
                                                   (5)
where,  C is the contour enclosing the n moving poles in the complex domain [3, 6-7].
3. The Sinh-Gordon Potential 
The Sinh-Gordon potential which is given by Eq.(2) is PT  symmetric if  1V  and 2V  are real. It is 
worth noting that, parity operation is given by pp  , xx    in the Hamiltonian whereas the 
time reversal remains same as the conventional  pp  ,  xx   and ii  . The QHJ equation
with 12  m , can be introduced as:
                           0]coshsinh[ 2
2  xVxVE
p         (6)
In order to transform Eq. (6) into a rational form, one can  change the variable xy cosh  in Eq.(6) 
and obtain the following relation:  
                   0])1([)(1)( 2
'22  yVyVEypyiyp     (7)
The coefficient of the 1
   in Eq.(7). Thus, in order to bring it to the Riccati type equation form, 
p  can be defined as;
 12  yip                                       (8)
And and another transformation is given as below,
 12 2  y
 .      (9)
Using Eqs.(8) and (9) in  Eq.(7), one can obtain a Riccati type equation as
          
   
  0
12222
2' 
 VyVyVE
      (10)
Assume that   has a finite number of moving poles in the complex y plane. In addition to,   has 
not only moving poles, but also fixed poles at 1y   and it is bounded at y . Then,   can be 
written in the form of,
11                            (11)
where 1b  and 
1b  are the residues at 1y . )(yPn  is a polynomial of degree n  and C  is a constant
due to the Liouville’s theorem (for more details, see [3] ) . At 1y ,   is expanded in a Laurent 
series as:
...)1(
1 
 yaa
                   (12)
Substituting Eq.(12) in Eq. (10), one obtains the values of residue 1b  as:
1 b                                               (13)
and also 
a   is found. For large y , 
  which are the values of C . If  the same 
procedure is applied to   for 1y
1 
 yaa
                   (14)
One can obtain the residue at 1y  as
                                    
1 b                                                          (15)
In order to the discuss the behaviour of   at infinity, one expands   as:
0 
                             (16)
Substitution of Eq.(16) in (10)  gives 
                            (17)
Behaviour of   approaches  to 
nbb  '11  for large y . Hence one can obtain [3],
 nbb '11                                          (18)
It can be seen that Eq.(18)  is positive. Therefore, one can choose positive value of   which is 
 and it leads to choose 
Ca   because of the physical solutions for the 
wavefunction. In order to obtain a QES condition, one must take all possible combinations of the 
residues '11 , bb . These combinations with a constraint  are given in table 3.1 as:
Table3.1. The QES condition and the n  for each '11 , bb .
1b 2b
11 bbn   Condition on
QES condition
     
1M M : odd,               1M 12  nM
3M M : odd,               3M 32  nM
2M M : even,               2M 22  nM
2M M : even,               2M 22  nM
                                                                                    
As it is seen from the table3.1 that sets 1 and 2 are valid if and only if M  is odd, sets 3 and 4 are valid 
if and only if M  is even. When it comes to the form of the wavefunction, using   dxpix exp)(
and writing p in terms of  , it is obtained as
  
exp)(
               (19)
Substitute Eq.(19) in (10) to obtain,
   
     .011
yVyVE
               (20)
Using Eq.(20), one can find the energy eigenvalues and corresponding wavefunctions for 3M  and 
2M  cases. For 3M  state, it can be seen that 1n  and  the )(yPn  is a first order polynomial 
given as   yP1 . The results  are illustrated in table3.2 and table3.3 respectively. 
Table3.2.  When 3M  and for set 1-2, the energy eigenvalues and wave-functions.
3M           '11 , bb              Energy, nE       Wave- function n
Set 1
11  bb , 1n
          
 
cosh)(
Set 2
11  bb , 0n
          20 E  xex
 sinh)(
where 1 and 
  are found for 0)sgn( 1 Vc  and 
 is found 
for 0)sgn( 1 Vc . Similarly for 2M case, the table3.3 is given as
Table3.3.  When 2M  and for set 3-4, the energy eigenvalues and wave-functions.
2M           '11 , bb              Energy, nE       Wave- function n
Set 3
11  bb , 0n
          
 1cosh)(
Set 4
11  bb , 0n
          
 1cosh)(
From table3.2 and table 3.3, the energy eigenvalues are obtained for the odd and even values of  
 for Sinh-Gordon potential.
If the complex forms of the potential given in Eq.(2) are written for 2 as below,
xiVxVxV 2cosh2sinh)( 2
1                                                       (21)
xVxiVxV 2cosh2sinh)( 2
1       (22)
the solutions of QHJ equation for these complex potentials can be discussed. Eq.(21) is PT
symmetric under parity reflections as x
, ii   and Eq.(22) is non- PT  symmetric.  If we 
take  Eq.(21) into consideration, values of  can be found by following the same procedure as,
     (23)
which also means the values of M  in the table3.1 are complex. The same procedure is valid for the 
potential relation in Eq.(22), too. Hence, these conditions don’t lead to physical solutions (for the 
wavefunctions). But, in ref.[3] the potential has the form given below
 22cosh)( iMxxV        (24)
which is PT  symmetric under  parity reflections x
, ii   and leads to real energy 
spectrum [3]. 
Conclusions
In this letter  quantum Hamilton-Jacobi formalism is briefly given and the generalized Sinh-Gordon 
potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. 
We have obtained the quasi exact solutions of eigenvalues and eigenfunctions of the potential for 
0, 1, 2n   states. The general Sinh-Gordon potential is applied to the coherent spin states. The 
eigenvalues and eigenfunctions comply with  the results of ref.[23], for  2/1,0S ( 1 ) spin 
states if the potential parameters are chosen appropriately. 
Interesting features of quantum expectation theory for PT -violating potentials may be affected by 
changing from complex to real systems.  Finally, we have pointed out that our exact results of the 
Sinh-Gordon potential may increase the number of applications in the study of various nonlinear 
potential applications in quantum systems.
Acknowledgements
This research was partially supported by the Scientific and Technological Research Council of 
Turkiye.
References
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Math. Gen., 38 (2005) 3213-3234; A. Mostafazadeh, J. Math. Phys. 46, (2005) 1-15.
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[23] O. B. Zaslavskii and V. V. Ulyanov, Sov. Pyhs. JETP  60 (1984) 991.

  The generalized Sinh-Gordon potential is solved within quantum Hamiltonian
Jacobi approach in the framework of PT symmetry. The quasi exact solutions of
energy eigenvalues and eigenfunctions of the generalized Sinh-Gordon potential
are found for n=0,1 states.

Introduction
The discovery of new class of physically significant spectral problems, called  quasi-exactly solvable 
(QES) models, has attracted much attention [1-3]. Several methods [4-5] for the generation of QES 
model have been worked out. These are the models for which a part of the bound state energy 
spectrum and corresponding wave functions can be obtained exactly. These models have been 
constructed and studied extensively by means of Lie algebraic approach. In order that a part of the 
spectrum can be obtained exactly, the parameters appearing in the relation must satisfy a condition 
known as the condition for quasi-exact solvability. 
Within the Quantum Hamiltonian Jacobi approach (QHJ) [6-7], it has been found to be an elegant and 
the simple method to determine the energy spectrum of exactly solvable models in quantum 
mechanics. The advantage of this method is that it is possible to determine the energy eigen-values 
without having to solve for the eigen-functions. In this formalism, a quantum analog of classical action 
angle variables is introduced [6-7] . The quantization condition represents well known results on the 
number nodes of the   wave function, translated in terms of logarithmic derivative, also is called 
quantum momentum function (QMF) [3, 6-7]. The equation satisfied by the QMF is a non-linear 
differential equation, called quantum Hamilton-jacobi equation leads to two solutions. The application 
of QHJ to eigen-values has been explored in great detail in ref. [3, 6-9].
In recent years, PT -symmetric Hamiltonians have generated much interest in quantum mechanics
[10-11]. Physical motivation for the PT - symmetric but non-Hermitian Hamiltonians have been 
emphasized by many authors [12-16]. Recently, Mostafazadeh [16, 17] has introduced a different 
concept  which is known as the class of pseudo-Hermitian Hamiltonians, and argued that the basic 
structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-
Hermiticity [17]. Following these detailed works, non-Hermitian Hamiltonians with real or complex 
spectra have been  analyzed by using various numerical  and   analytical techniques [18-22]. In the 
applications, Ranjani and her collaborators applied the QHJ formalism, to Hamiltonians with Khare-
Mandal potential and Scarf potential, characterized by discrete parity and time reversal PT
symmetries [3].
The aim of the present work is to calculate the energy eigenvalues and the corresponding 
eigenfunctions of the Sinh-Gordon potential field which is used in coherent spin states [23]. Originally 
the form of the potential is a hyperbolic one given as 
                                       
( ) sinh ( ) cosh
V x x B S x
         (1)        
We  study  general form of this hyperbolic effective potential which is called as Sinh-Gordon potential 
and is also in PT-symmetric form where the  potential parameters 1V and 2V  are real [23],
               xCoshVxSinhVxV  2
1)(                  (2)
in a Hamiltonian system generating quasi-exactly-solvable problems.
In this letter, we first give a brief description of the QHJ formalism. The Sinh-Gordon potential is 
studied in one dimension. In each of these cases the condition of quasi-exactly solvability is derived 
within the QHJ approach. The energy and eigenfunctions of the generalized hyperbolic effective 
potential   are obtained by using the QES condition.  
2. Quantum Hamiltonian-Jacobi Formalism
In the quantum Hamilton-Jacobi (QHJ) formalism, the logarithmic derivative of the wave function 
)(x is given  by [3, 6-7] :
          )(ln
ip                                            (3)
which is called as quantum momentum function QMF.  QMF plays an important role since it is stated 
analogous to the classical momentum function as, 
p   where S is the Hamilton’s characteristic
function and it is related to the wave function by 
x exp)( . Substituting )(x  in terms of S
in the Schrödinger equation  EH   and using the potential relation in Eq.(2), the following 
expression is obtained:
                                 )(2'2 xVEmpip   .                                    (4)
Leacock and Padgett [6-7] proposed the quantization condition for the bound states in order to find the 
eigenvalues. Quantum action variable is defined as QMF and exact quantization condition for the 
bound states of a real potential is given as [3, 6-7]
npdx
                                                   (5)
where,  C is the contour enclosing the n moving poles in the complex domain [3, 6-7].
3. The Sinh-Gordon Potential 
The Sinh-Gordon potential which is given by Eq.(2) is PT  symmetric if  1V  and 2V  are real. It is 
worth noting that, parity operation is given by pp  , xx    in the Hamiltonian whereas the 
time reversal remains same as the conventional  pp  ,  xx   and ii  . The QHJ equation
with 12  m , can be introduced as:
                           0]coshsinh[ 2
2  xVxVE
p         (6)
In order to transform Eq. (6) into a rational form, one can  change the variable xy cosh  in Eq.(6) 
and obtain the following relation:  
                   0])1([)(1)( 2
'22  yVyVEypyiyp     (7)
The coefficient of the 1
   in Eq.(7). Thus, in order to bring it to the Riccati type equation form, 
p  can be defined as;
 12  yip                                       (8)
And and another transformation is given as below,
 12 2  y
 .      (9)
Using Eqs.(8) and (9) in  Eq.(7), one can obtain a Riccati type equation as
          
   
  0
12222
2' 
 VyVyVE
      (10)
Assume that   has a finite number of moving poles in the complex y plane. In addition to,   has 
not only moving poles, but also fixed poles at 1y   and it is bounded at y . Then,   can be 
written in the form of,
11                            (11)
where 1b  and 
1b  are the residues at 1y . )(yPn  is a polynomial of degree n  and C  is a constant
due to the Liouville’s theorem (for more details, see [3] ) . At 1y ,   is expanded in a Laurent 
series as:
...)1(
1 
 yaa
                   (12)
Substituting Eq.(12) in Eq. (10), one obtains the values of residue 1b  as:
1 b                                               (13)
and also 
a   is found. For large y , 
  which are the values of C . If  the same 
procedure is applied to   for 1y
1 
 yaa
                   (14)
One can obtain the residue at 1y  as
                                    
1 b                                                          (15)
In order to the discuss the behaviour of   at infinity, one expands   as:
0 
                             (16)
Substitution of Eq.(16) in (10)  gives 
                            (17)
Behaviour of   approaches  to 
nbb  '11  for large y . Hence one can obtain [3],
 nbb '11                                          (18)
It can be seen that Eq.(18)  is positive. Therefore, one can choose positive value of   which is 
 and it leads to choose 
Ca   because of the physical solutions for the 
wavefunction. In order to obtain a QES condition, one must take all possible combinations of the 
residues '11 , bb . These combinations with a constraint  are given in table 3.1 as:
Table3.1. The QES condition and the n  for each '11 , bb .
1b 2b
11 bbn   Condition on
QES condition
     
1M M : odd,               1M 12  nM
3M M : odd,               3M 32  nM
2M M : even,               2M 22  nM
2M M : even,               2M 22  nM
                                                                                    
As it is seen from the table3.1 that sets 1 and 2 are valid if and only if M  is odd, sets 3 and 4 are valid 
if and only if M  is even. When it comes to the form of the wavefunction, using   dxpix exp)(
and writing p in terms of  , it is obtained as
  
exp)(
               (19)
Substitute Eq.(19) in (10) to obtain,
   
     .011
yVyVE
               (20)
Using Eq.(20), one can find the energy eigenvalues and corresponding wavefunctions for 3M  and 
2M  cases. For 3M  state, it can be seen that 1n  and  the )(yPn  is a first order polynomial 
given as   yP1 . The results  are illustrated in table3.2 and table3.3 respectively. 
Table3.2.  When 3M  and for set 1-2, the energy eigenvalues and wave-functions.
3M           '11 , bb              Energy, nE       Wave- function n
Set 1
11  bb , 1n
          
 
cosh)(
Set 2
11  bb , 0n
          20 E  xex
 sinh)(
where 1 and 
  are found for 0)sgn( 1 Vc  and 
 is found 
for 0)sgn( 1 Vc . Similarly for 2M case, the table3.3 is given as
Table3.3.  When 2M  and for set 3-4, the energy eigenvalues and wave-functions.
2M           '11 , bb              Energy, nE       Wave- function n
Set 3
11  bb , 0n
          
 1cosh)(
Set 4
11  bb , 0n
          
 1cosh)(
From table3.2 and table 3.3, the energy eigenvalues are obtained for the odd and even values of  
 for Sinh-Gordon potential.
If the complex forms of the potential given in Eq.(2) are written for 2 as below,
xiVxVxV 2cosh2sinh)( 2
1                                                       (21)
xVxiVxV 2cosh2sinh)( 2
1       (22)
the solutions of QHJ equation for these complex potentials can be discussed. Eq.(21) is PT
symmetric under parity reflections as x
, ii   and Eq.(22) is non- PT  symmetric.  If we 
take  Eq.(21) into consideration, values of  can be found by following the same procedure as,
     (23)
which also means the values of M  in the table3.1 are complex. The same procedure is valid for the 
potential relation in Eq.(22), too. Hence, these conditions don’t lead to physical solutions (for the 
wavefunctions). But, in ref.[3] the potential has the form given below
 22cosh)( iMxxV        (24)
which is PT  symmetric under  parity reflections x
, ii   and leads to real energy 
spectrum [3]. 
Conclusions
In this letter  quantum Hamilton-Jacobi formalism is briefly given and the generalized Sinh-Gordon 
potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. 
We have obtained the quasi exact solutions of eigenvalues and eigenfunctions of the potential for 
0, 1, 2n   states. The general Sinh-Gordon potential is applied to the coherent spin states. The 
eigenvalues and eigenfunctions comply with  the results of ref.[23], for  2/1,0S ( 1 ) spin 
states if the potential parameters are chosen appropriately. 
Interesting features of quantum expectation theory for PT -violating potentials may be affected by 
changing from complex to real systems.  Finally, we have pointed out that our exact results of the 
Sinh-Gordon potential may increase the number of applications in the study of various nonlinear 
potential applications in quantum systems.
Acknowledgements
This research was partially supported by the Scientific and Technological Research Council of 
Turkiye.
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