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4

Complex Classical Mechanics of a QES Potential

Bhabani Prasad Mandal

Department of Physics, Banaras Hindu University, Varanasi-221005, India∗

Sushant S. Mahajan

Department of Physics, Indian Institute of Technology,

Banaras Hindu University, Varanasi-221005, India†

(Dated: September 17, 2018)

We consider a Parity-time (PT) invariant non-Hermitian quasi-exactly solvable

(QES) potential which exhibits PT phase transition. We numerically study this

potential in a complex plane classically to demonstrate different quantum effects.

The particle with real energy makes closed orbits around one of the periodic wells

of the complex potential depending on the initial condition. However interestingly

the particle can have open orbits even with real energy if it is initially placed in

certain region between the two wells on the same side of the imaginary axis. On

the other hand when the particle energy is complex the trajectory is open and the

particle tunnels back and forth between two wells which are separated by a classically

forbidden path. The tunneling time is calculated for different pair of wells and is

shown to vary inversely with the imaginary component of energy. At the classical

level unlike the analogous quantum situation we do not see any qualitative differences

in the features of the particle dynamics for PT symmetry broken and unbroken

phases.

I. INTRODUCTION

Consistent quantum theory with unitary time evolution and probabilistic interpretation

for certain classes of non-Hermitian systems have been the subject of intrinsic research

∗Electronic address: bhabani.mandal@gmail.com†Electronic address: sushant.mahajan.itbhu@gmail.com

2

in frontier physics over the last one and half decade [1–3]. The huge success of complex

quantum theory [4? –18] has lead to its extension to many other branches of physics

[19–27]. In particular, its application to quantum optics is the most noticable, where

break down of PT-symmetry has been observed experimentally [22, 23]. More recently,

this formulation has been extended to classical systems [28–32]. Quantum mechanics

and classical mechanics are two completely different theories and provide profoundly dis-

similar description of physical systems. However, Bohr’s correspondence principle states

that quantum particles behave like classical ones when the quantum number is very high.

Keeping this in mind, classical systems have been investigated in a complex plane. Cor-

respondence between quantum and classical systems becomes more pronounced in the

complex domain [28, 29]. Remarkably, a particle with complex energy exhibits tunneling

like behavior which is usually realized in the quantum domain. This tunneling behavior of

a classical particle with complex energy is well demonstrated in Ref. [28, 29]. They have

shown that a classical particle can tunnel from one classically allowed region to another

allowed region separated by a classically forbidden path. Several other works in this field

are devoted to study the nature of the trajectory of a classical particle with complex en-

ergy [29–32]. Attempts have been made to study the effect of spontaneous PT-symmetry

breaking in the particle trajectories of analogous complex classical models. However the

study of complex classical system is only restricted to a few models.

The purpose of the present article is to extend these works further to investigate the

different behaviors of a complex classical system. We have chosen a complex classical sys-

tem corresponding to a QES [34–37], non-Hermitian PT-symmetric system [5] described

by the potential V (x) = −(ζ cosh 2x−iM)2. We numerically study the dynamics of a clas-

sical particle moving in the complex xy-plane subjected to this potential which consists of

periodic wells situated to the left and right side of the imaginary axis to demonstrate the

quantum tunneling effect and the trajectory of the particle in different situations. For real

energy, the particle makes closed orbits around one of the wells depending on the initial

condition. However, surprisingly the particle can have open orbits even with real energy

if it is initially placed in a certain region between the two wells on the same side of the

imaginary axis. On the other hand when the particle energy is complex, the trajectory

is open and the particle tunnels back and forth between two wells which are separated

by a classically forbidden path. In all the situations the particle trajectory never crosses

3

itself. At the classical level, unlike the analogous quantum situation, we do not see any

qualitative differences in the features of the particle dynamics for M-even ( PT symme-

try broken phase) and M-odd ( PT symmetry unbroken phase). The tunneling time is

calculated for different pairs of wells and is shown to vary inversely with the imaginary

component of energy, similar to a situation which is usually valid in the quantum domain.

Now, we state the plan of the paper. The QES potential and its solutions for broken and

unbroken PT symmetry cases are reviewed in Sec.II. We discuss the classical mechanics

of this potential in a complex plane and obtain the trajectories of the classical particle

in different situations in Sec.III. Variation of tunneling time with imaginary part of the

energy of the particle is discussed in sec.IV, and section V is kept for conclusions.

II. THE QES SYSTEM

The complex QES system is described by the Hamiltonian

H = p2 − (ζ cosh 2x− iM)2 (2.1)

where ζ is an arbitrary real parameter and M is an integer and we have considered

2m = h = 1. This Hamiltonian is symmetric under combined parity and time reversal

transformation. Parity transformation in this particular system is taken in the general

form as x → a−x, where a = iπ2. This system is shown to be a QES system. The first M

energy levels along with the corresponding eigenfunctions are calculated exactly for any

specific integer value of M . Further more, this QES system shows another remarkable

property relevant to PT symmetric non-Hermitian system, i.e. for even values of M , all

the eigenvalues are complex for any arbitrary value of ζ . On the other hand, for odd

M , all the eigenvalues are real if ζ ≤ ζc. In other words, system is always in broken PT

phase for even values of M , and shows PT phase transition when M is odd. The energy

eigenvalues for this system is calculated using Bender-Dunne (BD) polynomial methods

[38]. The zeros of BD polynomials give the energy eigenvalues for the QES system. Some

of the low lying levels are listed as follows.

M = 2 (2.2)

E± = 3− ζ2 ± 2iζ ; φ+ = A cosh x & φ− = B sinh x

4

M = 4

E1

± = 11− ζ2 − 2iζ ±√

1− iζ − ζ2; φ1

± = A sinh 3x+B sinh x

withA

B=

E − 7 + ζ2

2iζ

We can have PT phase transition when, M is odd

M = 1 (2.3)

E± = 1− ζ2; φ = A(constant)

M = 3

E± = 7− ζ2 ±√

1− 4ζ2; withφ± = A cosh 2x+ iB

E = 5− ζ2, with φ = C sinh 2x

withA

B=

4ζ

E − 9 + ζ2

Note that for odd M , energy eigenvalues are real subject to the condition ζ ≤ ζc and the

wave functions are also eigenstates of the PT operator. PT phase transition occurs at

ζ = ζc. This conclusion continues to be valid for higher odd values of M [5].

III. CLASSICAL MECHANICS IN THE COMPLEX PLANE

A classical particle with real energy E is not allowed to travel in the region where the

potential energy V (x) > E. However, this restriction is relaxed when we consider the

particle in a complex plane with complex energy E1 + iE2 as

E1 = (p21 − p22) + V1; E2 = 2p1p2 + V2 (3.1)

where complex momentum p = p1 + ip2 and we write the potential V = V1 + iV2. Now

since p1 and p2 can have any value from −∞ to ∞ there is no restriction as such on the

particle to be bound in a particular region of space. Particle is allowed to move anywhere

in the complex plane as long as the Eq. (3.1) is satisfied. This is the prime reason for

a classical particle with complex energy to travel through classically forbidden regions.

However, even though the particle can exist anywhere in the complex plane, it prefers the

region with lower energy. The particle follows a definite trajectory depending on initial

conditions. Open trajectories with local random walk type motion have been shown for

5

−5

0

5

−3−2

−10

12

3

0

10

20

30

40

50

Imaginary (Z)

Real (Z)

Pot

entia

l |V

(Z)|

−3 −2 −1 0 1 2 3−6

−4

−2

0

2

4

6

Real (Z)

Imag

inar

y (Z

)

n=1

n=2

n=0

n=1

n=−1

n=−1

n=0

n=−2

FIG. 1: Left: The magnitude of potential in the complex plane for ζ = 0.1 and M = 3. The

potential wells corresponding to the real energy orbits are distributed periodically, centered at

x = 2.04750, y = 4n+1

4π on the right, and x = −2.04750, y = 4n−1

4π on the left of the imaginary

axis. Right: The closed orbits traced by the particle when placed at different places in the

complex potential with Real Energy E = 0.8. This Fig. also indicates the positions of the wells.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

35

40

x

Pot

entia

l |V

(x)|

M=1M=2M=3M=4

FIG. 2: The double well potential for real x and its variation with the parameter M for ζ = 0.1

a classical particle with complex energy [28]. Depending on the value of the complex

energy, classical particle can be delocalized and move freely in the potential. However, a

classical particle with real energy generally moves in a closed orbit in the complex plane

[31].

We consider the classical system of a particle moving under the influence of the above

QES potential in a complex plane. The complexified potential V (z) = −(ζ cosh 2z− iM)2

6

−3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Re Z

Im Z

−4.5 −4 −3.5 −3 −2.5 −2 −1.5−1.5

−1

−0.5

0

Re Z

Im Z

FIG. 3: For ζ = 0.1, M = 3, E = 0.8. Trajectories of the particle when initial position is

Re(Z) = −2.04750 and Im(Z) = y. Left: y = n0 + 0.4740; Right: y = n0 + 0.52

−4.5 −4 −3.5 −3 −2.5 −2 −1.5−2

0

2

4

6

8

10

12

Re Z

Im Z

−3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6−2

0

2

4

6

8

10

12

14

16

Re Z

Im Z

FIG. 4: For ζ = 0.1, M = 3, E = 0.8. Trajectories of the particle when initial position is

Re(Z) = −2.04750 and Im(Z) = y. Left: y = n0 + 0.54; Right: y = n0 + 0.57

in the complex plane z = x + iy is shown in the Fig.1. The potential consists of a series

of wells on both left and right side of the imaginary axis [i.e. y-axis]. On the left of

the imaginary axis, the wells are positioned at y = 4n−1

4π and on the right of imaginary

axis at y = 4n+1

4π, where n is an integer, as well as a label for the wells. The Hermitian

counterpart of this potential V (x) = −(ζ cosh 2x − M)2 for different values of M with

fixed value of ζ(= 0.1) is plotted in Fig.2. The central barrier height increases with M

values. We consider a particle in this complex QES potential. The dynamics of this

particle are governed by the Hamilton’s equation as

z =∂H

∂p; −p =

∂H

∂z(3.2)

where H = p2

2m+ V (z). We find the trajectories in the complex plane for different com-

plex energies by solving the Eq. [3.2] for the above QES potential numerically. We

7

−3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.61.6

1.8

2

2.2

2.4

2.6

2.8

3

Re Z

Im Z

−4.5 −4 −3.5 −3 −2.5 −2 −1.51.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Re Z

Im Z

FIG. 5: For ζ = 0.1, M = 3, E = 0.8. Trajectories of the particle when initial position is

Re(Z) = −2.04750 and Im(Z) = y. Left: y = n0 + π + 0.4740; Right: y = n0 + π + 0.52

−4.5 −4 −3.5 −3 −2.5 −2 −1.52

3

4

5

6

7

8

9

10

11

12

Re Z

Im Z

−3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.62

4

6

8

10

12

14

16

18

20

Re Z

Im Z

FIG. 6: For ζ = 0.1, M = 3, E = 0.8. Trajectories of the particle when initial position is

Re(Z) = −2.04750 and Im(Z) = y. Left: y = n0 + π + 0.54; Right: y = n0 + π + 0.57

solve this system taking the units such that, 2m = h = 1. Since the imaginary part

of energy is thought to arise through quantum mechanical uncertainty, this choice of

units will give us the advantage of measuring the energy in multiples of h i.e. unit

energy= 1.0545X10−34Joules. To observe quantum effects, we have to go well beyond

the visibility of the naked eye, and hence we have chosen a length scale of one nanometer.

Choosing this system of units gives the units of mass and time to be milligram and second

respectively. Thus, our test particle has a mass of half code units, i.e. 0.5 milligrams.

This particle can safely be considered as a classical particle. All following discussions and

figures in this paper display the quantities in these units.

We see that particle orbits around the position of the different wells if the energy of

the particle is real and the particle is placed sufficiently close to the well [Figs. 3 & 5].

However, the particle can have open orbits even with real energy if not placed sufficiently

8

close to a well [Figs 4 & 6].

−4 −3 −2 −1 0 1 2 3−15

−10

−5

0

5

10

15

Re Z

Im Z

−5 −4 −3 −2 −1 0 1 2 3 4−40

−30

−20

−10

0

10

20

30

40

Re Z

Im Z

FIG. 7: The trajectory of a particle with energy E = 1+ i in potential for ζ = 0.1. Left : when

M = 2, the particle oscillates between the wells corresponding to n = +3 on the right n = −3

on the left of the imaginary axis. Right: when M = 3, the particle oscillates between the wells

corresponding to n = +10 on the right and n = −10 on the left of the imaginary axis.

−4 −3 −2 −1 0 1 2 3 4 5−80

−60

−40

−20

0

20

40

60

80

Re Z

Im Z

−5 −4 −3 −2 −1 0 1 2 3 4 5−150

−100

−50

0

50

100

150

Re Z

Im Z

FIG. 8: The trajectory of a particle with energy E = 1+ i in potential for ζ = 0.1. Left: when

M = 4, the particle oscillates between the wells corresponding to n = +21 on the right and

n = −21 on the left of the imaginary axis. Right: when M = 5, the particle oscillates between

the wells corresponding to n = +35 on the right and n = −35 on the left of the imaginary axis.

If we vary the distance of the starting point of the particle from the well along the

imaginary axis (y-axis), the closed orbit opens up after a certain value. For the first well

(n = 0 ) on the left of imaginary axis if particle is placed at y < (n0 + .52988875) or

y > (n0 − .52988875) we have closed orbit of the particle. Where n0 = −π

4, the position

of first well along y direction. However if the particle is not placed sufficiently close

y ≥ (n0 + .52988875) or y ≤ (n0 − .52988875) to the first well, the particle will have open

9

orbit even with real energy and will escape to infinity without tunneling to any other well.

We have illustrated this for the first and second well, i.e for n = 0 and 1 on the left side

of the imaginary axis. The second well is positioned at y = 3π

4≡ n1 along imaginary axis.

The particle escapes to infinity if it is placed beyond y = n1 ± .52988875.

The most interesting results that we have obtained for complex energy of the particle

are that:

1. It tunnels back and forth between the wells to the left and right of the imaginary

axis.

2. When placed inside a well, the particle spirals out until it enters the classically

forbidden region outside the well, then depending on the direction of its velocity, it spirals

into the first well (on the other side of the imaginary axis) that it encounters in its path.

The separation of the wells between which the tunneling takes place increases with M

values and also depends on the complex energy and on the parameter ξ. This spiraling

nature was also shown in Refs. [30, 31]

−2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Re Z

Im Z

−2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−1.1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

Re Z

Im Z

FIG. 9: The particle spiralling outwards from the well at n=0 on the left side of the imaginary

axis. Left: for E2 > 0 Right: for E2 < 0

3. When the particle is spiraling in, towards the center of a well, in all cases with

E2 > 0, we observe that its motion is clockwise, and, anticlockwise when it spirals out

from the well. Whereas for E2 < 0, the spiral motion is anticlockwise inwards and

clockwise outwards. The outward spiralling of the particle from the well at n=0 on the

left side of the imaginary axis is shown in figure 9. For E2 < 0, the right side well has

n < 0 and the left side well has n > 0.

4. For the results in figs 7-10, we tried 3 different initial conditions. (i) Placing the

10

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−15

−10

−5

0

5

10

15

Re Z

Im Z

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−25

−20

−15

−10

−5

0

5

10

15

20

25

Re Z

Im Z

FIG. 10: The trajectory of a particle with energy E = 1 + i in potential for ζ = 1. Left: when

M = 2, the particle oscillates between the wells corresponding to n = +3 on the right and

n = −3 on the left of the imaginary axis. Right: when M = 3, the particle oscillates between

the wells corresponding to n = +6 on the right and n = −6 on the left of the imaginary axis.

particle in the well on the left side, (ii) placing the particle in the well on the right side,

and (iii) placing the particle at the origin. All the three initial conditions give qualitatively

the same result. The only difference is that if the particle is placed at the origin, and

it is found to tunnel between two wells n=-a on the left and n=+a on the right, then

placing the particle initially in the well corresponding to n=-b on the left will make the

particle tunnel between the wells n=-b on the left and n=-b+2a on the right. Similarly, if

the particle is initially placing in the well n=+b on the right, then it will tunnel between

n=+b on the right and n=+b-2a on the left. For simplicity, we have used origin as the

starting point.

5. We have also observed that the trajectory of the particle in this complex potential

never crosses itself.

We have performed the numerical study for both odd and even values of M and with ζ

values below and above the critical value (ζc). From the quantum mechanical analysis of

this QES problem in Sec. II we know that when M is odd and ζ ≤ ζc we have unbroken

PT phase and QES energy eigenvalues are real. On the other hand PT symmetry breaks

spontaneously if (i) M is even with any ζ value or (ii) M is odd with ζ > ζc. For broken

PT symmetry, one can expect irregular trajectories with crossing points and disordered

behavior of the system. But, as is evident from the trajectories in the figs 7-10 we see in

the case of broken PT-symmetry, there is no crossing of trajectories in Non-PT Symmetric

11

−4 −3 −2 −1 0 1 2 3 4−40

−30

−20

−10

0

10

20

30

40

Re Z

Im Z

−3 −2 −1 0 1 2 3−80

−60

−40

−20

0

20

40

60

80

Re Z

Im Z

FIG. 11: The trajectory of a particle with energy E = 1 + i in potential for ζ = 1. Left: for

M = 4, the particle oscillates between the wells corresponding to n = +11 on the right and

n = −11 on the left of the imaginary axis. Right: for M = 5, the particle oscillates between

the wells corresponding to n = +19 on the right and n = −19 on the left of the imaginary axis.

potentials. Further, there are no qualitative differences in the trajectories of the particle

in PT-symmetric and Non-PT symmetric potentials classically, unlike the analogous sit-

uations in quantum theory. These results contradict the claims and speculations in Ref

[30]. However this is not very surprising as these results support the views of the Ref [33].

IV. TUNNELING TIME

The time taken by the particle to tunnel from one well to another depends on the

parameters M, ζ and the imaginary component of energy(E2) of the particle. We have

calculated the tunneling time, that is the time duration between two consecutive crossings

of imaginary axis by the particle (in opposite directions) for fixed M and ζ . The particle

spends different amount of time to the left and to the right of the imaginary axis, which

is not expected to be the case in classical mechanics. The tunneling time is calculated as

the average of the time spent by the particle on both sides of the imaginary axis. The

variation of the averaged tunneling time with E2 is listed in a tabular form [Table 1].

We observe that tunneling time varies inversely with E2 . This implies quicker tunneling

when E2 is more. The tunneling time is plotted versus E2 (in figure 12). The nature of

this curve satisfies the correspondence principle, since the particle shows more quantum

behaviour for higher E2. We would also like to point out that no tunneling is seen for

12

TABLE I: The variation of tunneling time across 2 wells with the imaginary component of

energy.

Imaginary Component of Energy Im(E) Tunneling time

0.3 54.19

0.5 32.42

0.8 20.1

1.0 15.99

1.2 13.14

1.5 10.42

1.7 9.203

2.0 7.739

2.2 7.008

2.5 6.097

2.7 5.635

3.0 5.054

3.2 4.745

3.5 4.329

3.7 4.058

4.0 3.76

4.2 3.601

4.5 3.355

4.7 3.22

5.0 3.025

5.2 2.902

5.5 2.763

5.7 2.673

6.0 2.541

6.2 2.47

6.5 2.375

6.7 2.313

13

0 1 2 3 4 5 6 70

10

20

30

40

50

60

Imaginary Component of Energy

Tun

ellin

g tim

e

FIG. 12: The variation of tunneling time with the imaginary part of energy for a particle initially

placed in a potential well corresponding to n = 0 of PT Symmetric potential (M = 3, ζ = 0.1)

on the right side of the imaginary axis. The real part of energy is fixed as 1 unit.

the particle with real energy as it either moves in closed orbits or it escapes to infinity

remaining on the same side of the imaginary axis. This tunneling behavior of a classical

particle in the complex plane is something which is generally observed in the domain of

quantum physics.

V. CONCLUSION

We have studied the complex classical mechanics of a system whose non-Hermitian

PT-invariant version is a QES system. We treat this model classically on a complex

plane to capture some of the strange behavior of the system. We find that the particle

tunnels back and forth between two wells (one on the left side and other on the right

of the imaginary axis). Positions of the wells between which it tunnels back and forth

depend on the values of the parameters M, ζ and the imaginary part of E. The distance

between the wells in which the particle tunnels, increases with M and it decreases when

ζ increases. Particle never tunnels between the wells which are located on the same side

of the imaginary axis. The time spent by the particle in the left well is different from the

time spent in the right well. The tunneling time is inversely proportional to the imaginary

part of the energy. More E2 implies more tunneling effect and in that situation tunneling

14

is between nearer wells. This can be understood by reducing E2 gradually. With very

less E2, the particle will tunnel between wells which are very far. And for zero imaginary

energy, the particle either stays inside the same well with a closed orbit, or tries to tunnel

between 2 wells, which are infinitely separated in the imaginary direction, resulting in

an open orbit. We do not observe any difference in the overall behavior of the particle

dynamics for PT-broken and unbroken situations.

Acknowledgments: One of us (BPM) acknowledges the financial support from Depart-

ment of Science and Technology (DST), Govt. of India under SERC project sanction

grant No. SR/S2/HEP-0009/2012 and the hospitality of the organizers of PHHQP11

held at APC, Paris where this work was presented.

[1] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).

[2] C. M. Bender: Rep. Prog. Phys. 70947 (2007) and references therein.

[3] A. Mostafazadeh: Int. J. Geom. Math. Mod. Phys. 7 1191 (2010) and references therein.

[4] C. M. Bender, S. Boettcher: Phys. Rev. Lett 89, 270401-1 (1998) . C. M. Bender, K.

Besseghir, H. F Jones and X. Yin: J. Phys. A , 42 355301 (2009); C. M. Bender, B. Tan,

J. Phys. A 39 1945 (2006); C. M. Bender, H. F. Jones: Phys. Lett. A 328 102 (2004).

[5] A. Khare and B. P. Mandal: Phys. Lett A 272 53 (2000).

[6] A. Mostafazadeh and F. Zamani: Ann. Phys. 321 (2006) 2183 ; ibid (2006) 2210. A.

Mostafazadeh and H. Mehr-Dehnavi : J. Phys. A 42 125303 (2009);

A. Mostafazadeh :Phys. Rev. Lett. 102 220402 (2009);

A.Mostafazadeh :J.Phys. A Math. Theor. 44 375302 (2009).

[7] Z. Ahmed : Phys. Lett. A 282 343; 287 295 (2001). Z. Ahmed: J. Phys. A: Math. Theor.

42 473005 (2001).

[8] A. Khare and B. P. Mandal: Spl issue of Pramana J of Physics 73,387 (2009).

[9] B. Basu-Mallick: Int. J of Mod. Phys. B 16 1875(2002);

B. Basu-mallick, T. Bhattacharyya, A. Kundu and B. P. Mandal :Czech. J. Phys. 54

5(2004). B. Basu-Mallick and B. P. Mandal: Phys. Lett.A 284 231(2001);

B. Basu-mallick, T. Bhattacharyya and B. P. Mandal: Mod. Phys. Lett. A 20 543(2004).

[10] B. P. Mandal, A. Ghatak: J. Phys. A, Math. Gen 45 444022 (2012).

15

[11] B. P. Mandal, S. Gupta: Mod. Phys. Lett. A 25 1723 (2010).

[12] A. Ghatak, B. P. Mandal: J. Phys. A Math. Gen. 45 355301 (2012).

[13] B. Bagchi, C. Quesne: Phys. Lett. A 273 256 (2000).

[14] K. Abhinav, P. K. Panigrahi: Annl. Phys. 326 538 (2011).

[15] B. P. Mandal, Mod. Phys. Lett. A 20, 655 (2005).

[16] A. Ghatak, R. D. Ray Mandal, B. P. Mandal, Ann. of Phys. 336, 540 (2013).

[17] M. Hasan, A. Ghatak, B. P. Mandal, Ann. of Phys. 344, 17, (2014).

[18] A.Ghatak , Z. A. Nathan, B. P. Mandal and Z. Ahmed: J. Phys. A, Math. Gen 45 465305

(2012).

[19] C. M. Bender and G. V. Dunne, J. Math. Phys. 40, 4616-4621 (1999); C. M. Bender, D. C.

Brody, and H. F. Jones, Phys. Rev. Lett. 93, 251601 (2004); C. M. Bender, D. C. , and H.

F. Jones, Phys. Rev. D 70, 025001 (2004); C. M. Bender, S. F. Brandt, J.-H. Chen, and Q.

Wang, Phys. Rev. D 71, 065010 (2005).

[20] J. L. Cardy, Phys. Rev. Lett. 54, 1345 (1985); J. L. Cardy and G. Mussardo, Phys. Lett.

B225, 275 (1989); A. B. Zamolodchikov, Nucl. Phys. B 348, 619 (1991); R. Brower, M.

Furman, and M. Moshe, Phys. Lett. B76, 213 (1978),

[21] C. M. Bender, V. Branchina, E. Messina, Phys. Rev. D 85, 085001 (2012).

[22] Z. H. Musslimani, K. G. Makris, R. El-Ganainy, D. N. Christodulides: Phys. Rev. Lett.

100 030402 (2008). C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodulides, M.

Segev and D. Kip: Nature (London) phys. 6 192 (2010). A. Guo, G. J. Salamo: Phys. Rev.

Lett. 103 093902 (2009). C. M. Bender, S. Boettcher and P.N. Meisinger : J. Math. Phys.

40 2201 (1999). C. T. West, T. Kottos and T.Prosen: Phys. Rev. Lett 104 054102 (2010).

C. M. Bender, B. K. Berntson, D. Parker and E. Samuel: arXiv: math-ph/1206.4972v1.

[23] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, T. Kottos: Phys. Rev. A 84 040101 (2011).

H. Ramezani, J. Schindler, F. M. Ellis, T. Kottos: Phys. Rev. A 85 062122 (2012). Z. Lin,

J. Schindler, F. M. Ellis, T. Kottos: Phys. Rev. A 85 050101(R) (2012).

[24] C. M. Bender, G. V. Dunne, P. N. Meisinger, M. Simsek: Phys. Lett A 281 (2001) 311-316.

C. M. Bender, M. Berryand, A. Mandilara: J. Phys. 35 (2002) L467. M. Znojil: J. Phys.

A 36 (2003) 7825. G. Levai: J. Phys. A 41 (2008) 244015.

[25] D. Bazeia, A. Das and L. Losano: Phys. Lett. B 673 283 (2009).

[26] S. Dey, A. Fring and L. Gouba: J. Phys A 45 (2012) 385302.

16

[27] B. P. Mandal, B. Mourya, and R. K. Yadav, Phys. lett. A, (2013) 1043.

[28] C. M. Bender, D. C. Brody, D. W. Hook, J. Phys. A: Math. Theor. 41, 352003 (2008);

C. M. Bender, Peter N. Meisinger, Qing-hai Wang, Phys. Rev.Lett.104:061601,2010, Annl.

Phys. 325, 2332 (2010); C. M. Bender, D. W. Hook, J. Phys. A: Math. Theor. 44, 372001

(2011); N. Turok, arXiv:1312.1772.

[29] S. Dey, A. Fring, Phy. Rev A 88, 022116 (2013)

[30] A. Nanayakkara, Czech. J. Phys. 54, 101 (2004) and J. Phys. A: Math. Gen. 37, 4321 (2004).

C. M. Bender, D. D. Holm, and D. W. Hook, J. Phys. A: Math. Theor. 40, F81 (2007). C.

M. Bender, D. D. Holm, and D. W. Hook, J. Phys. A: Math. Theor. 40, F793-F804 (2007).

C. M. Bender, D. C. Brody, and D. W. Hook, J. Phys. A: Math. Theor. 41, 352003 (2008).

C. M. Bender, D. W. Hook, and K. S. Kooner, J. Phys. A: Math. Theor. 43, 165201 (2010)

[31] Anjana Sinha, Eur. Phys. Lett. vol. 98 (2012) 60005

[32] A. Sinha, D. Dutta, P. Roy, Phys. Lett A , vol. 375 (2011) 452-457

[33] A.G. Anderson, C. M. Bender, U. I. Morone, Phys. Lett. A375, 3399; A Cavaglia, A Fring,

B. Bagchi, J. Phys. A: Math. Theor. 44, 325201 (2011).

[34] A. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, Inst. of Physics

Publishing, Bristol, (1994), and references therein.

[35] C.M. Bender and A. Turbiner, Phys. Lett. A173, 442 (1993). P. Assis and A. Fring, J.

Phys. A: Math. Theor. 42, 015203 (2009).

[36] A. Khare and B.P. Mandal, J. Math. Phys. 39, 3476 (1998).

[37] Y. Brihaye, A. Nininahazwe & B. P. Mandal, J. Phys. A40, 13063(2007).

[38] C. M. Bender, G. V. Dunne, J. Math. Phys. 37, 6 (1996).