PhD Thesis of Savita Kharab

A STUDY ON THE SOLUTIONS

OF

THE SCHRODINGER EQUATION

FOR

REAL AND COMPLEX POTENTIALS

A

THESIS

SUBMITTED TO THE FACULTY OF SCIENCE

KURUKSHETRA UNIVERSITY, KURUKSHETRA

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

PHYSICS

by

Savita Kharb

Under the Supervision

of

Dr.Fakir Chand

DEPARTMENT OF PHYSICS

KURUKSHETRA UNIVERSITY,

KURUKSHETRA(HARYANA)-136119

JULY-2011

i

Dedicated To My Teachers

ii

Acknowledgements

Above all, I bow my head before ‘God’, without whose blessing my present thesis

would not have existed and thanks for giving me patience and strength to over-

come the difficulties, which crossed my way in the accomplishment of this endeavor

and made me able to make my parents own dream come true. As such there is no

word in the world’s dictionary to express the real meaning of the word ‘ Guide’ but

true to say undoubtedly, it has been well expressed with realistic appropriateness

but for my advisor Dr.Fakir Chand, Department of Physics, Kurukshetra University,

Kurukshetra, under whose inspiration beckoning, I found myself in the position to

complete my heartily research work. I am extremely grateful to him for showing

personal interest, continuous encouragement, and moral support, generous and ex-

pert guidance throughout the course of my research work. I am equally delighted to

pen down my feeling for his wife Mrs. for her encouragement throughout the tenure

of this work. I am highly grateful to Prof. Nafa Singh and Prof. Shyam Kumar,

Prof. S.C.Mishra (Present Chairman), Dr. R. K. Moudgil, Dr. Sanjeev Aggar-

wal and Dr. Anu sharma,Dr.Manish,Ms.Suman,Department of Physics, Kurukshetra

University, Kurukshetra for their constant encouragement and for helping me as and

when required. I am very thankful to my colleagues Ms.Monika,Mr.Vishal,Dr.Vikas

Bhardwaj, Department of Applied Sciences, T.E.R.I. Kurukshetra for their help and

encouragement during this course of work. My special thanks are due to Mr. Anand

Kumar, Mr.Narender Kumar,Mr. Ramesh Kumar, Mr.Hitender,Mr.Sukhvinder and

Mr.Manjit, Research Scholars, Department of Physics Kurukshetra University, Ku-

rukshetra for their constant support and help. I am also thankful to all the staff

of Department of Physics Kurukshetra University, Kurukshetra for all type of help

during this work. I must pay my sincere gratitude to my parents who have made me

able to scale the heights of higher education.

iii

I am extremely thankful to my family members who directly or indirectly helped

me during this work. Last, but not the least important persons to be acknowledged

are my husband Dr.Rajesh Kharab and kids Ankit and Ashish for the degree of

patience they have shown during the course of this work

Kurukshetra

July, 2011 (Savita Kharb)

iv

List of Publications

Research Papaers in Journals

1.“Eigenvalue spectra of a PT-symmetric coupled quartic potential in two dimen-

sions”, F Chand, Savita and S C Mishra, Pramana - J. Phys. 75, 599 (2010).

2.“Searching critical point nuclei in Te- and Xe- isotopic hains using sextic oscillator

potential ”, S Kharb and Fakir Chand, (to appear in) Phys. At.Nucl. 74(2011)

(no.10).

3.“Solutions of the Schrodinger equation for PT-symmetric coupled quintic potential

in two dimensions ”, Savita and Fakir Chand, Commun.Theor.Phys.73(2011) 349-

361.

4.“The solution of the Schrodinger equation for for sextic potentials in two Dimen-

sions”, S Kharb, Fakir Chand , (To be communicated)

Research Papaers presented in Conferences/ Symposiums / Workshops

etc.

1.“The Solution of the Schrodinger equation for a complex coupled oscillator in

two dimensions ”, Devender Kumar, Savita Kharb, F Chand and S C Mishra, “In-

ternational Conference on Non-Hermitian Hamiltonians in Quantum Physics ” from

Jan. 13-16, 2009 held at Bhaba Atomic Research Center, Mumbai, India.

2.“The Solution of the Schrodinger equation of two dimensional PT-symmetric cou-

pled quartic potentials by the ansatz for eigen function method, Savita, Fakir Chand

and S.C.Mishra,“National Workshop on Nano technology and applied sciences” from

Nov. 26-27, 2009 held at H.C.T.M., Kaithal, India. Proceeding in Atti Della Foudazione

Giorgio Ronchi ANNo LXV, 811(2010).N.6.

Contents

1 Introduction 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Need of non-hermitian Hamiltonians . . . . . . . . . . . . . . . . . . 4

1.3 The Schrodinger Equation (SE) . . . . . . . . . . . . . . . . . . . . . 5

1.4 Different Methods for the Solution of the SE . . . . . . . . . . . . . . 6

1.4.1 Exact Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.2 Approximation Methods . . . . . . . . . . . . . . . . . . . . . 11

1.5 Various ways of Complexifying a Hamiltonian . . . . . . . . . . . . . 18

1.6 Concept of Non-Hermitian Hamiltonian . . . . . . . . . . . . . . . . . 20

1.7 Quantum Mechanical Theory for Hermitian Hamiltonians . . . . . . . 21

1.8 Quantum Mechanical Theory of Non-Hermitian Hamiltonians . . . . 23

1.9 Relation between Hermitian and PT - symmetric quantum theories . 26

2 The SE for Complex Systems 29

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 The Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.1 Solutions for PT -symmetric quartic potential . . . . . . . . . 32

2.3.2 Solutions for PT -symmetric quintic potential . . . . . . . . . 36

2.3.3 An Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Application of sextic oscillator potential 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 The Bohr Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 43

3.1.2 The Sextic Oscillator Potential . . . . . . . . . . . . . . . . . 45

3.1.3 Application to Te- and Xe-Isotopic Chains . . . . . . . . . . . 47

1

2 CONTENTS

4 Applications and Conclusions 51

4.1 Applications of Non-Hermitian Quantum Theory . . . . . . . . . . . 51

4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Chapter 1

Introduction

In this chapter we discuss the quantum theory of hermitian and non-hermitian Hamil-

tonians and various methods of complexifications. The non-hermitian Hamiltonians

can provide real eigenvalues,if it is PT - symmetric and its PT - symmetry is unbro-

ken. The different methods to solve the Schrodinger equation (SE) given in literature

are also discussed.

1.1 Introduction

The theory of quantum mechanics which is nearly one hundred years old, owing to a

large number of experimental verifications of its theoretical predictions, has become

an essential part of modern science. The theoretical understanding of several phe-

nomena in various branches of physics and chemistry viz. the phenomena pertaining

to resonance scattering in atomic, molecular, nuclear physics and in some chemical

reactions, requires an account of harmonic as well as anharmonic/nonlinear poten-

tials at both classical and quantum levels [1]. Moreover, the techniques developed

for these purposes may be used for the Schrodinger equation (SE) in the context of

other physical problems where the Schrodinger like equations appear [1]. However

the problem of anharmonic oscillators, or more generally that of polynomial interac-

tions, in one dimension in quantum theory continues to be an active area of research

despite a long history. This is mainly because these problems are, in general, not

exactly solvable and most of the work during these years has been devoted to the

development of various approximation techniques [1]. Although these problems could

be of interest in their own right but, interestingly, as a model, they also guide to

similar methods which could be of immediate use in quantum field theory. Although

the quadratic and quartic type anharmonic potentials have been studied rather ex-

3

4 CHAPTER 1. INTRODUCTION

tensively [2, 3, 4, 5, 6, 7, 8] in one dimension but not so many attempts have been

made to study the SE with a potential having higher order anharmonicity in two and

higher dimensions [9, 10, 11, 12, 13]. The study of the SE in two dimensions is of

interest in a number of physical problems, mainly in the fibre optics and quantum

chemistry [14].

In quantum mechanics, the hermiticity of Hamiltonian ensures real eigenvalues. But

for a system in which the position and momentum co-ordinates are taken as complex

makes the corresponding Hamiltonian complex or non-hermitian and thus eigenvalues

turn out to be complex. However, its PT -symmetric form exhibits real eigenvalues.

In the last decade many interesting investigations on the PT -symmetric quantum me-

chanics have generated a renewed interest in the analysis of non-hermitian potentials

[2, 5, 7, 8, 15, 16, 17]. The study of non-hermitian Hamiltonians demands special at-

tention because of their interesting applications in several areas of theoretical physics

like super conductivity, population biology, condensed matter physics and so on [8].

A Hamiltonian(H) is said to be PT -symmetric if it commutes with PT operator i.e.

H = HPT to retain the unitarity and real eigenvalues. The PT reflected Hamilto-

nian HPT is defined as HPT = (PT )H(PT ) with P as the space reflection or parity

operator and T as the time reversal operator. Therefore, PT - symmetry is an alter-

native condition to the conventional hermiticity. Thus, now it is possible to construct

infinitely many new Hamiltonians that would have been rejected in the past because

they were not hermitian in usual sense [2].

1.2 Need of non-hermitian Hamiltonians

General relativity and quantum mechanics (Q M) are the most important achieve-

ments of the twentieth century theoretical physics. The unification of Special Rel-

ativity and Q M, which is by far an easier task, has been the subject of intensive

research since late 1920’s. It has led to the formulation of various quantum field

theories. But most of the attempts have so far failed to produce a physical theory

offering concrete experimentally verifiable predications.The development of the spe-

cial relativistic quantum theories has also involved attempts at generalizing Q M. An

idea initially put forward by Dirac in 1942 and developed by Paul which is known as

the indefinite-metric quantum theories. Amore recent attempt at generalizing Q M is

due to Bender and his collaborators who adopted all its axioms except the one that

restricted the Hamiltonian to be hermitian. They replaced the latter condition with

the requirement that the Hamiltonian must have an exact PT -symmetric which in

1.3. THE SCHRODINGER EQUATION (SE) 5

particular assured the reality of its spectrum.

An important motivation for considering this so-called PT -symmetric Quantum Me-

chanics is provided by an interesting idea that is rooted in special relativistic local

quantum field theories (QFT). Among the most celebrated results of QFT is the CPT

-theorem. It states that every field theory satisfying the axioms of QFT is CPT -

invariant, where C is the charge-conjugation operator. Clearly replacing the axiom

that the Hamiltonian is Hermitian with the statement of the CPT -theorem might

lead to a generalization of QFT.

Pseudo-Hermitian operators is the one appearing in the context of the Dyson mapping

of Hermitian Fermionic Hamiltonians to equivalent quasi-Hermitian bosonic Hamil-

tonians. Pseudo-Hermitian Hamiltonian is used in modeling a delocalization phe-

nomenon relevant for the vortex pinning in superconductors. The application of

pseudo-Hermitian QM in dealing with the Hilbert space problem in relativistic QM

and quantum cosmology and the removal of ghosts in certain quantum field theories

relies on the construction of an appropriate (positive-definite) inner product on the

space of solutions of the relevant field equation. The basic idea behind the application

of pseudo-Hermitian QM in dealing with the Hilbert-space problem in relativistic QM

and quantum cosmology is that the relevant field equations whose solutions constitute

the statevectors of the desired quantum theory are second order differential equations

in a time variable. An interesting application of pseudo-Hermitian QM is its role in

dealing with the centuries-old problem of the propagation of electromagnetic waves

in linear dielectric media. It is the spectral properties of quasi-Hermitian operators

and their similarity to Hermitian operators that plays a key role. The use of the

machinery of pseudo-Hermitian QM in the study of effective quasi-Hermitian scatter-

ing Hamiltonians leads to a more reliable quantitative description of the scattering

problem. It also provides a better understanding of the approximation schemes used

in this context. Stenholm and Jakob explore the application of the properties of

pseudo- and quasi-Hermitian operators in the study of open quantum systems. In

Eslami-Moossallam and Ejtehadi have introduced the effective Hamiltonian for the

description of the dynamics of an anisotropic DNA molecule.

1.3 The Schrodinger Equation (SE)

In 1926, Schrodinger using the De Broglie’s idea of matter wave developed a rigorous

mathematical theory which has received the name of wave mechanics .The essential

feature of this theory is the incorporation of the expression for De Broglie’s wavelength


into the general classical wave equation. Schrodinger derived wave equation for a

moving particle which describes the behaviour of the wave function associated with

the microscopic particles and is known as the Schrodinger equation (SE). Consider

the SE in two dimensions as

Hψ(x, y) = Eψ(x, y), (1.1)

where H is the Hamiltonian operator given by

H =−h2

2m∇2 + V.

There are the two different equations given by Schrodinger, which are given as

(i) Time Independent Schrodinger equation (TIDSE) [44]

∇2ψ(x, y) +2m

h2 (E − V )ψ(x, y) = 0. (1.2)

For a free particle, V = 0, the TIDSE becomes

∇2ψ(x, y) +2m

h2 Eψ(x, y) = 0. (1.3)

(ii) Time Dependent Schrodinger equation (TDSE) [44]

(−h2

2m∇2 + V

)ψ(x, y) = ih

∂ψ(x, y)

∂ t, (1.4)

where ψ(x, y) is the wave function. The present study is based on the solutions of

the TIDSE for a number of nonlinear real and complex potentials in two dimensions.

1.4 Different Methods for the Solution of the SE

Since there is a great importance of harmonic and anharmonic potentials in the

study of various physical problems [1]. So to investigate such problems, a number

of exact and approximation methods have been developed to find the solution of the

SE by using certain methods. Methods available to solve the SE are either exact or

approximate, which are classified and described as in the following subsections

1.4.1 Exact Methods

The exact closed form solutions obtained by exact methods could be normalized and

well behaved at origin and at the infinity. But these methods provided either only

a few eigenvalues or else they became much involved as one proceeds to obtain the

complete eigenvalue spectra. So exactness of these solutions remain questionable,

1.4. DIFFERENT METHODS FOR THE SOLUTION OF THE SE 7

hence these solutions of the SE are partial or ‘quasi exact’ solutions. However, here

we shall call these solutions to be exact even if they correspond to a few eigenvalues.

The exact methods are classified as

(i) Eigenfunction Ansatz Method

First off all this method was adopted by Hautot [3] and then by many other authors

[4-7] for different potentials. In this method ansatz of wave function is assumed for

the solution of the Schrodinger equation .The unknown parameters in the ansatz

are obtained in terms of the parameters of the potential on rationalizing the SE.

Sometimes solutions are obtained with certain constraints on the potential parameters

as the number of parameters outnumber the unknown parameters in the ansatz.

The SE for a two Dimensional system is written (for h = m = 1 ) as

∂2ψ

∂ x2+∂2ψ

∂ y2+ (E − V (x, y))ψ = 0. (1.5)

Next we make an ansatz for eigenfunction of the form

ψ(x, y) = ϕ(x, y)e−g(x,y). (1.6)

On substituting eq.(1.6) in eq.(1.5), we obtain

gxx + gyy − (gx)2 − (gy)

2 + 2(E − V ) +1

ϕ(−2ϕxgx − 2ϕygy + ϕxx + ϕyy) = 0, (1.7)

where the subscripts to function g and ϕ indicate the differentiation w.r.t. the vari-

ables x and y.

From the structure of the above equation, it is clear that if the functions g and ϕ are

known for a given system, then rationalization of eq.(1.7) would directly provide the

energy eigenvalues and eq.(1.6) would then act as the wave function for the system.

(2) Quasi-Dynamical Lie-Algebraic Method:

Many exactly solvable problems [45, 46, 47] may be related to a finite dimensional

representation of Lie algebra. In this referance, Turbiner used the well known rep-

resentation of sl(2, R) or so(2, 1) [48] to treat the problem of sextic potential [45] in

the form:

J+ = z2 d

dz− 2jz, J0 = z

d

dz− j, J− =

d

dz, (1.8)

where z = x2 and j(j+1) is the eigenvalue of the Casimir operator. In fact, the

commutators of these operators give the standard relations

[J+, J−] = −2J0, [J0, J±] = ±J±. (1.9)


The main difficulty with this representation of the elements of sl(2, R) is that here it

is limited to a sextic potential, namely the potential of the form,

v(x) = a2x6 + 2abx4 + [b4 − (4n+ 3)a]x2; a, b ∈ R. (1.10)

Recently, de Souza Dutra and Boschi Filho [47] used a generalization of this repre-

sentation to obtain polynomial potentials of higher degrees via the algebra so(2, 1)

which is isomorphic to sl(2, R). The representation used is

T1 = α2x2−j(

d

dx)2 + α1x

1−j(d

dx) + α0x

−j,

T2 = − i

jx(

d

dx) − iσ, T3 = αxj.

These satisfy the so(2, 1) algebra viz,

[T1, T2] = −iT3 ; [T2, T3] = −iT1 ; [T1, T3] = −iT2. (1.11)

Provided only that

σ =1

2j[α1

α2

+ j − 1], λ = −(2α2j2)−1.

Here α1, α2 and j are the constants which are selected to match the different elements

of the Hamiltonian. Further, the work of de Souza and Boschi Filho provides a clue

to express the most general form of the Hamiltonian in terms of the generators of the

sl(2, R) or so(2, 1) algebras. Recently Burrows et.al [49] have suggested mathematical

details of this method in terms of so(3) or so(2, 1) algebras. It is suggested that the

Hamiltonian admits only a part of the bound state spectrum, when expressed as a

bilinear function of the generators of these algebras and the same can be calculated

analytically to provide a finite set of (quasi) exact solutions.

(3) Standard Differential Equation Methods

To use this method [1], the SE for a given potential is transformed into a standard

differential equation whose solution is already known in mathematics literature, by

a suitable transformation of both the dependent and independent variables. Very

often a differential equation satisfied by an orthogonal polynomial is obtained in

this method. Recently this method was used by Aly and Barut [50] to find the exact

solutions for three large classes of anharmonic potentials. In past several authors have

used general method for transformation of the SE into the hypergeometric equation.

However, for specific purpose, one still has to find special transformations to identify

the physical potentials. Aly and Barut have applied the variations of this technique to

a variety of anharmonic potentials and obtained the exact solutions for the following


energy-dependent potentials:

(i) V (x) = αx− kαx2 − 1

4a2x4,

(ii) V (x) =1

4αx2 − kαx3 − 1

4a2x6,

(iii) V (x) =1

2α(n− 1)xn−2 − 1

4α2x2n−2 − kαxn−1,

(iv) V (x) = −βkx+ (3

2α− 1

4b2)x2 − kαx3 − 1

4αβx4 − 1

4α2x6.

Hence the energy eigenvalue in the first three cases is (−k2) whereas in the last case,

it is (12β − k2).

(4) Method of Separation of Variables

Khare and Bhaduri [51] have recently obtained exact solution of the SE for several non

trivial but separable non central potentials by using this method in polar coordinates.

They have shown that those non central potentials for which the SE is separable are

analytically solvable, provided the separated problem for each of the variable belongs

to the class of exactly solvable one dimensional problems. In particular, the classes

of the potentials investigated in two and three dimensions are

V (r, θ) = U(r) + U1(θ)/r2,

and

V (r, θ, φ) = U(r) + U1(θ)/r2 + U2(φ)/r2 sin2 θ.

In two dimensions, they have also investigated a general case :

U(r) = 14ω2r2 and U1(θ) = G/ sin2 pθ.

For p = 3, an interesting application of these results are further used to the well known

but thus far difficult case of Calogero system [52] described by using the Hamiltonian:

H = −Σi(∂2/∂x2

i ) +1

12ω2Σ(xi − xj)

2 +1

2γΣi<j(xi − xj)

−2. (1.12)

In fact, for the three particles case, this system involves only two degrees of freedom

which may be mapped by the (r, θ) coordinates of a particle in a non central potential,

after the center of mass motion is factored out. It may be of interest to compare

qualitatively the exact solutions for the Calogero potential in eq.(1.12) and for the

harmonic plus inverse harmonic potential in one dimension. It was noticed that latter

case, the coupling of the inverse harmonic term is found to attain only certain discrete

values as a result of the normalization of the eigenfunction.

(5) Supersymmetric Factorization Method:

With the help of this method one can obtain exact analytic solutions of the SE [1, 53]


for one or few quantum states of general polynomial and non polynomial potentials. In

this method, the Hamiltonian satisfying the SE is

Hψn(x) ≡(− h2

2m

d2

dx2+ Veff (x)

)ψn(x) = Enψn(x). (1.13)

For the nth quantum state [57, 58]

H− = A+A−. (1.14)

The nth eigenstate of this later form of the Hamiltonian corresponds to eigen energy

E(−)n = 0. Here A+ and A− are given by

A± = ± h√2m

d

dx+W (x), (1.15)

where W(x) is the super-potential given by

W (x) = − h√2m

d

dxℓnψ(−)

n (x). (1.16)

From eqs.(1.14) and (1.15) the potential V− corresponding to H− is

V− = W 2(x) − h√2m

W ′(x). (1.17)

The nth eigenstate for this potential is obtained from eq.(1.16) as

ψ(−)n = exp

[−

√2m

h

∫ x

W (x′)dx′]. (1.18)

For the value of ‘n’which corresponds to the ground state, the relation given in

eqs.(1.14) and (1.18) correspond to an unbroken supersymmetry and there exist a

super-partner Hamiltonian H+ = A−A+ corresponding to H−. However for the ex-

cited states such a correspondence may not occur. So, for the purpose of obtaining

the exact solutions of eq.(1.13) only H− is important . Now from eqs.(1.13) and (1.17)

we have

V− = Veff + C,

or one may write

E(−)n = En + C,

where ‘C’ is a constant.

Since E(−)n = 0, by construction, then eigenvalue is given by

En = −C,


and subsequently, nth state eigenfunction ψn(x) becomes the same as ψ(−)n (x). As

far as the reproduction of the nodal structures of the wavefunction in this method

is concerned, it can be done by choosing appropriate W (x) which is consistent with

eq.(1.17). However this choice of W (x) may not be unique [54]. One can always write

another W1(x), which generates the same V− then

W1(x) = W (x) + f(x).

Adhikari et.al [53] used this method to study a generalized polynomial potential of

the form

Veff =2N∑j=3

bj−2xj−2 + (α/x) + ℓ(ℓ+ 1)h2/(2mx2), (b2N−2 > 0). (1.19)

The generalized nonpolynomial potential [55, 56] is

Veff = x2 + (m∑

j=0

λjxj−1)(

m∑j=0

gjxj)−1 + V0, (1.20)

where V0 is constant. Further Beckers et.al [55] have, in fact shown that the super-

symmetric three dimensional SE describing central given problem does not lead to

supersymmetric one dimensional radial equation in general.This situation is well sup-

ported by the work of Kostelecky and Nieto [56] within the framework of this method

exactly solvable potentials can be understood in terms of few basic ideas which in-

clude supersymmetric partner potentials, shape invariance and other transformations

[59].

1.4.2 Approximation Methods

In the starting of the quantum mechanics only very simple situations are solved with

the help of exact solution of the SE which can be solved exactly. But in practice, ex-

actly solvable problems are rare and for other cases, exact solution of the SE presents

great mathematical difficulties [1]. In order to discuss all such systems, various meth-

ods of approximate solutions are developed which provide more or less accurate energy

values and wavefunctions. As the exact solution of the SE can be obtained only for a

few potential variants, so the necessity of approximation methods were felt and they

are developed, which give direct solutions of the SE. In the absence of availability

of exact solutions of a problem in quantum mechanics, many approximation meth-

ods are employed and three main approximation methods are namely perturbation,

variational and WKB methods. Further, approximation methods are mainly used for


the one dimensional problems and for two and three dimensional problems are not

so frequently developed and inspite of the fact that the closed form solutions have

become available to check the accuracy and efficiency of these methods.

(i) WKB Method

It is the approximation method which is based on introducing correction to the clas-

sical description. The name of this method is an acronym for Wentzel-Kramers-

Brillouin. Other oftenly used acronym for the method include JWKB approximation

and WKBJ approximation where ‘J’ stand for Jeffreys. This method is named after

physicts Wentzel, Kramers and Brillouin who all developed it in 1926. In 1923,

mathematician Harold Jeffreys had developed a general method of approximating

linear, second order differential equations, which includes the SE. But since the SE

was developed two years later, and Wentzel, Kramers and Brillouin were apparently

unaware of this earlier work, Jeffreys is often neglected credit. This method is based

on a limiting transition from quantum to classical mechanics and it is applicable

when the mass is large, energy is high and potential is smooth (not applicable near a

classical turning point, where the momentum is zero.) .Thus, this method is helpful

when the behavior of micro object differ only slightly from classical behavior.Thus

this method is based on the observations that for a large value of momentum of a

particle moving in a sufficiently smooth field,the equation of motion of a particle differ

little from the classical Newtonian equation and a classical discussion and a quantum

system is approximately justified for the motion of particles with large momentum

in a potential field with a small gradient. This method is valid when the change in

momentum of the particle over a wavelength is much less than the momentum itself

and when the wavelength of the particle varies only slightly over distances of the order

of itself. Mathematically slowly varying potential can be expressed by the condition∣∣∣ 1

k2

dk

dx

∣∣∣≪ 1, (1.21)

k =2π

λ=

(2m[E − V ])1/2

h. (1.22)

From eqs.(1.21) and (1.22) we have

λdVdx

4π(E − V )≪ 1. (1.23)

In this method, solutions are based upon the expansion of the wavefunction, the first

term of which lead to classical results, second term to the old quantum theory results

and the higher terms to the results of the new quantum mechanics.


(ii) Perturbation Methods

This method was given by Schrodinger in 1926. The basic idea is taken from pertur-

bation theory of classical mechanics. This method is used to determine approximately

the eigen values and eigen functions of real systems, which do not differ too much

from the idealized systems for which exact solutions can be found. In these cases,

the approximation methods reduce to the evaluation of corrections to the exact solu-

tions. In these methods the Hamiltonian of the system is splitted into two parts, one

of which is an unperturbed Hamiltonian H0 corresponding to the simplified (unper-

turbed) systems while the other is perturbed term conveniently written as λv where

λ is real small dimensionless parameter and V is the perturbation function i.e.

H = H0 + λV

The perturbation may be considered using two different approaches. The first one,

known as stationary states theory and concerns itself with corrections to the un-

perturbed eigenvalues and eigenfunctions of H0. This approach is used when total

(perturbed) Hamiltonian is of prime physical interest. The second approach,known as

T D Perturbation theory, is used when the unperturbed Hamiltonian is of fundamen-

tal importance and the total Hamiltonian is modified in the course of time. In both

cases Perturbation constant λ must be small. Thus this method is an important tool

for describing real quantum systems, as it turns out to be very difficult to find exact

solutions to the SE for Hamiltonians of even moderate complexity. The Hamiltonians

to which we know exact solutions, such as Hydrogen atom, the quantum harmonic

oscillator and the particle in a box, are too idealized to adequately describe most

systems. For example, by adding a perturbative electric potential to the quantum

mechanical model of the Hydrogen atom, we can calculate the tiny shift in the spec-

tral lines of Hydrogen caused by the presence of an electric field (Stark effect). This

is only approximate because the sum of a Coulomb potential with a linear potential

is unstable although the tunneling time (decay rate) is very long.

(iii) Variational Method

This method is also known as Ritz method, after the name of Walter Ritz. The basis

of this method is variational principle. This method is specially applicable in chemical

problems.

In many cases, perturbation theory cannot be applied successfully as there may not

be closely related problems which are capable of exact solutions. The Helium atom

is such a case. No direct method of solving the wave equation has been found for this

atom and the application of the perturbation theory is unsatisfactory because the


first order approximation is not accurate enough, while it is troublesome to calculate

the higher order approximations. An approximation method which is conventionally

used for such systems is variational method. In quantum mechanics, this method

is used to find the approximation to the lowest energy state or ground state, and

some excited states. It is independent of the value of λ and applies even in the case

when the Perturbation is large. In this method a trial wavefunction Ψ is tested on

the system. This trial function is selected to meet the boundary conditions (and any

other physical constraints). Then calculate the lowest energy eigenvalue and ground

state of the system. However, the accuracy of this method is difficult to determine.

It is more useful in estimating energy eigenvalue rather than the eigen function.

(iv) Quantum Monte Carlo Method

Quantum Monte Carlo method is a large class of computer algorithm that simulates

quantum systems with the idea of solving the many body problems. This method

allows a direct representation of many body effects in the wave function, at the cost

of statistical uncertainty that can be reduced with more simulation time. For bosons,

there exist numerical exact and polynomial-scaling algorithms; for fermions, there ex-

ist very good approximations and numerically exact exponentially scaling Quantum

Monte Carlo algorithms, but none that are both.

Basically, any physical system can be described by the many body SE as long as the

constituent particles are non-relativistic, i.e., they are not moving near the speed of

light. So this method is useful to predict the behavior of any electronic system, in

almost every material in the world, if we could solve the SE. This knowledge has

important applications in fields from computer to biology. The difficulty in the SE

involves a function of three times the number of particles (in three dimensions) and

is difficult (and impossible in the case of fermions) to solve it in a reasonable amount

of time. Quantum Monte Carlo allows us to model a many body wavefunction of

our choice directly. The path integral Monte Carlo and finite temperature auxiliary

field Monte Carlo methods calculate the density matrix where as other most of the

methods aim at computing the ground state wavefunction of the system.

There are several quantum Monte Carlo flavors, each of which uses Monte Carlo in

different ways to solve the many body problems. The various types of quantum Monte

Carlo method are given as

Variational Monte Carlo: It is used in many sorts of quantum problems.

Diffusion Monte Carlo: This is the most common high-accuracy method for elec-

trons (chemical problems), since it comes quite close to the exact ground state energy


fairly efficiently. Also used for simulating the quantum behavior of atoms, etc.

Path Integral Monte Carlo: This method is applied where temperature is very

important, especially superfluid Helium.

Auxiliary Field Monte Carlo: This method is applied to lattice problems, as there

has been recent work on applying it to electrons in chemical systems.

Reptation Monte Carlo: Recent zero-temperature method related to Path inte-

gral Monte Carlo, with applications similar to Diffusion Monte Carlo but with some

different trade offs.

(v) Hartree Fock Method

This method has applications in the solution of the SE of atoms, molecules and solids.

It has also found wide spread use in nuclear physics. In older literature, this method

is called as Self-Consistent Field method. In computational physics and chemistry,

the Hartree Fock method is used to determine the ground state wavefunction and

ground state energy of a many body system. This method assumes that the exact,

N-body wavefunction of the system can be approximated by a single Slater deter-

minant (for fermion) or by a single permanent (i.e bosons) of N spin orbital. By

using the variational principle, we can derive a set of N coupled equation for the N

spin orbitals. Solutions of these equations yield the Hartree Fock wavefunction and

energy of the system, which are approximation of exact ones. The solutions to the

resulting non-linear equations behave as if each particle is subjected to the mean

field created by all other particles. The equations are almost universally solved by

means of an iterative, fixed-point type algorithm. The starting point for Hartree Fock

method is a set of approximate one electron wavefunction known as orbitals. For an

atomic calculation, these are typically the orbitals for a Hydrogen atom. However,

for a molecular or crystalline calculations, the initial approximate one- electron wave-

functions are typically a linear combination of atomic orbitals. These orbitals only

account for the presence of other electrons in an average manner. In the Hartree Fock

method, the effects of other electrons are accounted for in a mean field theory context.

The orbitals are optimized by requiring them to minimize the energy of the respective

Slater determinant. The resultant variational conditions on the orbitals lead to a new

one-electron operator- the Fock operator. At the minimum, the occupied orbitals are

eigen solutions to the Fock operator via a unitary transformation between themselves.

The Fock operator is an effective one-electron Hamiltonian operator being the sum

of two terms. The first is a sum of kinetic energy operator for each electron, the

internuclear repulsion energy, and a sum of nuclear electronic Coulombic attraction

terms. The second are Coulombic repulsion terms between electrons in a mean field


theory description; a net repulsion energy for each electron in the system, which is

calculated by treating all of the other electrons with in the molecule as a smooth

distribution of negative charge. Since the Fock operator depends on the orbitals used

to construct the corresponding Fock matrix, the eigenfunctions of the Fock operator

are in turn new orbitals which can be used to construct a new Fock operator. In this

way, the Hartree Fock orbitals are optimized iteratively until the change in total elec-

tronic energy falls below a predefined threshold. In this way, a set of self consistent

one-electron orbitals are calculated. The Hartree Fock electronic wavefunction is then

the Slater determinant constructed out of these orbitals. Following the basic postu-

lates of the quantum mechanics, the Hartree Fock wavefunction can then be used

to compute any desired chemical or physical property within the framework of the

Hartree Fock method and the approximations employed. Due to the electron-electron

repulsion term of the electronic molecular Hamiltonian involves the coordinates of two

different electrons, it is necessary to reformulate in an approximate way. Under this

approximation, all of the terms of the exact Hamiltonian except the nuclear-nuclear

repulsion term are re-expressed as the sum of one electron operators.

(vi) Density Functional Theory

DFT has been very popular for calculations in solid state physics since the 1970s.

It is versatile method available in condensed matter physics, computational physics

and computational chemistry used to investigate the electronic structure (principally

the ground state) of many body systems, in particular atoms, molecules and the con-

densed phases. With this theory, the properties of many electron systems can be

determined by using functionals i.e. functions of another functions, which in this

case is the spatially dependent electron density. Hence the name Density Functional

Theory (DFT) comes from the use of functionals of the electron density.

In many cases the results of DFT calculations for solid state systems agreed quite

satisfactorily with experimental data and the computational costs were relatively low.

However, DFT was not considered accurate enough for calculations in quantum chem-

istry until the 1990s, when the approximation used in the theory were greatly refined

to better model the exchange and correlation interactions. DFT is now a leading

method for electronic structure calculations in chemistry and solid-state physics.

Despite the improvement in DFT, there are still difficulties in using density functional

theory to properly describe intermolecular interactions, especially Vander Walls forces

(dispersion), charge transfer excitation, transition states, global potential energy sur-

face and some other strongly correlated systems; and in calculation of the band gap

in semiconductors. A poor treatment of dispersion renders DFT unsuitable (at least


when used alone) for the treatments of the systems which are dominated by disper-

sion ( i.e, interacting noble gas atoms) or where dispersion competes significantly with

other effects (i.e, in bio-molecules). The development of new DFT methods designed

to overcome this problem, by alteration to the functional or by the inclusion of addi-

tive terms, is a current research topic.

Within the framework of Kohan-Sham DFT, the intractable many-body problem of

interacting electrons in a static external potential is reduced to a tractable problem

of non-interacting electrons moving in an effective potential. The effective potential

includes the external potential and the effect of the Coulomb interactions between

the electrons i.e. exchange and correlation interactions. For the many-body electron

structure calculations, the nuclei of the treated molecules or clusters are seen as fixed

(the Born-Oppenheimer approximation), generating a static external potential ‘V’ in

which the electrons are moving. A stationary electronic state is then described by a

wavefunction

Hψ = [T + V + U ]ψ =[ N∑

i

(− h2

2m∇2

i ) +N∑i

V (ri) +N∑

i<j

U(ri, rj)]ψ = Eψ,

where H - electronic molecular Hamiltonian; N - number of electrons; T - N-

electron kinetic energy; V - N-electron potential energy from the external field, and

U - electron-electron interaction energy for the N-electron system. The operator T

and U are so-called universal operators as they are the same for any system, while

V is system dependent, i.e. non-Universal. The difference between having separable

single-particle problems and the much more complicated many-particle problem arises

from the interaction term U .

In the DFT the key variable is the particle density n(r), which for a normalized ψ is

given by

n(r) = N∫d3r2

∫d3r3......

∫d3rN ψ∗(r1, r2, .......rN)ψ(r1, r2, .......rN).

This relation can be reversed, i.e. for a given ground-state density n0(r), it is possible,

in principle, to calculate the corresponding ground-state wavefunction ψ0(r1, r2, .......rN).

In other words, ψ0 is a unique functional of n0

ψ0 = ψ[n0].

Consequently the ground state expectation value of an observable O is also a func-

tional of n0

O[n0] = ⟨ψ[n0] |O|ψ[n0]⟩.


In particular, the ground state energy is a functional of n0

E0 = E[n0] = ⟨ψ[n0] |T + V + U |ψ[n0]⟩.

Where the contribution of the external potential ⟨ψ[n0] |V |ψ[n0]⟩ can be written

explicitly in term of the ground-state density n0

v[n0] =∫

V (r)n0(r)d3r.

More generally, the contribution to the external potential ⟨ψ |V |ψ⟩ can be written

explicitly in terms of the density n

v[n] =∫

V (r)n(r)d3r.

Usually one starts with an initial guess for n(r), then calculate the corresponding V

and solve the the Kohan-Sham equation for the ϕi

− h2

2m+ V (r)]ϕi(r) = ϵiϕi(r).

From these, one can calculate a new density and starts again. This procedure is then

repeated until convergence is reached.

1.5 Various ways of Complexifying a Hamiltonian

The non-hermitian Hamiltonians are considered as complex Hamiltonian (H) as they

do not provide real eigenvalues,however,its PT - symmetric form exhibits real eigen-

values. Generally the Hamiltonian is expressed as the sum of kinetic energy and

potential energy and it is total energy operator.In one dimension, it is expressed as

H(x, p) =p2

2m+ V (x). (1.24)

For a physical system the first term is fix where as the second term V(x), can have

different functional forms depending upon the system under study. In the last decade,

many interesting investigations on non-hermitian potentials were carried out and these

studies showed that even complex Hamiltonian can generate real eigenvalues under

certain conditions. The Various ways of Complexifying a Hamiltonian are :

Method-I:- The easiest way to made Hamiltonian complex is, by assuming potential

function V (x) as complex [18], say V (x) = Vr(x) + iVi(x). Feshbach et.al [19] used

the simplest choice square-well potential, namely V (x) = V0 + iW0, in the optical

1.5. VARIOUS WAYS OF COMPLEXIFYING A HAMILTONIAN 19

model of the nucleus.

Method-III:- In this case two independent complex variables are defined as u =

x/b+ ip/c and v = x/b− ip/c where b and c are complex numbers which give rise to

a variety of possibilities regarding the nature of the transformation from (x, p) space

to (u, v) space, so that u, u∗, v, v∗ become the new degrees of freedom for describing

the system. Also for b = b∗ and c = c∗ , u and v become complex conjugate pairs

[32].

Method-IV:- In studying the second quantized version of the harmonic oscillator

in field-theory or the Heisenberg quantum mechanics of the oscillator , a real two di-

mensional phase space (x-p plane) is converted into a complex x-p plane say z-plane

s.t.

z = p+ iω0x, z∗ = p− iω0x,

and writes H(x, p) as H(z, z∗) for real x, p and ω0.

Method-II:- The Hamiltonian can also be made complex by just considering the

parameters in V (x) as complex, however x and p are kept as real [18].

Method-V:-Another way to convert hermitian HamiltonianH(x, p) to non-hermitian

Hamiltonian consists in, by considering each physical variable x and p as complex,

i.e.

x = x1 + ix2, p = p1 + ip2, (1.25)

x = x1 + ip2, p = p1 + ix2, (1.26)

where (x1, p1) and (x2, p2) are real and considered as canonical pairs [20]. So, one can

transform a real phase space (x, p) into a complex phase space (x1, p2, p1, x2) with

two additional degrees of freedom namely x2 and p2. The complex phase space [21]

is known as the extended complex phase space (ECPS) approach. So in this method,

both x and p are separately made complex.

Above two transformations given by eq.(1.25) are used in study of ion-acoustic waves

in plasma by Rao et.al [22], in developing a complex mechanics by Yang [23] ,in

coherent state studies by Xavier and de Aguiar [24] and by Kaushal et.al [25, 26,

27, 28, 29] in studying certain aspects of classical and quantum mechanics. During

complexification of H some impotant facts must be kept in mind:

1. It is type of the potential and its parameters which decides the nature of the

trajectory of the particle. So, complexification of potential parameters only allow

some variations around main track of the trajectory. On the other hand, the type


of the complexification inducted by eq.(1.26) brings in two additional variables

(x2, p2) in hyperspace. These varibles may or may not have any link with the

spatio-temporal cartesian grid and they follow the same rule as the physical

variables (x1, p1). Thus the variables (x1, p1) and their functions will account

for the physical reality in the nature where as the variables (x2, p2) and their

functions will account for the spurious effect (reality that is beyond the physical

reality ).

2. Some of the approaches based on the parametric complexification of H followed

recently [29, 30] to study the real eigenvalue spectra of non-hermitian Hamilto-

nians.

In methods-III and IV the complex phase space is obtained by assuming the variables

x and p as real and parameter space as complex. There are also other methods in

which x and p are separately made complex by inserting imaginary component in

each.

1.6 Concept of Non-Hermitian Hamiltonian

In literature, concept of non-hermitian Hamiltonian was first of all used by Wu in

1959 for calculating the ground state energy of “Bose sphere” [31] without divergence.

Remarkably, Hamiltonian possessed real eigenvalues without any justification for us-

ing such Hamiltonian.

In 1967, Wong pointed out that Hamiltonian losses its hermiticity due to external

interaction. So,a class of physically reasonable Hamiltonians would be perturbed type

and the generalised form is

H = H0 + H(1),

where H(1) = gH is the perturbation term in which ‘g’ is a parameter to vary the

influence of H and H0 is hermitian Hamiltonian. H is non-hermitian Hamiltonian

with the restriction that it may have only discrete spectrum unless part of its spectrum

coincides with that of H0. Later in 1969 Wu and Bender [32] discussed an example

of the anharmonic oscillator without explicitly discussing hermitian or non-hermitian

Hamiltonians. The Hamiltonian for such a model is

H =p2

2m+

1

2µ2x2 − gx4. (1.27)

The parameter g is extended into the complex plane, so the non-hermiticity of the

Hamiltonian is implicit. Then in 1975, Haydock and Kelly [33] used non-hermitian

1.7. QUANTUM MECHANICAL THEORY FOR HERMITIAN HAMILTONIANS 21

Hamiltonians to calculate the electronic structure of crystalline Arsenic and in 1980,

Stedman and Butler [21] first of all used time reversal operator in relation to complex

conjugation, as it is used in PT /CPT symmetry. Afterward a quantum decay process

with a non-hermitian Hamiltonian and Schrodinger wave equation was studied and

claimed that a decaying state could not have a sharp energy, and that the width of such

an energy level could be represented by an imaginary energy component. The common

factor in all these papers was that such theories were introduced on phenomenological

or heuristic grounds and they were fitted with experimental observations. In 1997,

Hatano and Nelson justified the use of a complex spectrum [34]. For a physically

acceptable non-hermitian theory the requirements are :

• The energy eigen spectrum must be completly real.

• Time evolution must be unitary.

• The set of eigenstates must be complete.

In light of above discussion the mathematical axiom of the hermiticity (H = H†) is

replaced by the physical transparent condition of space-time reflection (PT )- sym-

metry i.e. H = HPT and HPT ≡ (PT )H(PT ). So family of new PT - symmetric

Hamiltonians are-

H = p2 + x2(ix)ϵ, (1.28)

where, ϵ parameter is real. The energy eigenvalues of these Hamiltonians are real

and positive when ϵ ≥ 0 as shown by Bender et.al in 1998 [35] as they all are PT -

symmetric. But when ϵ < 0 there are complex eigenvalues. So it was concluded that

ϵ ≥ 0 is the parametric region of the unbroken PT - symmetry and that ϵ < 0 is the

parametric region of the broken PT - symmetry (see fig-1).

Non-hermitian Hamiltonians in eq.(1.28) can also think of either complex extension

of the harmonic oscillator Hamiltonian H = p2 + x2 or one can begin with any of

the hermitian Hamiltonians H = p2 + x2N , where N = 1, 2, 3....., and introduce the

parameter ϵ as follow: H = p2 + x2N(ix)ϵ with special case N = 1.

1.7 Quantum Mechanical Theory for Hermitian Hamiltoni-

ans

An obvious question for real and positive eigenvalues of PT - symmetric Hamilto-

nian is that, does a non-hermitian Hamiltonian such as given by eq.(1.28) define a


physical theory of quantum mechanics or is the reality and positivity of the spec-

trum merely an intriguing mathematical curiosity exhibited by some special classes

of complex eigenvalue problems. A physical quantum theory must have an energy

spectrum which is bounded below, possess a Hilbert space of state vectors which are

endowed with an inner product having a positive norm and unitary time evolution.

The simplest condition on the Hamiltonian H for the fulfilment of above requirements

is that the H be real and symmetric. However, One can allows H to be complex as

long as it is Dirac hermitian: H† = H.

In this section, we summarize the procedure for analyzing a theory defined by a con-

ventional hermitian quantum mechanical Hamiltonian.

1. Eigenfunctions and eigenvalues of H :- The time-independent Schrodinger

equation associated with given Hamiltonian(H) can be written and the eigenfunc-

tions Ψn(x) as well as eigenvalues En can be calculated. Generally above calcula-

tions are performed numerically as it is difficult to carry out calculations analytically.

2. Orthogonality of eigenfunctions :- For hermitian Hamiltonian H eigenfunc-

tion will be orthogonal w.r.t. the standard hermitian inner product if

(Ψ,Φ) ≡∫

Ψ(x)∗Φ(x)dx. (1.29)

The two eigenfunctions of H say Ψm(x) and Φn(x) are said to be Orthogonal if their

inner product associated with two different eigenvalues Em and En vanishes

(Ψm,Φn) = 0. (1.30)

3. Orthonormality of eigenfunctions :- We can normalize the eigenfunctions of

H due to its hermiticity and the norm of any vector is guaranteed to be positive, so

that the norm of every eigenfunction is unity then

(Ψn,Ψn) = 1. (1.31)

4. Completeness of eigenfunctions :- This theorem states that any (finite norm)

vector χ can be expressed as a linear combination of eigenfunctions of H in the Hilbert

space for linear operators

χ =∞∑

n=0

anΨn. (1.32)

Above statement corresponds to completeness in co-ordinate space and it is the re-

construction of the unit operator as a sum over the eigenfunctions

∞∑n=0

[Ψn(x)]∗Ψn(y) = δ(x− y). (1.33)

1.8. QUANTUM MECHANICAL THEORY OF NON-HERMITIAN HAMILTONIANS 23

5. Time evolution and unitarity :- The time evolution operator e−iHt is unitary

and preserves the inner product for a hermitian Hamiltonian[χ(t), χ(t)

]=[χ(0) eiHt, e−iHtχ(0)

]=[χ(0), χ(0)

]. (1.34)

6. Observables :- An observable is represented by a linear hermitian operator. The

outcome of a measurement is one of the real eigenvalues of this operator.

7. Assortment :- Besides these, a number of topics like classical and semiclassical

limit of quantum theory, probability and current density for perturbative and non-

perturbative calculations can also be considered for hermitian Hamiltonians .

1.8 Quantum Mechanical Theory of Non-Hermitian Hamil-

tonians

Following the steps outlined above now we study a non-hermitian Hamiltonian having

unbroken PT - symmetry. But in this case, we do not know the definition of inner

product. We will have to discover the correct inner product in the course of our

analysis which is determined by the Hamiltonian itself. The steps of procedure are

as follows:

1. Eigenvalues and eigenfunctions of H :- Here the assumption is that the

eigenvalues (En) can be found by using analytical or numerical methods and the

eigenvalues are all real, so it is equivalent to assuming the PT - symmetry of H is

unbroken. Hence all the eigenfunctions of H are also the eigenfunctions of PT opera-

tor. The zeros of PT - symmetric eigenfunctions have interesting complex interlacing

properties [36, 37, 38].

2. Orthogonality of eigenfunctions :- Orthogonality of the eigenfunctions can be

test iff we specify an inner product. Since we do not yet know what inner product to

use, so we can try to guess an inner product. By analogy, one can think that since the

inner product is appropriate for hermitian Hamiltonian (H = H†), so, a good choice

for an inner product associated with a PT - symmetric Hamiltonian (H = HPT )

might be

(Ψ,Φ) ≡∫

cdx[Ψ(x)]PT Φ(x) =

∫cdx[Ψ(−x)]∗Φ(x), (1.35)

where ‘c’ is a contour. With this definition of inner product, it can be shown by a

trivial integration by parts using time-independent SE that pairs of eigenfunctions of

H associated with different eigenvalues are orthogonal. However, for formulating a

valid quantum theory this guess for an inner product is not acceptable because the


norm of a state is not necessarily positive.

3. The CPT inner Product :- For a complex non-hermitian Hamiltonian having

an unbroken PT - symmetry, to construct an inner product with a positive norm a

new linear operator C that commutes with both H and PT will be constructed. As

C commutes with the Hamiltonian, it represents a symmetry of H and this symmetry

is denoted by the symbol C because the properties of C are similar to those of the

charge conjugation operator in particle physics [30]. The inner product w.r.t. CPTconjugation is defined as

⟨Ψ|χ⟩CPT =∫dx ΨCPT (x) χ(x), (1.36)

where ΨCPT (x) =∫dy C(x, y) Ψ∗(−y). It can be verified that this inner product

satisfies the requirements for the quantum theory defined by H to have a Hilbert

space with a positive norm and to be a unitary. The C operator is represented as

a sum over the eigenfunctions of H, but before doing so one must first show how to

normalize these eigenfunctions.

4. PT - symmetric normalization:- As the eigenfunctions Ψn(x) is common eigen-

function of both H and PT operator with eigenvalue λ = eiα, where λ and α depend

on ‘n’. Thus, PT -normalized eigenfunctions Φn(x) can be constructed and it is de-

fined by

Φn(x) ≡ e−iα/2Ψ(x). (1.37)

By this construction, Φn(x) is still an eigenfunction of both H and PT operator with

eigenvalue ‘1’and it can be easily shown both numerically and analytically that the

algebraic sign of the PT norm in (1.35) of Φn(x) is (−1)n for all n and for all values

of ϵ > 0 [39, 40]. Thus, we define the eigenfunctions so that their PT norms are

exactly (−1)n∫cdx[Φn(x)]PT Φn(x) =

∫cdx[Φn(−x)]∗Φn(x) = (−1)n, (1.38)

where c is a contour. In terms of these PT normalized eigenfunctions there is a simple

but unusual statement of completeness

∞∑n=0

(−1)nΦn(x)Φn(y) = δ(x− y). (1.39)

One can verify this statement of completeness , both numerically and analytically for

all ϵ > 0 [41, 42].

5. Construction of the C-operator :- For any Hamiltonian H with an unbroken

PT - symmetry there exist an additional symmetry due to its equal number of positive

1.8. QUANTUM MECHANICAL THEORY OF NON-HERMITIAN HAMILTONIANS 25

and negative norm states. The linear operator C that includes this symmetry can be

represented in coordinate space as a sum over the PT normalized eigenfunctions of

PT - symmetric Hamiltonians as

C(x, y) =∞∑

n=0

Φn(x)Φn(y). (1.40)

It can be noted that above equation is identical to the statement of completeness in

eq.(1.39) with exception that the factor (−1)n is absent. We can use eqs.(1.38) and

(1.39) to verify that the square of C is unity (C2 = 1)∫C(x, y)C(y, z)dy = δ(x− z). (1.41)

Thus, the eigenvalues of C are ±1. Also as C commutes with H so, the eigenstates of

H have definite value of C since C is linear. Specifically

C Φn(x) =∫dy C(x, y) Φn(x) =

∞∑m=0

Φm(x)∫dy Φm(y)Φn(y)

= (−1)nΦn(x). (1.42)

Again this new operator C resembles the charge-conjugation operator in the quantum-

field theory. However, the exact meaning of C is that it represents the measurement

of the sign of the PT norm in (1.38) of an eigenstate.

The operator P and C are different square roots of unity operator δ(x − y). That

is P2 =C2 = 1, but P = C because P is real and C is complex. The parity operator

in coordinate space is explicitly real [P(x, y)=δ(x+ y)], while the operator C(x, y) is

complex because it is sum of product of complex functions. The two operators P and

C do not commute. However C does commute with PT .

6. Positive norm and unitarity in PT - symmetric quantum mechanics :-

With the construction of the C operator now it is easy to use the new CPT inner

product defined in eq.(1.36). This new inner product is also phase independent as that

of PT inner product. Also, because the time evolution operator (as in the ordinary

quantum mechanics) is e−iHt and because H commutes with PT and CPT operators

respectively. Both the PT inner product and the CPT inner product remain time

independent as the states evolve. However, unlike the PT inner product, the CPTinner product is positive definite because C contributes a factor of ‘-1’ when it acts

on states with negative PT norm. In terms of the CPT conjugate, the completeness

condition will be∞∑

n=0

Φn(x)[CPT ,Φn(y)

]= δ(x− y). (1.43)


1.9 Relation between Hermitian and PT - symmetric quan-

tum theories

In 2003, Bender et.al [43] claimed that all hermitian Hamiltonian are PT - symmetric,

so PT - symmetry is a generalization of hermiticity .It can be shown : Let us consider

the general class of Hamiltonian as-

H = P 2 + V (x) (1.44)

For this H to be hermitian V (x) needs to be real. Showing that eq.(1.44) is T invariant

trivially because it is entirely real, so that complex conjugation has no effect, therefore

all hermitian Hamiltonians have parity. On this basis we can assume that parity

operator for a given system commutes with the Hamiltonian of the system , then we

can construct the parity operator in terms of the simultaneous eigenfunctions. Since

the set of the eigenfunctions for a given Hamiltonian is a complete set, then one may

write the Hamiltonian as a matrix represented in coordinate space

H(x, y) =∞∑

n=0

Enψ(x)ψ∗(y), (1.45)

or using Dirac notation

⟨x|H|y⟩ =∞∑

n=0

En⟨x|ψn⟩⟨ψn|y⟩. (1.46)

Then by analogy, one may construct the parity operator because the eigenfunctions of

the any quantum mechanical system are complete and orthonormal and it is confirmed

that the parity operator obeys the four properties listed below:

1. P is linear and hermitian.

2. P and H commute: [H, P ] = 0 (if H is P invariant).

3. P 2=1.

4. The nth eigenstate of H is also the nth eigenstate of P with eigenvalue, (−1)n.

Based upon this result, it can be concluded that all hermitian Hamiltonian are PT -

symmetric. This is a proposition which is elegant in its simplicity, but neither nu-

merical nor analytical verifications were given. But Bender et.al [43] contradict the

previous conclusion that all hermitian Hamiltonians have parity. They argued that

PT - symmetric quantum mechanics was not a generalization of hermitian quantum

mechanics. Instead one should start with a real symmetric Hamiltonian, then ex-

tend its matrix element into the complex domain in such a way that the appropriate

1.9. RELATION BETWEEN HERMITIAN AND PT - SYMMETRIC QUANTUM THEORIES27

requirements to determine the spectra and the time evolution (unitary quantum me-

chanics) are satisfied. An overlap of the class of hermitian Hamiltonians and the class

of PT - symmetric Hamiltonian (see fig-2) form a class of real symmetric Hamiltoni-

ans. Either matrix elements can be moved into complex domain with still hermitian

matrix and real spectra, or matrix elements can be generalizied for the case of PT -

symmetric Hamiltonians, on the condition that the PT operator commutes with the

Hamiltonian, thus ensuring real spectra, are the two ways to generalize the real sym-

metric Hamiltonian. Following either of these recipes will construct self consistent

theories of quantum mechanics, if one accepts the PT - symmetric quantum mechan-

ics.


In this context, Bender et. al [43] compared these cases for a finite D-dimensional

system, on the basis of how many parameters are required to describe each case, which

lead to quantitative comparison of size of each class. This comparison was stated as :

“for large D, a hermitian matrix would have D2 parameters, a PT - symmetric matrix

would have asymptotically 34D2 parameters and a real symmetric matrix would have

asymptotically 12D2 parameters”. The Hilbert space of physical states is specified

even before the Hamiltonian is known for the formulation of a conventional quantum

theory as defined by a hermitian Hamiltonians, . The inner product in this vector

space is defined with respect to Dirac hermitian conjugation. The Hamiltonian is

then chosen and the eigenvectors and eigenvalues of Hamiltonian are determined. In

contrast,in case of non-hermitian PT - symmetry before knowing the Hilbert space

and the associated inner product of the theory, Hamiltonian is determined dynami-

cally and then One must solve for the eigenstates of H as Hamiltonian depends on

the Hamiltonian itself .The C operator is defined and constructed in terms of the

eigenstates of the Hamiltonian and the inner product for a quantum theory is defined

by a non-hermitian PT - symmetric Hamiltonian.

The Hilbert space, which consists of all complex linear combinations of the eigen-

states of H, and the CPT inner product are determined by these eigenstates. Time

evolution is expressed by the operator e−iHt whether the theory is determined by a

PT -symmetric or ordinary hermitian Hamiltonian. To establish unitarity we must

show that as a state vector evolves, its norm does not change in time. If Ψ0(x) is any

given initial vector belonging to the Hilbert space extended by the energy eigenstates,

then it evolves into the state Ψt(x) at time ‘t’ according to Ψt(x) = e−iHT Ψ0(x). With

respect to the CPT inner product the norm of Ψt(x) does not change in time because

H commutes with CPT .

The Organization of the Thesis:

In the present study we are interested in exact solutions of the SE for different polyno-

mial potentials. For this purpose, we used the ansatz for the wavefunction method to

find the solutions of such systems. The complex system version of the ansatz method

along with some illustrative examples is given in chapter-II, while the searching for

critical point nuclei as an application of sextic potential is elaborated in chapter-

III. Finally, the relevant applications of non-hermitian Hamiltonian available in the

literature and concluding remarks are presented in chapter-IV.

Chapter 2

The SE for Complex Systems

A simple view of the quantum mechanical on complex Hamiltonians and their methods

are described in the first chapter. The PT - symmetric theory of non- hermitian and

other various methods to solve SE are also discussed. As a matter of fact the problem

of polynomial interactions in one dimension in quantum theory is an interesting area

of research. Hence here we would like to extend this work in two dimensions. Using

the eigenfunction ansatz method, we have solved the quartic and quintic potentials.

2.1 Introduction

In the last decade many interesting investigations on the PT -symmetric quantum me-

chanics have generated a renewed interest in the analysis of complex (non-hermitian)

potentials [1, 2, 3, 4, 5, 6, 7]. These studies show that a non-hermitian Hamiltonian

can generate real and bounded eigenvalues except when PT -symmetry is sponta-

neously broken in which case its complex eigenvalues should come in conjugate pairs.

Therefore, now it is possible to investigate the eigenvalue spectra of a number of

non-hermitian Hamiltonian systems by imposing the PT -symmetric condition.

Most of such studies, however, are restricted to only for one dimensional systems

and their generalization in higher dimensions is needed for studying some nontrivial

applications. With this motivation, recently we have carried out some studies of two

dimensional complex systems [8, 9, ?] within the framework of an extended com-

plex phase space. Here, with the same spirit, we find the quasi-exact solutions of the

Schrodinger wave equation (SE) for a generalized PT -symmetric complex quartic and

quintic potentials. In quantum mechanics, the quasi-exact solvable systems are those

for which it is possible to find a finite number of (i.e. some specific or isolated) exact

eigenvalues and the corresponding eigenfunctions in closed form. The non-hermitian

29

30 CHAPTER 2. THE SE FOR COMPLEX SYSTEMS

PT -symmetric quartic Hamiltonian, called ‘wrong sign’ Hamiltonian, of the form

H =p2

2m− gx4, g > 0, (2.1)

and its some variants have been studied extensively in the past [7, ?, ?, ?, ?, ?]. The

Hamiltonian (2.1) is particularly interesting due to its analogy with a −ϕ4 quantum

field theory to model the dynamics of Higg’s sector of the Standard model. But

the main difficulty with (2.1) is that the wavefunctions don’t vanish exponentially as

|x| → ∞, but oscillate on positive and negative real x-axes. However, when a cubic

term iλx3 is added to (2.1), the wavefunctions decay exponentially on real x-axis and

also in the interior of the Stoke’s wedges. Thus the presence of the imaginary cubic

term in (2.1) enables one to calculate the C-operator perturbatively which is essential

to develop a consistent quantum theory to deal with non-hermitian Hamiltonians.

Although there are various ways for obtaining complex Hamiltonians [12, ?], but

here in the present work, a complex quartic and quintic Hamiltonian are derived by

choosing potential coupling parameters as complex. The same scheme of complexi-

fication has also been used in [13] for studying a family of one dimensional complex

PT -symmetric sextic potentials [?, ?, ?].

In literature, there exist several methods for solving the SE for dynamical systems

[14, ?], however a technique known as the ansatz for the eigenfunction method has

been explored for obtaining ground and excited state energies of a variety of real po-

tentials [15, ?, ?, ?]. Very recently, Midya and Roy [16] investigated the quasi-exact

solutions of the position dependent mass Schrodinger equation for one dimensional

sextic potential using this technique. Further the same method has also been suc-

cessfully used to obtain the eigenvalue spectra of a number of non-hermitian complex

potentials [6, 8, 9, ?, 17].

The study of non-hermitian Hamiltonians demand special attention due to their some

interesting applications in several areas of theoretical physics like superconductivity,

population biology, quantum cosmology, condensed matter physics, quantum field

theory. Specially quintic potentials have been used in the studies of vibrations of the

atomic and molecular systems, to model optical bistability in a dispersive medium

where the refractive index is optical intensity dependent, to improve image sharpen-

ing in the presence of noise, to represent partially folded intermediates in proteins, to

model a magnetoelastic beam in the nonuniform field of permanent magnets [11](and

ref. theirin) and to study shape evolution in isotope chains [12]. Various aspects of

quintic potentials, in real as well as in complex forms, are studied by many authors

[9, 11, 12, 13, 14, ?, 15, 16, 17, 18].

2.2. THE METHOD 31

2.2 The Method

Here we describe the essential steps of the ansatz for the eigenfunction method for

obtaining the solutions of the SE for two dimensional systems.

The SE is written (for h = m = 1) as

∂2ψ

∂ x2+∂2ψ

∂ y2+ 2(E − V (x, y))ψ(x, y) = 0. (2.2)

Next we make an ansatz for eigenfunction of the form

ψ(x, y) = ϕ(x, y)e−g(x,y). (2.3)

On substituting eq.(2.3) in eq.(2.2), we obtain


2 + 2(E − V ) +1

ϕ(−2ϕxgx − 2ϕygy + ϕxx + ϕyy) = 0, (2.4)

where the subscripts to function g and ϕ indicate the differentiation w.r.t. the vari-

ables x and y.

From the structure of the above equation, it is clear that if the functions g and ϕ

are known for a given system, then rationalization of eq.(2.4) would directly pro-

vide the energy eigenvalues and eq.(2.3) would then act as the wavefunction for

the system. However, the results for the ground state can be obtained by setting

ϕ(x, y) = constant. Therefore, for the ground state solutions, eq.(2.4) reduces to


2 + 2(E − V ) = 0. (2.5)

Hence to implement this scheme to get the solution of the SE for a specific potential

essentially requires some suitable forms of g(x, y) and ϕ(x, y). For polynomial type

of potentials, these may be assumed polynomials as well.

So in the next section we consider a two dimensional coupled quartic potential and find

its eigenvalue spectra under a suitable ansatz for eigenfunction by solving eq.(2.4).

2.3 The Illustrative Examples

Using the method described above, we first find the solution of the SE for various

states by considering a simple and well known example of a PT -symmetric quartic

and quintic potential along with their some variants in two dimensions.


2.3.1 Solutions for PT -symmetric quartic potential

Let us consider a general two dimensional coupled quartic complex potential

V (x, y) = a10x+ a01y + a20x2 + a02y

2 + a11xy + a30x3

+a03y3 + a12xy

2 + a21x2y + a22x

2y2 + a31x3y

+a13xy3 + a40x

4 + a04y4, (2.6)

where the parameters aij are constants. The potential (2.44) will be PT -symmetric,

if a10, a01, a12, a21, a30, a03 ∈ i ℜ and a20, a02, a11, a22, a31, a13, a40, a04 ∈ ℜ.

For the present system, the ansatz for the function g(x, y) is made as

g(x, y) = α10x+ α01y + α20x2 + α02y

2 + α11xy

+α12xy2 + α21x

2y + α30x3 + α03y

3, (2.7)

where the coefficients α10, α01, α12, α21, α30, α03 ∈ i ℜ and α20, α02, α11 ∈ ℜ are

chosen to ensure the PT -symmetry of the wavefunction.

Now to solve the SE for the potential (2.44), in the following subsections, we assume

the polynomial forms of ϕ(x, y) for the first four states.

as

(i)ϕ(x, y) = 1, (ii) ϕ(x, y) = x+ y + γ1,

(iii) ϕ(x, y) = x2 + y2 + a3xy + a2y + a1x+ a0.

For the complex potential (2.44), γ1 is imaginary in (ii), whereas a1 and a2 are imag-

inary, but a0 and a3 are real in (iii).

The Ground State Solutions: For the ground state solution, take ϕ(x, y) = 1.

Thus, using eqs.(2.44) and (2.45) in eq.(2.5) and on rationalization, we get the fol-

lowing set of algebraic equations

E = α02 + α20 −1

2(α2

10 + α201), (2.8)

a01 = α10α11 + 2α02α01 − 3α03 − α21, (2.9)

a10 = α01α11 + 2α20α10 − 3α30 − α12, (2.10)

a12 = 4α02α12 + 3α11α03 + 2α20α12 + 2α11α21, (2.11)

a21 = 4α20α21 + 3α11α30 + 2α02α21 + 2α11α12, (2.12)

2.3. THE ILLUSTRATIVE EXAMPLES 33

a11 = 2(α20α11 + α02α11 + α21α10 + α01α12), (2.13)

a22 = 2α221 + 2α2

12 + 3α12α30 + 3α21α03, (2.14)

a02 = 3α03α01 + α10α12 + 2α202 + α2

11/2, (2.15)

a20 = α21α01 + 3α30α10 + 2α220 + α2

11/2, (2.16)

a03 = 6α02α03 + 2α11α12, (2.17)

a30 = 6α20α30 + 2α11α21, (2.18)

a31 = 6α21α30 + 2α12α21, (2.19)

a13 = 6α12α03 + 2α12α21, (2.20)

a40 = (9α230 + α2

21)/2, (2.21)

a04 = (9α203 + α2

12)/2, (2.22)

among the potential coupling parameters, wavefunction parameters and the energy

E. The above equations provide solutions of potential parameters aij in terms of the

wavefunction parameters αij. Therefore for various choices of αij, different aij will be

obtained. In this way one can obtain the eigenvalue spectra of a family of complex

quartic potentials in two dimensions. Note that the energy eigenvalue, eq.(2.46), is

real as α20, α02 ∈ ℜ.It is worth to mention that the potential (2.44) essentially forms a two dimensional

quartic ‘wrong sign’ Hamiltonian because the potential coefficients a31, a13, a40 and

a04, eqs. (2.57)-(2.60), are negative numbers as α12, α21, α03 and α30 are assumed

imaginary constants.

First Excited State Solution: Now we consider the second case, for which ϕ(x, y)

is given as

ϕ(x, y) = x+ y + γ1, (2.23)

where γ1 is a complex constant. So, using eqs.(2.44), (2.45) and (2.74) in eq.(2.4), we

get a set of seventeen equations out of which twelve equations are same as given by

eqs.(2.49)- (2.60) and the remaining five equations are written as

E = γ1(α21 + a01 + 3α03 − α10α11 − 2α10α02) + α20 + α11 − α210 + 3α02, (2.24)


E = γ1(α12 + a10 + 3α30 − α10α11 − 2α10α20) + α02 + α11 − α210 + 3α20, (2.25)

E = (α20 + α02 − α210)/γ1 − 2α10, (2.26)

a10 = −α12 − α21 − 6α30 + α11α01 + 2α10α20, (2.27)

a01 = −α12 − α21 − 6α03 + α11α10 + 2α01α02. (2.28)

In this case we get three relations for energy E. These expressions will provide unique

and real energy eigenvalues when

α01 = α10, α20 = α02, α12 = α21 = −3α30 = −3α03. (2.29)

Thus under these restrictions, the energy eigenvalue E and the value of γ1 are com-

puted as

E = 4α20 + α11 − α210, (2.30)

γ1 =−2α10

α11 + 2α20

. (2.31)

Note that the energy is real and discrete as α20, α11 ∈ ℜ.

Second Excited State Solutions: For obtaining the energy eigenvalues of the

second excited state of the system (2.44), we take the third choice of ϕ(x, y) as

ϕ(x, y) = x2 + y2 + a3xy + a2x+ a1y + a0, (2.32)

where a0 and a3 are real numbers while a1 and a2 are pure complex constants.

Again inserting eqs. (2.44), (2.45) and (2.32) in eq. (2.4), we obtain a set of twenty

algebraic equations. Out of these, six relations involving energy and two for a10 and

a01 are different from the eqs.(2.49)- (2.60). These relations are given as

a10 = −α12 − a3α21 − 9α30 + 2α10α20 + α11α01, (2.33)

a01 = −a3α12 − α21 − 9α03 + α11α10 + 2α01α02. (2.34)

aij under different choices of ϕ(x, y) one must consider - a3 = 2, α21 = −3α30, α12 =

−3α03. There are also conditions on a0, a1, a2 ,which provide values of these constants

- which will be real for a1 = a2and

The six relations involving E provide the energy equation as

E = 6α20 + 2α11 − (α210 + α2

01)/2, (2.35)


under the same condition (2.29). The coefficients a0, a1, a2 and a3 are calculated as

a0 =(2α20 + α11) + (α01 + α10)

2

(2α20 + α11)2, (2.36)

a1 = a2 =α10 + α01

α20 + α11/2, a3 = 2. (2.37)

In this case, the energy eigenvalues also come out to be real and discrete.

Third Excited State Solutions: Finally for obtaining the energy eigenvalues

of the third excited state of the system (2.44), the fourth choice of ϕ(x, y) is made as

ϕ(x, y) = x3 + y3 + β21x2y + β12xy

2 + β11xy + β20x2 + β02y

2 + β10x+ β01y + β0,(2.38)

where β11, β20, β02 and β0 are considered as real numbers while β21, β12, β10 and β01

as complex constants.

Again inserting eqs. (2.44), (2.45) and (2.90) in eq. (2.4), we obtain a set of twenty

four algebraic equations. Out of these, ten relations involving energy and two for

a10 and a01 are different from the eqs.(2.49)- (2.60). a10 = −α12 − a3α21 − 9α30 +

2α10α20 + α11α01, (2.38)

a01 =

-a3α12 − α21 − 9α03 + α11α10 + 2α01α02.(2.39)aij under different choices of ϕ(x, y)

one must consider - a3 = 2, α21 = −3α30, α12 = −3α03. There are also conditions on

a0, a1, a2 ,which provide values of these constants - which will be real for a1 = a2and

These ten relations involving E again reduce to a single energy equation

E = 8α20 + β21α11 + β11α21 − 12α30 − α210, (2.40)

under the restriction (2.29). Further, the solutions of various βij in terms of αij may

be obtained by selecting β21 = β12, β20 = β02 and β10 = β01 = 1. The coefficients

β20, β21, β11, β0 are given as

β20 =16α20 + 2α11(8 + α−1

30 ) + 3

12α30 + 2α10 − 18, β21 = 3. (2.41)

β11 =−3

4− α11

2α30

+9(16α20 + 2α11(8 + α−1

30 ) + 3)

4(6α30 + α10 − 9), (2.42)

β0 =2α10 − 2β20

6α20 + 3α11 + β11α21 − 12β20α30

, (2.43)

In this case, the energy eigenvalue also comes out to be real and discrete.


2.3.2 Solutions for PT -symmetric quintic potential

Consider a general two dimensional coupled PT -symmetric quintic potential

V (x, y) = a10x+ a01y + a20x2 + a02y

2 + a11xy + a30x3 + a03y

3 + a12xy2

+a21x2y + a22x

2y2 + a31x3y + a13xy

3 + a40x4 + a04y

4 + a50x5

+a05y5 + a14xy

4 + a41x4y + a23x

2y3 + a32x3y2 + a06y

6 + a60x6

+a15xy5 + a51x

5y + a24x2y4 + a42x

4y2 + a33x3y3, (2.44)

where the coupling parameters aij are constants. The potential (2.44) will be PT -

symmetric for aij ∈ i ℜ ∀ i + j = odd, i, j = 0, 1, .., 5 and aij ∈ ℜ ∀ i + j =

even, i, j = 0, 1, .., 6.

For the present system, the ansatz for the function g(x, y) is made as

g(x, y) = α10x+ α01y + α20x2 + α02y

2 + α11xy + α12xy2 + α21x

2y

+α30x3 + α03y

3 + α22x2y2 + α31x

3y + α13xy3 + α40x

4 + α04y4, (2.45)

where αij ∈ i ℜ ∀ i + j = odd, i, j = 0, 1, 2, 3 and αij ∈ ℜ ∀ i + j = even, i, j =

0, 1, 2, 3, 4 are chosen for a PT -symmetric wavefunction.

For determining energy eigenvalues for various energy states, we here assume the

polynomial forms of ϕ(x, y) for the first four states.

The Ground State Solutions: For the ground state solutions, consider ϕ(x, y) = 1.

Thus, using eqs.(2.44) and (2.45) in eq.(2.4) and on rationalization of the resultant

expression, we get the following set of algebraic equations

2E0 = 2α02 + 2α20 − α210 − α2

01, (2.46)

a01 = α10α11 + 2α02α01 − α21 − 3α03, (2.47)

a10 = α01α11 + 2α20α10 − 3α30 − α12, (2.48)

a11 = 2(α20α11 + α02α11 + α21α10 + α01α12) − 3α13 − 3α31, (2.49)

2a02 = −12α04 − 2α22 + 6α03α01 + 2α10α12 + 4α202 + α2

11, (2.50)

2a20 = −12α40 − 2α22 + 6α30α10 + 2α21α01 + 4α220 + α2

11, (2.51)

a12 = 4α02α12 + 3α11α03 + 2α20α12 + 3α13α01 + 2α10α22 + 2α11α21, (2.52)

a21 = 4α20α21 + 3α11α30 + 2α02α21 + 3α31α10 + 2α01α22 + 2α11α12, (2.53)

a22 = 2α221 + 2α2

12 + 3α12α30 + 3α21α03 + 4α22(α20 + α02) + 3α11(α13 + α31), (2.54)

a03 = 6α02α03 + α11α12 + 4α01α04 + α10α13, (2.55)

a30 = 6α20α30 + α11α21 + 4α10α40 + α01α31, (2.56)


a31 = 6α21α30 + 2α12α21 + 6α20α31 + 2α11α22 + 4α11α40 + 2α02α31, (2.57)

a13 = 6α12α03 + 2α12α21 + 6α02α13 + 2α11α22 + 4α11α04 + 2α20α13, (2.58)

2a40 = 9α230 + α2

21 + 2α11α31 + 16α20α40, (2.59)

2a04 = 9α203 + α2

12 + 2α11α13 + 16α02α04, (2.60)

a50 = 12α30α40 + α31α21, (2.61)

a05 = 12α03α04 + α13α12, (2.62)

a51 = 12α31α40 + 2α31α22, (2.63)

a15 = 12α13α04 + 2α13α22, (2.64)

a23 = 6α12α13 + 6α03α22 + 4α21α04 + 3α30α13 + 3α12α31 + 4α21α22, (2.65)

a32 = 6α21α31 + 6α30α22 + 4α12α40 + 3α03α31 + 3α21α13 + 4α12α22, (2.66)

a33 = 6α22α31 + 6α13α22 + 4α13α40 + 4α31α04, (2.67)

a41 = 9α30α31 + 2α21α22 + 2α12α31 + 8α21α40, (2.68)

a14 = 9α03α13 + 2α12α22 + 2α21α13 + 8α12α04, (2.69)

2a42 = 16α22α40 + 6α31α13 + 4α222 + 9α2

31, (2.70)

2a24 = 16α22α04 + 6α31α13 + 4α222 + 9α2

13, (2.71)

2a60 = 16α240 + α2

31, (2.72)

2a06 = 16α204 + α2

13. (2.73)

The above equations provide solutions of potential parameters aij in terms of the

wavefunction parameters αij. Therefore for various choices of αij, different aij can be

obtained and hence a family of quintic potentials. Note that the energy eigenvalue,

eq.(2.46), is real as α20, α02 ∈ ℜ.First Excited State Solutions: For the first excited state, we consider ϕ(x, y) as

ϕ(x, y) = β10x+ β01y + β0, (2.74)

where the parameters β10 and β01 are complex constants and β0 is real one. So,

using eqs.(2.44), (2.45) and (2.74) in eq.(2.4), we get a set of thirty two equations

out of which twenty two are same as given in eqs.(2.52)- (2.73) and the remaining ten

equations are written as

E1 = α02 + α20 − (α210)/2 − (α2

01)/2 + β10α10/β0 + β01α01/β0, (2.75)

E1 = α02 + 3α20 − (α210)/2 − (α2

01)/2 + β01α11/β10, (2.76)

E1 = α20 + 3α02 − (α210)/2 − (α2

01)/2 + β10α11/β01, (2.77)

a20 = 3α10α30 + α01α21 − 10α40 + 2α220 + α2

11/2 − α22 − α31β01/β10, (2.78)


a02 = 3α01α03 + α10α12 − 10α04 + 2α202 + α2

11/2 − α22 − α13β10/β01, (2.79)

a20β01 + a11β10 = β01(−3α22 + α211/2 + α01α21 + 3α10α30 + 2α2

20 − 6α40)

+β10(2α20α11 + 2α02α11 + 2α01α12 − 6α31 + 2α10α21 − 3α13), (2.80)

a02β10 + a11β01 = β10(−3α22 + α211/2 + α10α12 + 3α01α03 + 2α2

02 − 6α04)

+β01(2α02α11 + 2α20α11 + 2α10α21 − 6α13 + 2α01α12 − 3α31), (2.81)

a20β0 + a10β10 = β10(α01α11 − 6α30 + 2α10α20 − α12) − β01α21

+β0(α01α21 − 6α40 + 2α220 − α22 + α2

11/2 + 3α10α30), (2.82)

a02β0 + a01β01 = β01(α10α11 − 6α03 + 2α01α02 − α21) − β10α12

+β0(2α202 − 6α04 − α22 + α2

11/2 + 3α01α03 + α10α12), (2.83)

a01β10 + a10β01 + a11β0 = +β10(2α01α02 − 3α21 − 3α03 + α10α11)

+β0(2α20α11 − 3α31 + 2α01α12 − 3α13 + 2α10α21 + 2α02α11)

+β01(2α10α20 − 3α12 − 3α30 + α01α11). (2.84)

The first three expressions will provide a unique and real energy eigenvalue expres-

sion and the remaining seven relations connecting a01, a10, a02, a20, and a11 reduce to

eqs.(2.47)- (2.51) under the conditions:

α04 = α40, α20 = α02, α12 = −α21 = −3α03, α13 = α31,

2α22 = −3α31, α13 = −4α04, α30 = −α03. (2.85)

The energy eigenvalue E1 is written as

E1 = 4α20 + α11 − (α210)/2 − (α2

01)/2, (2.86)

β10 = β01 = arbitrary, β0 =β10(α10 + α01)

2α20 + α11

. (2.87)

The energy is real and discrete as α20, α11 ∈ ℜ.

Second Excited State Solutions: For obtaining the energy eigenvalues of the

second excited state of the system considered in this work, we take the third choice

of ϕ(x, y) as

ϕ(x, y) = x2 + y2 + β11xy + β10x+ β01y + β0, (2.88)

where β11, β01 and β10 are considered as real numbers while β0 as complex one.

Again inserting eqs. (2.44), (2.45) and (2.88) in eq. (2.4), we obtain a set of thirty

seven algebraic equations. Out of these, six relations involving energy and nine for

a01, a10, a02, a20, and a11 are different from the eqs.(2.47)- (2.51). Again the six


relations involving E reduce to a single energy equation

E2 = 6α20 + 2α11 − (α210 + α2

01)/2, (2.89)

and the relations for a01, a10, a02, a20, and a11 will be same as given in eqs.(2.47)-

(2.51) under the conditions given in eq.(2.85).

Further the coefficients β’s are given as

β11 = β01 = β10 = 2, β0 =α10 + α01 − 1

2α20 + α11

. (2.90)


Third Excited State Solutions: Finally for obtaining the energy eigenvalues of

the third excited state, we assume

ϕ(x, y) = β30x3 + β03y

3 + β21x2y + β12xy

2 + β11xy

+β20x2 + β02y

2 + β10x+ β01y + β0, (2.91)

where β11, β20 and β02 are considered as real numbers while β03, β30, β21, β12, β10, β01

and β0 as complex constants.

Again inserting eqs. (2.44), (2.45) and (2.91) in eq. (2.4), we obtain a set forty four

algebraic equations. Ten relations involving energy and twelve expressions for a10,

a01, a11, a20 and a02 are different from eqs.(2.47)- (2.51).

Once again to obtain a single energy relation and same relations for a10, a01, a11, a20

and a02 as given in eqs.(2.47)- (2.51), we here assume, for simplicity, β02 = β20 = 1

and β03 = β30 = ι in addition to the restriction (2.85). Thus the third excited state

energy is given by

E3 = 8α02 + 3α11 − (α210 + α2

01)/2, (2.92)

Finally the solutions of various βij are given as

β01 = β10 =6 + ι(α11 + 2α01 + 2α10)

4α20 + 3α11

, β11 = 2,

β12 = β21 = 3ι, β0 =ι[β10(α01 + α10) − 2]

6α20 + 3ια11

, (2.93)


2.3.3 An Illustration

From the general results obtained in the above section one can develop eigenvalues

and eigenfunctions for some specific potentials. A number of such potentials can


be constructed by choosing the parameters αij of function g(x, y). But while fixing

αij one should not skip the condition (2.85). Let us consider one such potential by

selecting α11 = α02 = α20 = α04 = α40 = 1, α22 = 6, α13 = α31 = −4, α01 = −α10 = ι,

α03 = −α30 = ι, and α21 = −α12 = 3ι, which immediately leads to a potential

V (x, y) = 5ι(x− y) − 15.5(x2 + y2) + 40xy − 11ι(x3 − y3) − 30x2y2

−18ι(xy2 − x2y) + 20(x3y + xy3) − 5(x4 + y4) − 24ι(x5 − y5)

−96ι(xy4 − x4y) − 340ι(x2y3 − x3y2) + 16(y6 + x6)

−96(xy5 + x5y) + 240(x2y4 + x4y2) − 320x3y3. (2.94)

Thus for this particular potential, the solutions are given as

E0 = 3, ψ0 = e−g(x,y), (2.95)

E1 = 6, ψ1 = (x+ y)e−g(x,y), (2.96)

E2 = 9, ψ2 = [x2 + y2 + 2(xy + x+ y) − 1/3]e−g(x,y), (2.97)

E3 = 12, ψ3 = [ιx3 + ιy3 + 3ιx2y + 3ιxy2 + 2xy + x2 + y2

+(6 + ι)(x+ y)/7 − (2 + 4ι)/15]e−g(x,y), (2.98)

where g(x, y) = −[ι(x − y) − x2 − y2 − xy + 3ι(xy2 − x2y) + ι(x3 − y3) − 6x2y2 +

4(x3y + xy3) − x4 − y4]. Note that the energy levels are evenly spaced and nth state

energy becomes En = 3n where n is a positive integer.

Chapter 3

Application of sextic oscillator

potential

In the previous chapter we have discussed the solutions of the SE for complex systems.

With the same spirit, here we discuss the application of Sextic oscillator potential.

3.1 Introduction

The sextic oscillator potential, which belongs to the class of quasi exactly solvable

potential, consists of features which are advantageous in describing transitions be-

tween different nuclear shapes and in identifying critical point nuclei. Atomic nuclei

are known to exhibit shape phase transitions from one kind of collective behaviour to

another when the number of protons and/or neutrons is altered. In fact, a particular

nuclear shape is a result of the balance between the short-range interaction, which

favours the spherical shape, and the long-range quadrupole-quadrupole interaction,

which induces deformation. Owing to the dependence of this balance on the nucleon

numbers, there may occur a transition from one equilibrium shape to another one as

we proceed along an isotopic chain. In general the shape phase transition of some

order occurs through some critical point.

Theoretically, the shape as well as the shape phase transition can be well described

either within the collective model, which consists in solving the concerned Bohr Hamil-

tonian, or within the interacting boson model wherein the nucleus is assumed as a

system containing bosons, pairs of nucleons coupled to angular momentum L = 0 and

L = 2. However the solution of Bohr Hamiltonian, which uses the potential picture

to describe the collective excitation in terms of shape variables β and γ is a rather

effective method. Further a potential which is γ-independent reduces the problem of

41

42 CHAPTER 3. APPLICATION OF SEXTIC OSCILLATOR POTENTIAL

solving a Schrodinger-like radial equation. The potential depends on various param-

eters characterizing the given nucleus. Changing these parameters gives rise to shape

phase transition through a critical point.

So far various potential models and the corresponding solutions of the Schrodinger-like

equation associated with the Bohr Hamiltonian have been considered. For instance,

in case of a second order shape phase transition from spherical to deformed config-

uration a 5-dimensional infinite well is used, since the potential is expected to be

flat at the point of a second order shape phase transition. For this potential the

β-equation becomes a Bessel equation with eigenfunctions proportional to the Bessel

functions while the spectrum is determined by the zeros of the Bessel functions. Orig-

inally, Bohr used the harmonic oscillator potential for which the eigenfunctions are

proportional to Laguerre polynomials, and the spectrum has a simple form. The

Davidson potential leading to eigenfunctions proportional to Laguerre polynomials

has also been used. Besides these, other solutions in this framework corresponding to

a potential well of finite depth, Coulomb-like potential, Kratzer-like potential, Sextic

oscillator potential, a linear potential, a hybrid model employing a harmonic oscilla-

tor for L ≤ 2 and an infinite square well potential for L ≥ 4 has also been obtained.

For some of the potentials exact solutions exist while for others approximations are

used.

Owing to the aforesaid reason we here concentrate only on the solution of Bohr Hamil-

tonian for sextic oscillator potential with a special emphasis on the identification of

critical point nuclei in Te- and Xe-isotopic chains. Very recently, a systematic study

of the evolution from spherical to deformed γ-unstable shapes in the Ru, Pd and Cd

isotope chains, all having proton number just below the Z = 50 proton shell closure,

by using the sextic oscillator as a γ-independent potential in the Bohr equation has

been performed by Levai and Arias. Here we apply the same procedure to identify

the critical point nuclei in Te- and Xe-isotopic chains, which contain the number of

protons above the Z = 50 shell closure.

The Bohr Hamiltonian is being described in detail in Section .2 and the γ-independent

sextic oscillator potential along with its evolution in the parameter space is discussed

in Section .3. The results of the present study with detailed discussions are reported

in Section .4.

3.1. INTRODUCTION 43

3.1.1 The Bohr Hamiltonian

According to the phenomenological nuclear collective model due to Bohr and Mottel-

son, the radius of nuclear surface is expressed, in polar co-ordinates, as an expansion

in spherical harmonics such that

R(θ, ϕ) = R0[1 +∑λ,µ

Yλ,µ (θ, ϕ)αλ,µ ]. (3.1)

where R0 is the equilibrium radius and R(θ, ϕ)is the distance of a point on the surface

from the origin. The expansion coefficients αλ, µ are the coordinates of a multidimen-

sional space in which a point represents a deformed surface. For the nuclear radius

to be real the required condition is αλ,µ = (−1)µα∗λ,−µ. Since the monopole (λ = 0)

and dipole (λ = 1) degrees of freedom correspond to the change in volume and the

c.m. motion of the nucleus, one confines to quadrupole (λ = 2) degree of freedom.

In fact quadrupole states are the most fundamental collective type of the low-lying

excitations in nuclei. For very small values of α‘s the quadrupole deformed surface

is an ellipsoid which is randomly oriented in space. In case of (λ = 2), the eq.(3.1)

reduces to

R(θ, ϕ) = R0[1 +2∑

µ=−2

Y2,µ(θ, ϕ)α2,µ]. (3.2)

The five coordinates α2,µ appearing in this equation may be mapped onto a set of

five variables a0, a2, θ1, θ2, θ3 where in two parameters are related to the extent of

deformation while the other three to the angular orientation of the ellipsoid. This

mapping from the space fixed co-ordinate system to body fixed co-ordinate system

may be achieved through the following transformation

α2,µ =∑µ

α2,µD2µ,ν(θi). (3.3)

With D2µ,ν(θi) as the transformation function for the spherical harmonics of sec-

ond order and θi being the triad of Eulerian angles θ, ϕ, ψ describing the relative

orientation of the axes. Further it is more convenient to choose a body-fixed co-

ordinate system whose axes coincide with the principal axis of the ellipsoid so that

a2,1 = a2,−1; a2,2 = a2,2. And the five variables αλ, µ are replaced by the three Eulerian

angles θi and the two real internal coordinates a2,0 and a2,2. For further simplification

it is appropriate to consider a new set of co-ordinates β, γ, θi called the Hill-Wheeler

co-ordinates, in which the subset β, γ defines a two dimensional polar co-ordinate

system with in the five dimensional quadrupole deformation space, namely:

a2,0 = βcosγ, a2,2 = a2,−2 = (β/c2)sinγ (3.4)


With∑

µ[α2,µ]2 = β2. Now, in quadrupole deformation space the Bohr Hamiltonian

built with the generalized co-ordinates and momenta is written as

H = T + V =∑µ

[(1/2B2)[πµ]2 + (c2/2)[τµ]2] (3.5)

Where B2 and c2 are the mass and stiffness parameters respectively and π[µ are the

conjugate momenta associated to αµ. In general, the Bohr Hamiltonian consists of

three terms -

H = Tvib + Trot + V. (3.6)

The first term, Tvib, is related to the K.E. of shape vibration, the second term, Trot,

to the K.E. of rotation while the third one,V , to the restoring potential. The two

K.E.terms are given in terms of parameters β and γ as

Tvib =−h2

2B2

[1

β4

∂

∂ ββ4 ∂

∂ β+

1

β2

1

sin3γ

∂

∂ γsin3γ

∂

∂ γ]and (3.7)

Trot =−h2

2B2

[1

4β2

3∑k=1

Q2k

[sinγ − 2π3k]2

] (3.8)

With Qk as the angular momentum operator in the variables θi. Thus,in terms of the

parameters β and γ the Bohr Hamiltonian is written as

H =−h2

2B2

[1

β4

∂

∂ ββ4 ∂

∂ β+

1

β2

1

sin3γ

∂

∂ γsin3γ

∂

∂ γ− 1

4β2

3∑k=1

Q2k

[sinγ − 2π3k]2

] + V (β, γ)(3.9)

If the potential V (β, γ) is assumed to be independent of variable γ , i.e. V (β, γ) =

U(β, γ) then the Hamiltonian is separable and the β-dependent part can be separated

by the substitution ψ(β, γ, θi) = ββ2ϕ(γ, θi) in the equation

Hψ(β, γ, θi) = Eψ(β, γ, θi)

. This substitution leads to a form similar to the usual radial Schrodinger equation

for ϕ

−d2ϕ

dβ2+ [

(τ + 1)(τ + 2)

β2+ u(β)]ϕ = ϵϕ (3.10)

and the following differential for (Φ) :

[− 1

sin3γ

∂

∂ γsin3γ

∂

∂ γ+

1

4

∑k

Q2k

[sinγ − 2π3k]2

]Φ(γ, θi) = ΛΦ(γ, θi) (3.11)

Above Λ = τ(τ + 3) , τ = 0, 1, 2..... , ϵ = 2B2

−h2E u(β) = 2Bh2 U(β). The eq.(3.10) is

either exactly solvable or quasi-exactly solvable depending on the form of potential

u(β). The quantum mechanical potentials for which only a finite portion of energy


spectra and associated eigenfunctions can be found exactly in the closed form are

said to be quasi-exactly solvable. For these potentials particular analytic solutions

can be found when the potential parameters obey certain constraining conditions.

One such quasi-exactly solvable potential is the sextic oscillator potential which is of

particular importance in describing the transition between different shape phases of

atomic nuclei. The ansatz for wave function is one of the simplest method to solve

the Schrodinger equation analytically for the sextic oscillator potential . This method

consists in assuming a particular form of the eigenfunction for a given potential to

obtain a set of equations that leads to exact solutions provided that certain relations

between the parameters of the potential hold.

3.1.2 The Sextic Oscillator Potential

Consider the following quasi-exactly solvable sextic oscillator potential

u(β) = (b2 − 4ac)β2 + 2abβ4 + a2β6 + uπ0 (3.12)

where the superscript π = ± takes into account the fact that the potential is slightly

different for even and odd values of τ . The potential depends on three parameters

namely a, b and cπ when the constant uπ0 is taken as zero. The parameters cπ appearing

in the potential uβ are related to quantum number τ through 2cπ = (τ + 2M + 7/2)

with c+ for even values of τ and c− for odd values of τ . Here M is a non-negative

integer such that M + 1 represents the number of solutions that can be obtained

exactly which renders above potential quasi- exactly solvable. The exact solutions

providing eigen functions and eigen values of the above potential corresponding to

M = 0 and M = 1 are discussed in detail in. For cπ to remain constant it is needed

that when M is decreased/increased by one unit, the τ must be increased/decreased

by two units. Clearly the sequence of (M, τ) values (K, 0), (K − 1, 2), (K − 2, 4)

corresponds to the solutions forc+ = K + 7/4 and (K, 1), (K − 1, 3), (K − 2, 5)to the

solution for c− = K + 9/4. It is worth mentioning that the extrema of this potential

depend on the sign of the coefficients of β2andβ4 terms while the coefficient β6 of is

always positive. For b2 > 4acπ and b > 0 i.e. for b > 2√acπ there is a minimum

at β = 0 and the potential increases monotonously with β. When b2 < 4acπ or

−2√acπ < b < 2

√acπ a minimum appears for β > 0 while for b2 > 4acπ and b < 0

i.e. for b < −2√acπ there appears minimum at β = 0, then a maximum and then

the second minimum with increase in β . To locate the exact position of the maxima

and minima of the potential set u′(β) = 0 that is


[width=3in]Fig.1.eps

Figure 3.1: evolution

2(b2 − 4acπ)β + 8abβ3 + 6a2β5 = 0

or

2β[(b2 − 4acπ) + 4abβ2 + 3a2β4] = 0

Now depending on the cases discussed above either β = 0 or

(b2 − 4acπ) + 4abβ2 + 3a2β4 = 0

leading to

β2 = 1/3a[−2b ±√b2 + 12acπ]. Thus the shape of the potential is quite sensitive to

the values of the parameters a, b and cπ . In fact, the (a, b) plane can be divided into

three different regions corresponding to different shapes of the potential. A typical

diagram showing the specific potential shapes in different domain of parameter (a, b)

space along with the critical (or transitional) parabola a = b2/11 is shown in Fig. 1.

It is clear from this figure that for (a, b) very far away in the right of the critical

parabola the minimum of the potential occurs sharply at β = 0 and the potential

gets flattened near β = 0 as one approaches to the parabola. The minimum at β = 0

corresponds to a spherical shape. For (a, b) point well inside the parabola there occurs

a minimum at β > 0 which indicates that in this region the equilibrium shape of the

nucleus becomes deformed. Thus there is a transition from spherical to deformed

shape when the right side of the parabola is crossed and that is why the parabola is a

critical (or transitional) parabola. In the left side region of the parabola there occurs

a maximum first and then a minimum.

Now the solution of eq. (3.10) for the potential in eq.(3.12) obtained in an algebraic

method are written as

Φα(β) = Nαβpα(1 + dαβ

2)e−q4β4− p

2β2

(3.13)

Where α = (ξ, τ). The coefficient pα = τ + 2 and dα is expressed in terms of the

parameters a and b.

The normalization constant is given by

Nα = (2a)2pα+1

8

√2

Γ(pα + 1/2)[U(

1

4(2pα + 1),

1

2,b2

2a)

+(pα + 1/2)2dα

2a1/2U(

1

4(2pα + 3),

1

2,b2

2a)

+(pα + 1/2)(pα + 3/2)(dα

2a1/2)2

U(1

4(2pα + 5),

1

2,b2

2a)]−

12 (3.14)


Where U are Kummer,s functions, which are clearly related to the well known con-

fluent hyperglometric functions.

3.1.3 Application to Te- and Xe-Isotopic Chains

The region lying in the close vicinity of the proton shell closure at Z = 50 appears to

contain the best possible candidates of critical point nuclei for spherical to γ-unstable

deformed shape phase transition. Very recently, the Ru, Pd and Cd-isotopic chains,

lying below the Z = 50 proton shell closure, were investigated to find the critical

point nuclei in these chains by Levai and Arias and found that Ru104, Pd102 and

Cd106,108 are good candidates for critical nuclei while Cd116 was proposed as a new

critical nucleus. In the present work we consider the Te- and Xe-isotopic chain, which

is above the Z = 50 proton shell closure, to identify the critical point nuclei using

sextic oscillator potential formulation.

In a typical application of the sextic potential in Bohr Hamiltonian it is sufficient to

consider the simple M = 0 and M = 1 cases. For M = 1 and τ = 0 , there exists

two solutions one without any node and other with one node which are represented

as ϕn,τ = ϕ0,0 and ϕ1,0 where n is number of nodes and the corresponding energy

eigen values are E0,0 = 7b−2√b2 + 10a and E0,0 = 7b+2

√b2 + 10a respectively. The

same potential is obtained by considering M = 0 and τ = 2 and in this case there is

only single solution ϕn,τ = ϕ0,2 with E0,0 = 9b as the energy eigen value. Similarly

for odd τ values, ϕ0,1 and ϕ1,1 are the solutions for (M = 1; τ = 1) and ϕ0,3 for

(M = 0; τ = 3) with the corresponding energy eigen values as E0,1 = 9b−2√b2 + 14a

, E1,1 = 9b+ 2√b2 + 14a and E0,3 = 11b , respectively.

The experimental values of the energies of first few 0+, 2+, 4+and6+ low lying levels,

the important ingredients needed in the analysis, are taken from ISOTOPE EX-

PLORER and are listed in Tables 1 and 2 for Te and Xe isotopes respectively.

Table 1. Eperimental values of energies (in keV) of first few low lying levels of

Te isotopes. The subscript are the quantum numbers n and τ respectively and the

quantity in the parenthesis represents the spin and parity of the state. (the best fit

values of the parameters a and b are also given in last two columns)


Isotope E0,1(2+) E0,2(4

+) E0,3(6+) E1,0(0

+1 ) a b

Te110 1403 1917 2228 658 6208 207

Te114 709 1484 2217 1348 13841 183

Te120 560 1162 1776 1103 8830 145

Te124 603 1248 1747 1657 7762 149

Te126 666 1361 1776 1873 10418 156

Te128 743 1497 1811 1979 11044 165

Te130 839 1633 1815 1965 9218 173

Table 2. Same as Table 1 but for Xe isotopes.

Isotope E0,1(2+) E0,2(4

+) E0,3(6+) E1,0(0

+1 ) a b

Xe122 331 828 1466 1149 7178 112

Xe124 354 879 1549 1269 6339 119

Xe126 389 942 1635 1314 8842 127

Xe128 443 1033 1737 1583 9416 136

Xe130 536 1204 1944 1793 11368 155

Xe132 668 1440 2112 13122 17 6

The states 2+, 4+ and 6+ are associated with (n, τ) = (0, 1), (0, 2) and (0, 3) model

states respectively. The level 0+ plays a very crucial role in the interpretation of

quadrupole collectivity in nuclei thus its identification needs particular attention.

The best way to identify 0+ level is based on the B(E2) values. However, when

the information about B(E2) values are not available then the fact that the radially

excited 0+ state, corresponding to quantum numbers (n, τ) = (1, 0), is lower than the

0+ state, corresponding to quantum number (n, τ) = (0, 3), belonging to the multiplet

with excitation may be used to identify 0+ level.

Then the best fit value of free parameter a is taken as the average of the values

calculated by using the following expressions

a =(9E0,3−11E0,1)2−4E2

0,3

6776, ,a =

(9E1,0−7E0,2)2−4E20,2

3240

a =(81E0,1−E0,2)2−4E2

0,2

4536and a =

(11E1,0−7E0,3)2−4E20,3

6776,

while that of b as the average of values calculated using b = E0,3

11and b = E0,2

9relations.

The so obtained best fit values of a and b for Te and Xe isotopes are listed in Tables

1 and 2 respectively.

In Figs. 2 and 3, we have plotted the best fitted values of the parameters a and b

in the (a, b) plane for Te- and Xe- isotopic chains respectively along with the critical

parabola.

Fig. 2. The best fit values of the potential parameters in (a, b) plane for various


Te isotopes.

Fig. 3. Same as Fig. 2 but for Xe isotopes.

It may be clearly observed in these figures that the points indicating the isotopes in

(a, b) plane all remain in the domain to the left of the critical parabola. The nucleus

Te110 in Te-isotopic chain being closest to the critical parabola may be considered

as a critical point nucleus. Besides the A = 124 isotopes of both Te as well as Xe

nuclei also offer candidature of being critical nuclei above Z = 50 proton shell closure.

Chapter 4

Applications and Conclusions

In the 2nd chapter we have discussed the the complex Hamiltonian systems along with

some interesting examples to find the solutions of the SE in two dimensions where as

in 3rd chapter, we have discussed quasi-exactly solvable sextic oscillator potential and

its application to Te- and Xe-isotopic chain to find critical nuclei point in this chain.

Although the results are obtained in this work for some particular systems, yet their

study help to understand many theoretical phenomena in physics and chemistry (

structural phase transition, polaron formation in solid, concept of false vacua in field

theories, model for various molecules, fibre optics etc.). Now, we briefly describe the

application of non hermitian theory and summarize the finding of the present work

along with some concluding remarks.

4.1 Applications of Non-Hermitian Quantum Theory

In 1959 Wu showed that the ground state of a Bose system of hard spheres is described

by a non-hermitian Hamiltonian [42]. He found that the ground state energy of this

system is real and guessed that all the energy levels are real. Hollowood showed that

the non-hermitian Hamiltonian of a complex Toda lattice has real energy levels [20].

In studies of the Lee-Yang edge singularity [102, 103, 104, 105] Cubic non-hermitian

Hamiltonian of the form H = p2 + ix3 is used. In all these cases a non-hermitian

Hamiltonian having a real spectrum appeared mysterious at that time, but now the

explanation is simple. In every case the non-hermitian Hamiltonian is PT -symmetric

and is constructed so that the position operator x or the field operator Φ is always

multiplied by ‘i’. To describe magnetohydrodynamic systems [108, 109] PT - sym-

metric Hamiltonians are used. Also to study non-dissipative time-dependent systems

interacting with electromagnetic fields [110] PT - symmetric Hamiltonians are used.

51

52 CHAPTER 4. APPLICATIONS AND CONCLUSIONS

In this section we briefly describe different areas of quantum mechanics in which non-

hermitian PT - symmetric Hamiltonians play an important and vital role.

1. Supersymmetric PT - symmetric Hamiltonians:- The discovery of the PT -

symmetric Hamiltonians in quantum mechanics enable the combination of PT - sym-

metry and Supersymmetry [91] in the context of quantum field theory. In [91] it

was shown that one can easily construct two dimensional quantum field theories by

introducing a PT - symmetric superpotential of the form

S(ϕ) = −ig(iϕ)1+ϵ.

The resulting quantum field theories exhibit a broken parity symmetry for all δ >

0.However supersymmetry remains unbroken, which is verified by showing that the

ground state energy density vanishes and that the fermion-boson mass ratio is unity.

Droley et.al checks the connection between supersymmetric and broken PT - symme-

try in quantum mechanics in [124].

2. New PT - symmetric quasi-exactly solvable Hamiltonians :- A quantum-

mechanical Hamiltonian whose finite portion of energy spectrum and associated eigen-

functions can be found exactly and in closed form [111]is called quasi-exactly solvable

(QES). An integer parameter J is involved in QES potential; for positive values of J

one can find exactly the first J eigenvalues and eigenfunctions, typically of a given par-

ity. QES systems are classified using an algebraic approach in which the Hamiltonian

is expressed in terms of the generators of a Lie algebra [112]. Before the discovery

of non-hermitian PT - symmetric Hamiltonians, the lowest-degree one-dimensional

QES polynomial potential that was known a sextic potential having one continuous

parameter as well as a discrete parameter ‘J’. A simple case of such a potential is

[113]

V (x) = x6 − (4J − 1)x2. (4.1)

For this potential the Schrodinger equation

ψ′′(x) + (E − V (x))ψ(x) = 0

has even parity solution of the form

ψ(x) = e−x4/4J−1∑k=0

ckx2k. (4.2)

the coefficients ck for 0 ≤ k ≤ J − 1 satisfy the recursion relation

4(J − k)ck−1 + Eck + 2(k + 1)(2k + 1)ck+1 = 0 (4.3)

4.1. APPLICATIONS OF NON-HERMITIAN QUANTUM THEORY 53

where c−1 = cj = 0 the linear equation (4.3)have a non trivial solution for c0, c1,−−− − −, cJ−1, if the determinant of the coefficients vanishes. This determinant is a

polynomial of degree ‘J’ in the variable E, for each integer J. The roots of this poly-

nomial are all real and they are the J quasi-exact energy eigenvalues of the potential

(4.1). An entirely new class of QES quartic polynomial potentials having two con-

tinuous parameters in addition to the discrete parameter J is introduced after the

discovery of PT - symmetry and the Hamiltonian has the form [114]

H = p2 − x4 + 2iax3 + (a2 − b2)x2 + 2i(ab− J)x, (4.4)

where a and b are real and J is a positive integer. The spectra of this family of

the Hamiltonians are real, discrete and bounded below. Like the eigenvalues of the

potential given in eq.(4.1), the lowest J eigenvalue of H are the roots of a polynomial

of degree J. The eigenfunction ψ(x) satisfies PT - symmetric boundary conditions, it

vanishes in the stoke wedges. The eigenfunction satisfies

−ψ(x) + [−x4 + 2iax3 + (a2 − 2b)x2 + 2i(ab− J)x]ψ(x) = Eψ(x). (4.5)

We obtain the QES portion of the spectrum of H in eq.(4.4) by making the ansatz

ψ(x) = exp(−fr13ix3 − fr12ax2 − ibx)PJ−1(x, )

where

PJ−1(x, ) = xJ−1 +∑J−2

k=0ckx

k. (4.6)

is a polynomial in x of degree J-1. On putting ψ(x) into eq.(4.5), dividing off the

exponential and collecting power of x, a polynomial in x of degree J − 1 is obtained.

setting the coefficients of xk(1 ≤ k ≤ J − 1) to 0 gives a system of J-1 simultaneous

linear equations for the coefficients ck(0 ≤ k ≤ J − 2). We solve these equations and

insert the value of ck into the coefficient of x0. This gives a polynomial QJ(E) of

degree J in the eigenvalue E. The coefficients of this polynomial are functions of the

parameters a and b. The first two polynomials are

Q1 = E − b2 − a, (4.7)

Q2E2 − (2b2 + 4a)E + b4 + 4ab2 − 4b+ 3a2. (4.8)

The roots of QJ(E) are the QES portion of the spectrum of H. The polynomials

QJ(E) simply dramatically when we substitute

E = F + b2 + Ja and K = 4b+ a2. (4.9)


The new polynomials then have the form

Q1 = F, (4.10)

Q2F2 −K, (4.11)

Q3F3 − 4KF − 16, (4.12)

Q4F4 − 10KF 2 − 96F + 9K2, (4.13)

Q5F5 − 20KF 3 − 336F 2 + 64K2F + 768K. (4.14)

The roots of these polynomials are all real only for K≥Kcritical, where Kcritical is a

function of J. This beautiful family of QES Hamiltonians is only considered due to

the discovery of PT - symmetric quantum mechanics otherwise it would never hap-

pened because the quartic term has a negative sign. The Hamiltonians in eq.(4.4)

would have been rejected because it would have been assumed that the spectra of such

Hamiltonians would be unbounded below in the absence of PT - symmetric Hamilto-

nians with positive spectra.

3. Quantum Brachistochrone :-The similarity transformation maps the non

hermitian PT - symmetric Hamiltonian H to a hermitian Hamiltonian h . The two

Hamiltonians, H and h, have the same eigenvalues, but this does not mean they de-

scribe the same physics. To illustrate the difference between, H and h, we show how

to solve the quantum brachistochrone problem for PT - symmetric and for hermitian

quantum mechanics, and we show that the solution to this problem in these two for-

mulations of quantum mechanics is not the same.

The fancy word brachistochrone means ‘shortest time’. Thus the quantum brachis-

tochrone problem is defined as follow : Suppose we are given initial and final quantum

states |ψI⟩ and |ψF ⟩ in a Hilbert space. There exist infinitely many Hamiltonians H

under which |ψI⟩ evolves into |ψF ⟩ in some time t:

|ψF ⟩ = e−iHt/h|ψI⟩. (4.15)

The problem is to find the Hamiltonian H that minimize the evolution time t subject

to the constraint that ω, the difference between the largest and smallest eigenvalues of

H, is held fixed. The shortest evolution time is denoted by τ . In hermitian quantum

mechanics there is an unavoidable lower bound τ on the time required to transform

one state into another. Thus, the minimum time required to flip unitarily a spin-up

state into a spin-down state of an electron is an important physical quantity because

it gives an upper bound on the speed of a quantum computer. Bender have shown

[56] that the hermitian quantum mechanics can be extended into the complex domain

4.1. APPLICATIONS OF NON-HERMITIAN QUANTUM THEORY 55

while retaining the reality of the energy eigenvalues, the unitarity of the time evolu-

tion and the probabilistic interpretation. It has been found that within this complex

framework a spin-up state can be transformed arbitrarily quickly to a spin-down state

by a simple non-hermitian Hamiltonian [121].

The optimal evolution time τ for a hermitian version of two dimensional brachis-

tochrone problem hamiltonian required to transform(

10

)to the orthogonal state

(01

)is 2πh/ω [122]. However for PT - symmetric Hamiltonians, the evolution time re-

quired to transform(

10

)to(

01

)is (2α− π)h/ω. Optimizing this result over allowable

values for α, we find that as α approaches π/2, the optimal time τ tends to zero. This

result is quite general and even holds for broad classes of non-hermitian Hamiltonians

[123]

The hermitian and non-hermitian PT - symmetric Hamiltonians share the properties

that (i) the evolution time is given by 2πh/ω and (ii)∆H ≤ ω/2. The key difference

is that a pair of states such as(

01

)is 2πh/ω are orthogonal in a hermitian theory, but

have a separation in the PT - symmetric. This is because the Hilbert space metrics

of the PT - symmetric quantum theory depends on the Hamiltonian. Hence, it is

possible to choose the parameter α to create a wormhole-like effect in the Hilbert

space

The result established here provide the possibility of performing experiments that

distinguish between hermitian PT - symmetric Hamiltonians. If practical implemen-

tation of complex PT - symmetric Hamiltonians were feasible, then identifying the

optimal unitary transformation would be particularly important in the design and

implementation of fast quantum communication and computation algorithms. Of

course, the wormhole-like effect can be realized if it is possible to switch rapidly

between hermitian PT - symmetric Hamiltonians by means of similarity transforma-

tions. It is conceivable that so much quantum noise would be generated that there is

a sort of quantum protection mechanism that places a lower bound on the time re-

quired to switch Hilbert spaces. If so, this would limit the applicability of the Hilbert

space wormhole to improve quantum algorithms.

4. Complex crystals :- An experimental signal of a complex Hamiltonian might

be found in the context of condensed matter physics. Consider the complex crystal

lattice whose potential is

V (x) = i Sin x. (4.16)


The optical properties of complex crystal lattices were first studied by Berry and

O’Dell, who referred to them as complex diffraction gratings [115]. While the Hamil-

tonian H = p2 + isin x is not Hermitian, it is PT - symmetric and all of its energy

bands are real. At the edge of the bands the wavefunction of a particle in the case of

ordinary crystal lattices is fermionic (4π-periodic) where as for complex crystal it is

bosonic (2π-periodic). The discriminant for a hermitian sin(x) potential is plotted in

figure-3 and discriminant for a non-hermitian i sin(x) potential is plotted in figure-4.

The difference between these two figures is subtle. In figure-4 the discriminant does

not go below-2 and thus there are half as many gaps [116]. Direct observation of such

a band structure would give unambiguous evidence of a PT - symmetric Hamiltonian.

complex periodic potentials having more elaborate band structures have also been

found.

5. Other applications:- There are some other applications of non-hermitian theories

as

1. In the Hatano-Nelson model of non-hermitian magnetic field all eigenvalues are

real for anisotropy (non-hermiticity) parameter below a critical value.

2. Non hermitian Hamiltonian [34, 92]is used to explain quantum decay. As a

consequences of uncertainity principle, a decay state could not have sharp energy

and that the width of such energy levels could be represented by an imaginary

component.

3. In Nuclear physics one studies standard Schrodinger Hamiltonian with complex

valued potentials [127], which in this connection are called optical or average

nuclear potentials.

4. Non hermitian interactions are also discussed in field theories, for example, when

studying Lee Yang zeros. Even in recent studies on localization-delocalization

transition in superconductors [47] and in the theoretical description of diffraction

of atoms by standing light waves non-hermitian hamiltonian are of interest.

5. In the theory of reaction-diffusion systems, many models have been constructed

for systems described by matrices that can be non-hermitian [125, 126] and PT -

symmetric.

6. The non-hermitian operator is used to treat scattering phenomena in atomic and

molecular physics [128].

7. In Bose sphere [28] the non-hermitian Hamiltonian represents the low lying energy

levels of a system.

4.2. CONCLUSIONS 57

8. To calculate the electronic structure of crystalline Arsenic.The chemical pseudo

potential theory often give rise to a non-hermitian representation of interaction

between localized electronic orbits [32].

9. Hamiltonian rendered non Hermitian by an imaginary external field is used to

study Delocalization transition in condensed matter systems such vortex flux line

deppining in type-II superconductors [6], population biology [81, 82], quantum

cosmology, quantum field theory.

10. It is unknown whether any non-hermitian Hamiltonian can be used to describe

any experimentally observable phenomena, although these had already been used

to describe interacting systems [42].

In 2004, Weigert noted some applications of non-hermitian Hamiltonians in the

description of absorptive optical media, inelastic scattering from nuclei and other

loss mechanism on the atomic or molecular level. He also noted that non-

hermitian theories had recently been ”rediscovered’ within particle physics.

4.2 Conclusions

In the present work, we have calculated the energy eigenvalues of a family of two

dimensional PT -symmetric coupled quartic and quintic potentials using a suitable

ansatz for the wavefunction scheme, also we have discussed quasi-exactly solvable

sextic oscillator potential and its application to Te- and Xe-isotopic chain to find

critical nuclei point in this chain. The SE is rationalized for suitable forms of g(x, y)

and ϕ(x, y). The parameters of potential and ϕ(x, y) are expressed in terms of g(x, y)

parameters. This can provide a number of coupled quartic potentials for various

choices of g(x, y) parameters αij. The energy eigenvalues are computed for the first

four states. Although considering higher order polynomials for ϕ(x, y), other higher

excited state solutions for the quartic potential can be computed but expansion of

algebra makes the calculations difficult. The eigenvalue spectra found to be real and

discrete. The solutions found in this study are exact with certain constraints on the

potential parameters. The role of these constraint is very crucial not only in deciding

the ground and higher excited states but also in obtaining the bound states for the

system. The constraints are selected in such a way that under a particular choice,

from the general relations of eigenvalues and eigenfunctions, one can easily reproduce

the well known relations of eigenvalue and eigenfunction for a simple harmonic oscilla-

tor. The number of constraints increases as the order of anharmonicity increases. We

have also extended this problem for higher order anharmonicity i.e., quintic (sextic)


potentials and the solutions so obtained satisfy the well known systems. In addition

to ground state solutions, the first, second and third excited state solutions are also

calculated in chapter -II.

The method used here is quite general to calculate the eigen spectra and the asso-

ciated eigenfunctions for a non-hermitian Hamiltonians. To find the eigenvalues and

the corresponding eigenfunctions for the given system, one can make more than one

choices among ansatz parameters and can obtain mathematically correct results for

Er, Ei and ψ(x, y) for a given potential in various states. For the PT - symmetric

potentials the eigen spectra is real i.e. imaginary parts of the eigen spectra reduce to

zero. Thus it is the beauty of this method that it provides another degree of freedom

to obtain the real eigen spectra for non-hermitian Hamiltonians. This particular ob-

servation is in consonance with many other studies of PT - symmetric potentials [36].

Thus the present approach may be utilized as an alternative means to study PT -

symmetric complex systems.

Although the eigenfunction ansatz method is straight forward and easy to implement.

But it is more suitable to derive eigenvalues and eigenfunctions for a few initial states

and becomes a bit tedious for higher states due to the involved mathematics and more

constraints on eigenfunctions ansatz parameters.

In third chapter, we have searched the critical nuclei for spherical to γ- unstable de-

formed shape phase transition in Te- and Xe-isotopic chains, lying above the Z = 50

proton shell closure, within the framework of the sextic oscillator formalism. It has

been found that Te110, Te124 and Xe124 are the good candidates for the critical point

nuclei in the isotopic chains considered here.

Bibliography

[1] R.S. Kaushal, “Classical and Quantum Mechanics of Noncentral Potentials”,

Narosa Publishing House, New Delhi (1998).

[2] L.I. Schiff, “Quantum Mechanics”, McGraw-Hill International Ed. (1987).

[3] A.S. Davydov, “Quantum Mechanics”, translated, edited and with additions by

D. Ter Haar, Pergamon, Oxford (1965).

[4] R.S. Kaushal, J. Opt. Soc. Am. A8 (1991) 1245.

[5] H.H. Aly and A.O. Barut, Phys. Lett A145 (1990) 299.

[6] R.S. Kaushal, Ann. Phys. 206 (1991) 90.

[7] E.T. Whittaker, “A Treatise on the Analytical Dynamics of Particles and Rigid

Bodies”, 4th ed. (Dover, New York, 1944).

[8] B. Bagchi, F. Cannata and C. Quesne, Phys. Lett. A269 (2000) 79.

[9] R.S. Kaushal, J. Phys. A: Math.Gen 38 (2005) 3897.

[10] C.M. Bender, Rep. Prog. Phys. 70 (2007) 947.

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

[12] C.M. Bender, Phy. Rep. 40 (1999) 315.

[13] R.S. Kaushal and H.J. Korsch, Phys. Lett. A276 (2000) 47.

[14] A. Khare, B.P.Mandal, Phys. Lett. A272 (2000) 53; Pramana J. Phys 73 (2009)

387.

[15] F. M. Fernandez, R. Gujardiola, J. Ross and M. Znojil, J. Phys. A: Math. Gen.

31 (1998) 10105.

[16] N. Moiseyev, Phys. Rep. 302 (1998) 211.

[17] C.M. Bender, S. Boettcher and P.N. Meisinger, J. Math. Phys. 40 (1999) 2201.

[18] B. Bagchi and R.Roychoudhury, J. Phys. A: Math. Gen. 33 (2000) L1.

[19] A.L.Xavier Jr. and M.A. M.de Aguiar, Ann. Phys. 252 (1996) 458.

59

60 BIBLIOGRAPHY

[20] T. Hollowood, Nucl. Phys. B384 (1992) 523.

[21] N. Hatano and D.R. Nelson, Phys. Rev. Lett. 77 (1996) 570.

[22] N. Hatano and D.R. Nelson, Phys. Rev. B56 (1997) 8651.

[23] C.M. Bender, D.W. Hook and L.R. Mead, J. Phys. A: Math. Theor. 41 (2008)

392005.

[24] E. Caliceti, F. Cannata and S. Graffi, J. Phys. A: Math. Gen. 39 (2006) 10019.

[25] E. Caliceti, S. Graffi, Rend. Lincei Mat.Appl. 19 (2008) 163.

[26] B.P. Mandal, Mod. Phys. Lett. A20 (2005) 655.

[27] Z.H. Musslimani, K.G. Makris, R. El-Ganainy and D.N. Christodoulides, Phys.

Rev. Lett. 100 (2008) 030402.

[28] C.M. Bender, Brod Joachim, Refig Andre and E. Reuter Moretz, J. Phys. A:

Math. Gen. 37(2004) 10139.

[29] H. Feshbach, C.E. Porter and V.F. Weisskopf, Phys.Rev. 96 (1954) 488.

[30] R.K. Colegrave, P. Croxson, and M.A. Mannan, Phys. Lett. A131 (1988) 407.

[31] R.S. Kaushal, Pramana J. Phys 73 (2009) 287.

[32] R.S. Kaushal and Shweta Singh, Ann. Phys. 288 (2001) 253.

[33] N.N. Rao, B. Buti and S.B. Khadkikar, Pramana J. Phys. 27 (1986) 497.

[34] C. D. Yang, Chaos, Solitons and Fractals 33 (2007) 1073.

[35] R.S. Kaushal, J. Phys. A: Math. Gen. 34 (2001) L709.

[36] R.S. Kaushal and Parthasarathi, J. Phys. A: Math. Gen. 35 (2002) 8743.

[37] Parthasarathi and R.S. Kaushal, Phys. Scr. 68 (2003) 115.

[38] Shweta Singh and R.S. Kaushal, Phys. Scr. 67 (2003) 181.

[39] Parthasarathi, D. Parashar and R.S. Kaushal, J. Phys. A: Math. Gen. 37 (2004)

781.

[40] D. Bessis and J. Zinn-Justin, J. Math. Phys. 40 (1999) 678.

[41] M. Znojil and G. Levai, Phys. Lett. A264 (1999) 108.

[42] T.T. Wu, Phys. Rev. 115 (1959) 1390.

[43] J. Wong, J. Math. Phys. 8 (1967) 115.

[44] T.T. Wu and C.M. Bender, Phys. Rev. 184 (1969) 25.

[45] R. Haydock and M.J. Kelly, J. Phys. C: Solid State Phys. 8 (1975) 197.

[46] G.E. Stedman and P.H. Butler, J. Phys. A: Math. Gen. 13 (1980) 3125.

BIBLIOGRAPHY 61

[47] D.R. Gilson “Research Thesis” University of Hull (2008).

[48] F.H.M. Faisal and J.V. Moloney, J. Phys. B: At.Mol.Phys. 14 (1981) 3603.

[49] W. Stefan, J. Phys. A: Math. Gen. 39 (2006) 235.

[50] W. Stefan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) S416.

[51] M. Znojil, P. Siegl and G. Levai, Phys. Lett. A373 (2009) 1921.

[52] G. Levai, Pramana J.Phys. 73 (2009) 329.

[53] A. Mostafazadeh, J. Math. Phys. 43 (2002) 205, 2814, 3944; 36 (2003) 7081.

[54] C.M. Bender, D.C. Brody and H.F. Jones, Phys. Rev. Lett. 89 (2002) 270401.

[55] Z. Ahmed, C.M. Bender and M. V. Berry, J. Phys. A: Math. Gen. 38 (2005)

L627.

[56] C.M. Bender, S. Boettcher and V.M. Savage, J. Math. Phys. 41 (2000) 6381.

[57] A. Eremenko, A. Gabrielov and B. Shapiro, Preprint math-ph/0703049 (2007).

[58] H. Hezari, Preprint math-ph/0703028 (2007).

[59] C.M. Bender, D.C. Brody and H.F. Jones, Phys. Rev. Lett. 92 (2004) 119902;

98 (2007) 40403.

[60] G.A. Mezincescu, J. Phys. A: Math. Gen. 33 (2000) 4911.

[61] C.M. Bender and Q. Wang, J. Phys. A: Math. Gen. 34 (2001) 3325.

[62] C.M. Bender, P. N. Meisinger and Q. Wang, J. Phys. A: Math. Gen. 36 (2003)

1029; 36 (2003) 6791.

[63] R.S. Kaushal, Pramana J. Phys 42 (1994) 315.

[64] M. Znojil and P.G.L. Leach, J.Math. Phys. 33 (1992) 2785.

[65] A.P. Hautot, Phys. Lett. A38 (1972) 305.

[66] Z. Ahmed, Phys. Lett. A282 (2001) 343; A294 (2002) 287.

[67] Y.P. Varshni, Phys. Lett. A183 (1993) 9.

[68] L.A. Singh, S.P. Singh and K.D. Singh, Phys. Lett. A148 (1990) 389.

[69] A.V. Turbiner, Comm. Math. Phys. 118 (1988) 467.

[70] A.V. Turbiner and A.G. Ushveridze, Phys.Lett. A126 (1987) 181.

[71] A. de Souza Dutra and H.Boschi Filho, Phys. Rev. A144 (1991) 4721.

[72] P.E.G. Assis and A. Fring, J. Phys. A: Math.Theor. 42 (2009) 015203.

[73] B.L. Burrows, M. Cohen and T. Feldman, J. Math. Phys. 35 (1994) 5572.

62 BIBLIOGRAPHY

[74] R. Adhikari, R. Dutt and Y.P. Varshni, Phys. Lett. A141 (1989) 1.

[75] E. Witten, Nucl. Phys. B185 (1981) 513.

[76] F. Cooper and B. Freedman, Ann. Phys. 146 (1983) 262.

[77] M.M. Nieto, Phys. Lett. B145 (1984) 208.

[78] J. Beckers, N. Debergh and L. Ntibashirakandi, J. Math. Phys. 36 (1995) 641.

[79] A. Kostelecky and M.M. Nieto, Phys. Rev. Lett. 53 (1984) 2285.

[80] F. Cooper, A. Khare and U.P. Sukhatme, Phys. Rep. 251 (1995) 267.

[81] A. Khare and R.K. Bhaduri, J. Phys. A27 (1994) 2213.

[82] F. Calogero, J. Math. Phys. 10 (1969) 2191.

[83] S.N. Behra and A. Khare, Lett. Math. Phys. 4 (1980) 153.

[84] S.N. Behra and A. Khare, Pramana J. Phys 15 (1980) 501.

[85] A. Khare and S.N. Behra, Pramana J.Phys. 14 (1980) 327.

[86] D. Amin, Phys. Today 35 (1982) 35; Phys. Rev. Lett. 36 (1976) 323.

[87] S. Colemann,“Aspects of symmetry,” selected Erice lectures (Cambridge Univ.

Press, Cambridge,1988) P 234.

[88] Fakir Chand and S.C. Mishra, Pramana J.Phys. 68 (2007) 891.

[89] R.S. Kaushal, Phys. Rev. A46 (1992) 2941.

[90] F.T. Hioe, D. MacMillan and E.W. Montroll, Phys. Rep. 43 (1978) 305.

[91] M. Hashimoto, Opt. Comm. 32 (1980) 383.

[92] R. Damburg, R. Propin and S. Graffi, Int. J. Quant. Chem. 21 (1982) 191.

[93] H Goldstein, “Classical Mechanics”, 2nd ed. Addition Wesley (1980).

[94] E. C. G. Sudarshan and N. Mukunda, “Classical Dynamics: A Modern Prospec-

tive”, Wiley, New York (1974).

[95] Ram Mehar Singh, Fakir Chand and S.C. Mishra, Pramana J. Phys. 72 (2009)

647.

[96] Fakir Chand, S.C. Mishra and Ram Mehar Singh, Pramana J. Phys. 73 (2009)

349.

[97] Z. Ahmed, Pramana J. Phys. 73 (2009) 323.

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

[99] C.M. Bender, Euro Contemp. Phys. 46 (2005) 277.

BIBLIOGRAPHY 63

[100] Fakir Chand, Ram Mehar Singh, Narender Kumar and S.C. Mishra, J. Phys.

A: Math.Theor. 40 (2007) 10171.

[101] Ram Mehar Singh, Fakir Chand and S.C. Mishra, Comm. Theor. Phys. 51

(2009) 397.

[102] M.E. Fisher, Phys. Rev. Lett. 40 (1978) 1610.

[103] J.L. Cardy, Phys. Rev. Lett. 54 (1985) 1345.

[104] J.L. Cardy and G. Mussardo, Phys.Lett. B225 (1989) 275.

[105] A.B. Zamolodehikov, Nucl. Phys. B348 (1991) 619.

[106] R. Brower, M. Furman and M. Moshe, Phys.Lett. B76 (1978) 213.

[107] B. Harms, S. Jones and C.I. Tan, Nucl. Phys. 171 (1980) 392; Phys. Lett. B91

(1980) 291.

[108] U. Guenther, F. Stefani and M. Znojil, J. Math. Phys. 46 (2005) 063504.

[109] U. Guenther, B.F. Samsonov and F. Stefani, J. Phys. A: Math. Theor. 40 (2007)

F169.

[110] C.F. de.M. Faria and A. Fring, Laser Phys. (2006) Preprint quant-ph/0609096.

[111] A.G. Ushveridze “Quasi-Exactly Solvable Models in Quantum Mechan-

ics”(1993) IOP Publishing.

[112] A.V. Turbiner, Sov. Phys.-JETP 67 (1988) 230 ; Contemp. Math. 160 (1994)

263.

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

[114] C.M. Bender and S. Boettcher, J. Phys. A: Math.Gen. 31 (1998) L273.

[115] M.V. Berry and D.H.J. O’Dell, J. Phys. A: Math. Gen. 31 (1998) 2093.

[116] C.M. Bender and G.V. Dunne and P.N. Meisinger, Phys. Lett. A252 (1999)

272.

[117] A. Khare and U.P. Sukhatme, Phys. Lett. A324 (2004) 406.

[118] A. Khare and U.P. Sukhatme, J. Math. Phys. 46 (2005) 082106.

[119] A. Khare and U.P. Sukhatme, J. Phys. A: Math.Gen. 39 (2006) 10133.

[120] A. Khare and U.P. Sukhatme, J.Math.Phys. 47 (2006) 062103.

[121] C.M. Bender, D.C. Brody, H.F. Jones and B. Meister, Phys. Rev. Lett. 98

(2007) 40403.

[122] D.C. Brody, D.W. Hook, J. Phys. A: Math. Gen. 39 (2006) L167.

64 BIBLIOGRAPHY

[123] P.E.G. Assis and A. Fring, preprint quant-ph/0703254 (2007).

[124] P. Dorey, C. Dunning and R. Tateo, J. Phys. A: Math. Gen. 34 (2001) L391.

[125] M. Henkel, “Classical and Quantum Nonlinear Integrable Systems: Theory and

Applications” (2003) IOP Publishing.

[126] F.C. Alcaraz, M. droz, M. Henkel and V. Rittenberg, Ann. Phys. 230 (1994)

250.

[127] J.P. Killingbeck and J.Georges, J. Phys. A: Math. Gen. 36 (2003) R105.

[128] Y.B. Aryeh, Ady Mann and Itamar Yaakov, J. Phys. A: Math. Gen. 37 (2004)

12059.

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