Acta Polytechnica


https://doi.org/10.14311/AP.2022.62.0090
Acta Polytechnica 62(1):90–99, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

RATIONAL EXTENSION OF MANY PARTICLE SYSTEMS

Bhabani Prasad Mandal

Banaras Hindu University, Institute of Science, Physics Department, Lanka, 221005–Varanasi, Uttar Pradesh
India
correspondence: bhabani.mandal@gmail.com

Abstract.
In this talk, we briefly review the rational extension of many particle systems, and is based on

a couple of our recent works. In the first model, the rational extension of the truncated Calogero-
Sutherland (TCS) model is discussed analytically. The spectrum is isospectral to the original system and
the eigenfunctions are completely expressed in terms of exceptional orthogonal polynomials (EOPs). In
the second model, we discuss the rational extension of a quasi exactly solvable (QES) N-particle Calogero
model with harmonic confining interaction. New long-range interaction to the rational Calogero model
is included to construct this QES many particle system using the technique of supersymmetric quantum
mechanics (SUSYQM). Under a specific condition, infinite number of bound states are obtained for
this system, and corresponding bound state wave functions are written in terms of EOPs.

Keywords: Exceptional orthogonal polynomials, rational extensions, many particle systems, SUSYQM.

1. Introduction
Orthogonal polynomials play very useful and important roles in studying physics, particularly in electrostatics
and in quantum mechanics. In quantum mechanics, only a few of the commonly occuring bound states problems,
which have a wide range of applications and/or extensions, are solvable. Such systems generally bring into
physics a class of orthogonal polynomials.These classical orthogonal polynomials have many properties common,
such as (i) each constitutes orthogonal polynomials of successive increasing degree starting from m = 0, (ii) each
satisfy a second order homogeneous differential equations, (iii) they satisfy orthogonality over a certain interval
and with a certain non-negative weight function, etc. In 2009, new families of orthogonal polynomials (known
as exceptional orthogonal polynomials (EOP)) related to some of the old classical orthogonal polynomials were
discovered [1–3]. Unlike the usual classical orthogonal polynomials, these EOPs start with degree m = 1 or
higher integer values and still form a complete orthonormal set with respect to a positive definite inner product
defined over a compact interval. Two of the well known classical orthogonal polynomials, namely Laguerre
orthogonal polynomials and Jacobi orthogonal polynomials, have been extended to EOPs category. Xm Laguerre
(Jocobi) EOP means the complete set of Laguerre (Jacobi) orthogonal polynomials with degree ≥ m. m is
positive integer and can have values of 1, 2, 3, . . . Attempts were made to also extend the classical Hermite
polynomials [4]. Soon after this remarkable discovery, the connection of EOPs with the translationally shape
invariant potential were established [5–9]. The list of exactly solvable quantum mechanical systems is enlarged
and the wave functions for the newly obtained exactly solvable systems are written in terms of EOPs. Such
systems are known as rational extension of the original systems. The study for the exactly solvable potentials
has been boosted greatly due to this discovery of EOPs over the past decade [10–37].

There are several commonly used approaches to build the rationally extended models, such as SUSYQM
approach [38, 39], Point canonical transformation approach [40, 41], Darboux-Crum transformation approach [42,
43], group theoretical approach [44], etc. These approaches have been used to study different problems in this
field leading to a discovery of a large number of new exactly solvable systems, which are isospectral to the
original system and the eigenfunctions are written in terms of EOPs. Further, quasi-exactly solvable (QES)
systems [45–49] and conditionally exactly solvable (CES) systems [50, 51] attracted attention in literature due
to the lack of many exactly solvable systems. Several works have been devoted to the rational extension of
these QES/ CES systems [22, 24, 37]. Nowadays, the parity time reversal (PT) symmetric non-Hermitian
systems [52–62] are among the exciting frontier research areas. Rational extensions have also been carried out for
non-Hermitian systems [6, 19, 29–32]. Even though most of the rational extensions are for the one dimensional
and/or one particle exactly solvable systems, the research in this field has also been extended to many particle
systems [24, 25, 27]. We have done several works on rational extensions for many particle systems. In one of
the works, the well known Calogero-Wolfes type 3-body problem on a line was extended rationally to show
that exactly solvable wave functions are written in terms of Xm Laguerre and Xm Jacobi EOPs [26]. However,
this article is based on two of our earlier works on rational extension of many particle systems [24, 25], which

90

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vol. 62 no. 1/2022 Rational extension of many particle systems

were central to the talk presented during the AAMP meeting. In the first work [25], we discuss the rational
extension of the truncated Calogero-Sutherland model using a PCT approach. We indicate how to obtain
rationally extended solutions, which are isospectral to the original system in terms of Xm Laguerre EOPs. In
the second model [24], we discuss the rational extension of a QES N-particle Calogero model with a harmonic
confining interaction. New long-range interactions to the rational Calogero model are included to construct this
QES many particle system using SUSYQM. The wavefunctions are expressed, again, in terms of exceptional
orthogonal Laguerre polynomials.

Now, we present the organisation of the article. In the next section, we present the TCS model and its
solutions in brief to set the things for the section 3, where we consider the rational extension of the TCS model.
In section 4, the QES solutions for the rationally extended Calogero type many particle system are presented.
Section 5 is reserved for conclusions.

2. TCS model
In his work, Jain-Khare (JK) [63] exactly solved some variant of Calogero-Sutherland model (CSM) on the full
line by taking only the nearest and next-to-nearest neighbor interactions through 2-body and 3-body interactions.
Later, Pittman et al. [64] generalized this model by considering an N-body problem on a line with harmonic
confinement with tunable inverse square as well as the three-body interaction extends over a finite number of
neighbors and were able to solve it exactly. This model is known as truncated Calogero-Sutherland model (TCS).
N-body TCS model [64], where particles are interacting through 2-body and 3-body potentials, is given by

H =
N∑

i=1

[
−

1
2
∂2

∂x2i
+

1
2
ω2x2i

]
+

∑
i<j

|i−j|≤r

λ(λ − 1)
| xi − xj |2

+
∑

i<j<k
|i−j|≤r
|j−k|≤r

λ2(xi − xj )x · (xj − xk)x
|xj − xj |2|xj − xk|2

(1)

with λ ̸= 0 and x = (x1,x2, . . . ,xN ) ∈ RN . The 2- body interaction is attractive for 0 < λ < 1 and is repulsive
for λ ≥ 1. Here, r is the integer parameter and for r = 1, this system reduces to that of the Jain-Khare [63]
model. However, for r = N − 1, it corresponds to the CSM [65–67] model.

Using standard techniques in the case of many particle systems, the time independent Schrodinger equation
(TISE) corresponding to the above system can be written in radial and angular parts as

Φ′′(ρ) +
(
N + 2s − 1 + λr(2N − r − 1)

)
1
ρ

Φ′(ρ) + 2(E −
1
2
ω2ρ2)Φ(ρ) = 0 , (2)

(where ρ =
∑N

i x
2
i , is the radial coordinate and the prime denotes the differentiation with respect to its

arguments and this convention is adopted throughout this manuscript) and
[ N∑

i=1

∂2

∂x2i
+ 2λ

N −1∑
i<j

1
xi − xj

(
∂

∂xi
−

∂

∂xj

)]
Ps(x) = 0. (3)

(where the function Ps denotes the homogeneous polynomial of angular variables of degree s = 0, 1, 2, . . . ). To
obtaine these Eqs. we have substituted the wave function

Ψ(x) =
∏
i<j

(xi − xj )λ Φ(ρ) Ps(x) (4)

in the TISE, HΨ = EΨ .
This model is solved exactly and the solution is given by,

the spectrum: En = ω
(
2n + s +

N

2
+

λr

2
(2N − r − 1)

)
, (5)

and the corresponding radial wave function in terms of classical Laguerre polynomials is given as

Φ(ρ) ≃ exp(−
ωρ2

2
)L(α)n (ωρ

2); n = 0, 1, 2, . . . (6)

where α =
(
s − 1 + N2 +

λr
2 (2N − r − 1)

)
. This result is consistent with JK and CSM models in the appropriate

limit [64]. In the next section, we will extend this model by adding some interaction terms. Then, the extended
model will be cast as a rational extension of the TCS model.

91



Bhabani Prasad Mandal Acta Polytechnica

3. Extended TCS model (ETCS)
We would like to find another system related to the TCS model, which is isospectral to the TCS and possibly,
its wave functions are written in terms of EOPs. To find the rational extension of the TCS model, we start by
adding a new interaction term, [25]

He = H +
(α1 + α2ω2ρ2)
(β1 + β2ω2ρ2)2

= H + Vnew, (7)

where α1,2 and β1,2 are unknown constants and will be fixed later. We would like to show that this He will
correspond to the rational extension of this TCS model for some specific values of the parameters α1,2 and β1,2.
Since Vnew depends only on radial coordinate, ρ, only the radial equation will be modified and angular equation
will be the same as in the case of the TCS model. The radial part is obtained having the same substitution as
in Eq. (4) in the earlier section for the Hamiltonian He

Φ′′ext(ρ) +
(
N + 2s − 1 + λr(2N − r − 1)

) 1
ρ

Φ′ext(ρ) + 2
(
E − (

1
2
ω2ρ2 + Vnew)

)
Φext(ρ) = 0, (8)

with Ps(x) satisfying the same generalised Laplace equation as in Eq. (3). Note that here, a prime on Φext(ρ)
indicates a derivative with respect to ρ.
We further substitute,

Φext(ρ) = f(ρ)ζ(g(ρ)), (9)

in Eq. 8 where f(ρ) and g(ρ) are two undermined functions and ζ(g) is a special function to obtain

ζ′′(g) +
(

2f′(ρ)
f(ρ)g′(ρ)

+
g′′(ρ)
g′(ρ)2

+
τ

ρg′(ρ)

)
ζ′(g) +

1
g′(ρ)2

(
f′′(ρ)
f(ρ)

+
τf′(ρ)
ρf(ρ)

+ 2(Eext − Vext)
)
ζ(g) = 0, (10)

where, τ =
(
N + 2s − 1 + λr(2N − r − 1)

)
and Eext is exactly same as En given in Eq. (5)

We now compare this differential equation satisfied by ζ(g(ρ)) with the differential equation satisfied by the
X1 Laguerre polynomial L̂

(α)
n (g)

L̂
′′ (α)
n (g(ρ)) −

(g − α)(g + α + 1)
g(g + α)

L̂
′ (α)
n (g(ρ)) +

1
g

(
(g − α)
(g + α)

+ n − 1
)

L̂
(α)
n (g(ρ)) = 0; n ≥ 1, (11)

to obtain (with n → n + 1,)

Vext =
1
2
ω2ρ2 +

4ω
(2ωρ2 + τ − 1)

−
8ω(τ − 1)

(2ωρ2 + τ − 1)2
, (12)

and

f(ρ) ≃ (g′(ρ))−
1
2 ρ−

α
2 exp

(
1
2

∫ g
[−

(g − α)(g + α + 1)
g(g + α)

]dg
)
. (13)

for a given g(ρ) as defined in the case of a conventional model

g(ρ) = ωρ2; α =
τ

2
−

1
2
. (14)

The energy eigenvalues Eext for the new system with the potential in Eq. 12 turn out to be the same as that of
the conventional TCS model as discussed in Section 2 and are given by Eq. (5). However, the corresponding
eigenfunction Φext(ρ) is completely different. Using f(ρ) and replacing ζ(g) → L̂

(α)
n+1(g) in Eq. (9), the expressions

for the energy eigenfunctions are obtained in terms of X1 exceptional orthogonal Laguerre polynomials (L̂
(α)
n+1(g))

as

Φext(ρ) ≃
exp(− ωρ

2

2 )
(2ωρ2 + α)

L̂
(α)
n+1(ωρ

2); n = 0, 1, 2, . . . (15)

Note that the X1 Laguerre polynomial (L̂
(α)
n+1(g)) is related to the classical Laguerre polynomials by

L̂
(α)
n+1(g) = −(g + α + 1)L

(α)
n (g) + L

(α)
n−1(g). (16)

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The constant parameters α1,2 and β1,2 for which the Hamiltonian (7) is exactly solvable can easily be determined
by comparing Eqs. (7) and (12), and one finds that

α1 = −4ω(τ − 1); α2 = 8,
β1 = τ − 1; and β2 = 2/ω. (17)

In the special cases of r = 1 and r = N − 1, we then obtain the rational extension of the JK model and the
CSM, respectively.

Xm case:
Similar to the X1 case, we redefine Eqs. (8) and (9) by replacing Φext(ρ) → Φm,ext(ρ) and f(ρ) →

fm(ρ),ζ(g) → ζm(g), respectively. Now the differential Eq. (8) will also be m dependent and can be written as

Φ′′m,ext(ρ) +
(
N + 2s − 1 + λr(2N − r − 1)

) 1
ρ

Φ′m,ext(ρ) + 2
(
E − (

1
2
ω2ρ2 + Vm,new)

)
Φm,ext(ρ) = 0. (18)

Now, we proceed with the steps as in the case of X1, by substituting Φm,ext(ρ) = fm(ρ)ζm(g(ρ)) in the above
equation to obtain the differential equation for ζm(g), which is exactly same as in Eq. (10).Then, we compare
that equation with the Xm exceptional Laguerre differential equation

L̂
′′(α)
n,m (g(ρ)) + Qm(g)L̂

′(α)
n,m(g(ρ)) + Rm(g)L̂

(α)
n,m(g(ρ)) = 0, (19)

with

Qm(g) =
1
g

[
(α + 1 − g) − 2g

L
(α)
m−1(−g)

L
(α−1)
m (−g)

]

and Rm(g) =
1
g

[
n − 2α

L
(α)
m−1(−g)
L

(α)
m (−g)

]
(20)

to get (replacing n by n + m)

Vm,new = −2ω2ρ2
L

(α+1)
m−2 (−g)

L
(α−1)
m (−g)

+ 2ω(α + ωρ2 − 1)
L

(α)
m−1(−g)

L
(α−1)
m (−g)

+ 4ω2ρ2
(
L

(α)
m−1(−g)

L
(α−1)
m (−g)

)2
− 2mω, (21)

and

f(ρ) ≃ (g′(ρ))−
1
2 ρ−

α
2 exp

(
1
2

∫ g
Qm(g)dg

)
. (22)

We note that the spectrum for the potential in Eq. (21) is exactly same as that for the potential in Eq. (12) and
for a usual TCS system. However, the eigen functions in all three cases are different. For a usual TCS model,
these are in terms of classical Laguerre polynomials, in case of the potential in Eq. (12), that is, in the X1 case,
these are in terms of X1 Laguerre polynomials and finally the wave functions for the system with the potential
in Eq. (21) are in terms of Xm Laguerre polynomials. The wave functions for the system described by the
potential in Eq. (21) are given by

Φm,ext(ρ) ≃
exp(− ωρ

2

2 )
L̂

(α−1)
m (−ωρ2)

L̂
(α)
n+m(ωρ

2); n,m = 0, 1, 2, . . . (23)

where the Xm Laguerre polynomial (L̂
(α)
n+m(g)) is related to the classical Laguerre polynomials by

L̂
(α)
n+m(g) = L

(α)
m (−g)L

(α−1)
n (g) + L

(α−1)
m (−g)L

(α)
n−1(g). (24)

As expected, for m = 1, the above results reduce to the corresponding X1-case, while for the m = 0 case one
gets back the conventional TCS model. In the next section, we will consider a rational extension of Calogero
like many particle systems.

93



Bhabani Prasad Mandal Acta Polytechnica

4. QES many particle system
In this section, we would like to discuss a rational extension of Calogero like many particle systems [24]. We
start with a many particle Calogero like Hamiltonian with an arbitrary potential U(

√
Nρ), which depends only

on the ‘radial’ coordinate ρ

H = −
N∑

i=1

∂2

∂x2i
+

N∑
i<j

λ

(xi − xj )2
+ U(

√
Nρ); λ ≥ −

1
2
, ρ =

√√√√ 1
N

N∑
i<j

(xi − xj )2 (25)

Our aim is to construct a possible structure of U(
√
Nρ), for which the many particle model is exactly solvable.

To this end, we follow the standard method [65–67] and take a trial wavefunction for ψ(x) in a sector of the
configuration space corresponding to a definite ordering of particles (e.g., x1 ≥ x2 ≥ · · · ≥ xN ) as

ψ(x) =
∏
i<j

(xi − xj )a+
1
2 Pk,q (x)ϕ(ρ), with a =

1
2

√
1 + 2λ (26)

and obtain the radial part differential equation,

−[ϕ′′(ρ) + 2(k + b + 1)
1
ρ
ϕ′(ρ)] + [U(

√
Nρ) − E]ϕ(ρ) = 0 (27)

with b = N (N −1)2 a +
N (N +1)

4 − 2. The angular part is described by Pk,q (x), which are translationally invariant,
symmetric, k-th order homogeneous polynomials satisfying the differential equations

N∑
j=1

∂2Pk,q (x)
∂x2j

+
(

a +
1
2

) ∑
j ̸=k

1
(xj − xk)

(
∂

∂xj
−

∂

∂xk

)
Pk,q (x) = 0 . (28)

Note that the index q in Pk,q (x) can take any integral value ranging from 1 to λ(N,k), where λ(N,k) is the
number of independent polynomials. The existence of such translationally invariant, symmetric and homogeneous
polynomial solutions of (28) has been discussed in the original work by Calogero [65, 66].

For our purpose, we will look for a solution of the radial part and proceed with the substitution

ϕ(ρ) = ρ−(l+1)χ(ρ), with l = k + b (29)

in Eq. (27) to obtain

−
d2

dρ2
χ(ρ) + Uk(

√
Nρ)χ(ρ) = Eχ(ρ) (30)

where Uk(
√
Nρ) is k dependent (through l) and is written as

Uk(
√
Nρ) =

l(l + 1)
ρ2

+ U(
√
Nρ) . (31)

Our aim here is to find a solution of Eq. (30) with a possible general structure of U(
√
Nρ). This will provide an

exact solution of a many particle system given in Eq. (25). This can be done in various ways, but we would like
to use the technique of SUSYQM, details of which can be found in [38, 39]

We consider a specific superpotential of the form [22]

W(ρ) = ρ +
2g1ρ

1 + g1ρ2
+
α + 1
ρ

,g1(α) =
2

2α + 3
. α ∈ R+ (32)

for which one of the partner potentials is a radial oscillator. The partner potentials can be calculated as,

V+(ρ) = ρ2 +
α(α + 1)

ρ2
+ 2α + 7 , (33a)

V−(ρ) = ρ2 +
(α + 1)(α + 2)

ρ2
−

4g1
1 + g1ρ2

+
8g21ρ2

(1 + g1ρ2)2
+ 2α + 5 . (33b)

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The potential V+ is the potential for a radial oscillator model with a constant term. The complete solution for
this potential is given by

E+n = 4
(
n + α +

5
2

)
, n = 0, 1, 2, . . . (34)

χ+n (ρ) =
√

n!
Γ(n + α + 32 )

ρα+1e−ρ
2/2Lα+1/2n (ρ

2) . (35)

Where Lα+1/2n (ρ2) is a usual Laguerre polynomial. Using the results of SUSYQM, we can obtained the solution
for the partner potential V−(ρ) given in Eq. (33b) as

E−n = 4
(
n + α +

5
2

)
, (36)

which is the same as in the case of V+, the radial oscillator potential. The radial part of the wave function is
written as

χ−n (ρ) =
√

4(n!)
E+n Γ(n + α + 32 )

ρα+2e−ρ
2/2 1

L
α+1/2
1 (−ρ2)

L̂
α+3/2
n+1,1 (ρ

2). (37)

This solution for V−(ρ) is possible only when the parameter g1 depends on α in a particular fashion and hence,
the model with V− is conditionally exactly solvable. Note that V+(ρ) can be used to generate the exactly
solvable Calogero model with a harmonic confining interaction as in this case

U(
√
Nρ) = Uk(

√
Nρ) −

l(l + 1)
ρ2

= ρ2 +
α(α + 1) − l(l + 1)

ρ2
, (38)

apart from an overall constant term. Now, l = k + b and α are free parameters and we can chose the parameter
α = l such that U(

√
Nρ) = ρ2 is independent of k. Calogero has shown that the many particle system (25) with

U(
√
Nρ) = ρ2 can be solved exactly for all possible values of k. However, unlike this case, V−(ρ) represents

a QES many particle system as we explain below.

U(
√
Nρ) = ρ2 +

(α + 1)(α + 2) − l(l + 1)
ρ2

−
4g1

1 + g1ρ2
+

8g21ρ2

(1 + g1ρ2)2
+ 2α + 5 , (39)

In this case, α depends on g1 and hence can’t be chosen as such that U(
√
Nρ) is independent of k. Hence, the

many particle system with U(
√
Nρ) is QES as it is solvable only for a particular value of k. The eigenvalues are

En = 4
(
n + α +

5
2

)
, n = 0, 1, 2, . . . . (40)

and the corresponding (unnormalized) QES eigenfunctions in terms of X1 Laguerre polynomials are written as

ψn(x) = ρα−l+1e−ρ
2/2 1

L
α+1/2
1 (−ρ2)

L̂
α+3/2
n+1,1 (ρ

2)
∏
i<j

(xi − xj )a+
1
2 Pk̃,q (x) . (41)

Note that the solution of the angular part Pk̃,q (x) is only for a specific value of the degree of the polynomial,
i.e. for k = k̃. We can’t find the solution of the radial part for k ̸= k̃. Thus, the solution is not complete, we
have obtained a part of it and thus, in this sense, we called the solutions QES. Now, we have obtained another
potential given in Eq. (33b), which is isospectral to a radial oscillator potential but the wave functions are
completely different and are written in terms of X1 Laguerre polynomials. Thus, we have achieved the rational
extension of Calogero like many particle system with a U(

√
Nρ) given in Eq. (39).

We would like to point out that exactly same result can also be obtained using the other method, like PCT
method. Furthermore, using the PCT approach, one can obtain the most general rationally extended radial
oscillator potential

Vm(ρ) = ρ2 +
l(l + 1)
ρ2

−
4ρ2Ll+3/2m−2 (−ρ

2)
L

l−1/2
m (−ρ2)

+ 2(2ρ2 + 2l − 1)
4ρ2Ll+1/2m−2 (−ρ

2)
L

l−1/2
m (−ρ2)

+ 8ρ2[
4ρ2Ll+3/2m−2 (−ρ

2)
L

l−1/2
m (−ρ2)

]2 − 4 (42)

95



Bhabani Prasad Mandal Acta Polytechnica

whose bound state solutions for the energy eigenvalues En = (4n + 2l + 3) n = 0, 1, 2, . . . and eigenfunctions
are written in terms of Xm exceptional Laguerre Polynomials as

χn,m(ρ2) = [
(n − m)!

(l + 1/2 + n)Γ(l + 1/2 + n − m)
]1/2

xl+1 exp(−ρ2/2)
L

l−1/2
m (−ρ2)

L̂
l+1/2
n+m,m(ρ

2) (43)

Where L̂l+1/2n+m,m(ρ2) is Xm exceptional orthogonal Laguerre polynomial, m = 0 corresponds to usual Laguerre
polynomials. Now, we note that m = 1 corresponds to the case we discus, in the context of Calogero Model as
the potential

V1 = ρ2 +
l(l + 1)
ρ2

−
8

2ρ2 + 2l + 1
+

32ρ2

(2ρ2 + 2l + 1)2
(44)

calculated from Eq. (42) is the same as our V−(ρ) in Eq. (33b) when l = (α + 1) and g1 = 2(2α+3) apart from an
overall constant term. The solution obtained through the PCT approach in Eq. (43) for m = 1 is exactly the
same as we obtained through the supersymmetric approach.

5. Conclusions
In this article, we have reviewed two of our old works [24, 25] on a rational extension of many particle systems.
In the first model [25], we have considered the TCS model with pairwise 2-body and 3-body interactions, and
using the well known PCT approach, we have extended the model rationally. The spectrum is isospectral to the
original TCS system and the wave functions are written in terms of Xm Laguerre polynomials. This means that
we have a family of isospectral systems for m = 1, 2, . . . related to the TCS model with different potentials.
The eigen functions are different and are written in terms of EOPs. In the other model, we have considered the
QES Calogero like many particle system, and using SUSYQM techniques, obtained the general structure of
the QES potential for which we can find the QES solutions. The wave functions are written in terms of EOPs.
We have considered broken SUSYQM in our approach. We feel it would be of interest to investigate similar
many particle systems for which supersymmetry is unbroken.

Acknowledgements
The author acknowledges the support from MATRIX project (Grant No. MTR/2018/000611), SERB, DST Govt. of
India and the Research Grant for Faculty under IoE Scheme (number 6031) of BHU.

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	Acta Polytechnica 62(1):90–99, 2022
	1 Introduction
	2 TCS model
	3 Extended TCS model (ETCS)
	4 QES many particle system
	5 Conclusions
	Acknowledgements
	References