EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 2005-2008
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Real Spectra in Logarithmic model PT-symmetry
operators: Iso-spectra in Logarithmic PT-symmetry

Biswanath Rath1, Rabab Jarrar2, Hussein Shanak2, Rania Wannan3,
Jihad Asad2,∗

1 Department of Physics, Maharaja Sriram Chandra Bhanj Deo University, Takatpur,
Baripada -757003, Odisha, India
2of Physics, Faculty of Applied Sciences, Palestine Technical University- Kadoorie, Tulkarm,
P 305,Palestine
3 Department of Applied Mathematics, Faculty of Applied Sciences, Palestine Technical
University- Kadoorie, Tulkarm, P 305, Palestine

Abstract. We reflect real spectra of new logarithmic model PT-symmetry operators with singular
and non-singular in nature. We also notice the iso-spectral nature between inverted and non-
inverted PT-symmetry potentials. Present numerical result give good agreement with previous
results.

2020 Mathematics Subject Classifications: 47B40, 81Q10, 81Q60

Key Words and Phrases: Quantum mechanics, Logarithmic Potentials, PT-symmetry operators

1. Introduction

Real spectra in quantum operators are confined to Hermiticity(H = H†) as well as
PT-symmetry [1]([H,PT ] = 0).Here P stands for parity operator having the properties:
PxP−1 = −x; PpP−1 = −p. Similarly T stands for the time reversal operator having the
propertiesTxT−1 = x;TpT−1 = −p and TT−1 = −i. In the Hermiticity(more precisely
self-adjoint operator), it ia possible to find two Hamiltonians, which are iso-spectral to
each other[2]. For example[2,3]

h(1) = p2 + V0(1− e2|x|/a) (1)

and
h(2) = p2 + v0(1− e−2|x|/a) (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4860

Email addresses: biswanathrath10@gmail.com (B. Rath),r.jarrar@ptuk.edu.ps (R. Jarrar),
h.shanak@ptuk.edu.ps (H. Shanak), r.wannan@ptuk.edu.ps (R. Wannan),
j.asad@ptuk.edu.ps (J. Asad)

https://www.ejpam.com 2005 © 2023 EJPAM All rights reserved.



B. Rath et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 2005-2008 2006

If one clearly analyzes one is scattering in nature and the other is confining nature [2,3].
Till now no such models are reflected in nature. In a recent paper Bender etal [4] have

suggested a new class of logarithmic PT-symmetry potentials as

h1 = p2 + x4 log(ix) (3)

h2 = p2 − x4 log(ix) (4)

h3 = p2 − x4 log(x2) (5)

and reflected energy spectrum of H1 only. Further authors reported analytical calcu-
lation of energy level using WKB approach[4]

En

[logEn)]1/3
∼

[
Γ(7/4)(n+ 1/2)

√
π

Γ(5/4)
√
2

]4/3

(6)

does not give encouraging results when compared with numerical results. This motivates
the present author to calculate spectra of H1,2 and suggest new models on logarithmic
potentials. Apart from this aim is to find out whether iso-spectral Hamiltonians are
possible in PT-symmetry operators.

2. Logarithmic new models

Here we consider different models as follows
Quadratic Logarithmic model V (x) = −x2 log( i

x)
The Hamiltonian considered here is

Hquadratic
1 = p2 − x2 log(

i

x
) (7)

Quartic inverted model V (x) = −x4 log( i
x)

Here we consider the Hamiltonian

Hquartic
2 = p2 − x4 log(

i

x
) (8)

Below we present few energy levels

3. Logarithmic models

Here we consider the recently prposed models

h1 = p2 + λx4 log(ix) (9)

h2 = p2 − λx4 log(ix) (10)

h3 = p2 − λx4 log(x2) (11)

and present complete spectra in table-2 for λ = 1.



B. Rath et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 2005-2008 2007

TABLE.I: PT-symmetry inverted logarithmic model potentials

n H1 Present H2

0 1.326 591 6 1.249 087 3
3 9.173 294 3 13.738 280 4
6 17.734 002 2 31.665 810 8
9 26.633 530 1 52.993 926 2
12 76.113 329 2 76.974 762 5

4. Method of calculation

Here we use matrix diagonalisation method [5] to solve the eigenvalue relation

H|Ψ >= E|Ψ > (12)

where
|Ψ >=

∑
Am|m > (13)

Here |m > satisfy the relation

[p2 + x2]|m >= (2m+ 1)|m > (14)

Numerical results obtained using MDM are tabulated in table.1.

5. Conclusion

In this report, we present numerical convergent energy levels of new model PT-invariant
Hamltonians using matrix diagonalisation method[5]. Further we feel the present method
can be used confidently to realize real spectra study in similar Hamiltonians of interest.
Lastly we the spectra of H2 and h1 are the same. In brief

V (x) = x4 log(ix) → V (x) = −x4 log(
i

x
) (15)

Hence these two potentials can be considered as iso-spectral models in PT-symmetry.
Lastly we do not find any numerical results to present in table-1 for a comparison with
the present numericals. Interested readers can consider other values of λ.

Acknowledgements

The authors R. Jarrar, H. Shanak, R. Wannan and J. Asad would like to thank
Palestine Technical University- Kadoorie for supporting them financially.



REFERENCES 2008

TABLE.2: PT-symmetry logarithmic model

n h1 Present h1Previous [4] Previous (WKB)[4]

0 1.249 08 1.249 09 0.546 27
3 13.738 27 13.738 3 7.314 80
6 31.665 82 31.665 8 16.697 9
9 52.993 79 52.993 9 27.695 6
12 76.976 08 76.974 8 39.932 4

n h2 Present Previous[4] Previous(WKB)[4]

0 0.109 1
1 6,959 6
2 8.257 1
3 18.039 4

n h3 Present Previous[4] Previous(WKB)[4]

0 0.025 4
1 4.977 7
2 9.237 1
3 16.478 6

References

[1] C.Bender and S.Boettcher Real spectra in non-Hermitian Hamiltonian having PT-
symmetry, Phys.Rev.Lett,(1997),80,5243-5246.

[2] H.F.Jones,Comment on: Solvable model bound states in the continuum (BIC)in One
dimension(2019, 94, 105214),Phys.Scr,(2021),96,087001.(see ref-2)

[3] Z.Ahmed and H.F.Jones,Scattering states and bound states of exponential potentials,
arxiv:2102:06095v1(see ref-3).

[4] C.Bender,A.Felski,S.P.Klevansky and S.Sarkar,PT-symmetry and Renormalisation in
Quantum Field Theory,arxiv:2103.14864v1.

[5] B.Rath,Real spectra in some negative potentials: Linear and nonlinear one dimen-
sional PT-invariant quantum systems,Eur.Phys.Journal.Plus.(2021), 136,493.