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

TL;DR

This paper explores the spectral properties of logarithmic PT-symmetry operators, revealing iso-spectral relationships between different potential types and confirming findings with numerical results.

Contribution

It introduces a new logarithmic model of PT-symmetry operators and demonstrates their iso-spectral nature, expanding understanding of spectral properties in this context.

Findings

Real spectra are reflected in the new logarithmic PT-symmetry model.

Iso-spectrality exists between inverted and non-inverted potentials.

Numerical results align well with previous studies.

Abstract

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

Tables

n	$H_1$ Present	$H_2$
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

n	$h_1$ Present	$h_1$Previous [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	$h_2$ Present	Previous[4]	Previous(WKB)[4]
0	0.109 1
1	6,959 6
2	8.257 1
3	18.039 4
n	$h_3$ Present	Previous[4]	Previous(WKB)[4]
0	0.025 4
1	4.977 7
2	9.237 1
3	16.478 6

Equations

$$h^{(1)}=p^2+V_0(1-e^{2|x|/a})$$

$$h^{(2)}=p^2+v_0(1-e^{-2|x|/a})$$

$$h_1=p^2+x^4\log (ix)$$

$$h_2=p^2-x^4\log (ix)$$

$$h_3=p^2-x^4\log (x^2)$$

$$\frac{E_{n}}{[\log{E_{n})}]^{1/3}}\sim\Biggl{[}\frac{\Gamma(7/4)(n+1/2)\sqrt{\pi}}{\Gamma(5/4)\sqrt{2}}\Biggr{]}^{4/3}$$

$$H_1^{quadratic}=p^2-x^2\log (\frac{i}{x})$$

$$H_2^{quartic}=p^2-x^4\log (\frac{i}{x})$$

$$h_1=p^2+\lambda x^4\log (ix)$$

$$h_2=p^2-\lambda x^4\log (ix)$$

$$h_3=p^2-\lambda x^4\log (x^2)$$

$$H|\Psi >=E|\Psi >$$

$$|\Psi >=\sum A_m|m>$$

$$[p^2+x^2]|m>=(2m+1)|m>$$

$$V(x)=x^4\log (ix)\to V(x)=-x^4\log (\frac{i}{x})$$

Full text

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

Biswanath Rath $\dagger$,Rabab Jarrar$*$, Hussein Shanak$88$, Jihad Asad$***$ and Rania Wannan$\bullet$

$\dagger$* Department of Physics, Maharaja Sriram Chandra Bhanj Deo University, Takatpur, Baripada -757003, Odisha, India. e.mail:biswanathrath10@gmail.com*

$*;**;***$: Department of Physics,Faculty of Applied Sciences, Palestine Technical University, Kadoorie ,Tulkarm P 305,Palestine. e.mail:j.asad@ptuk.edu.ps ;r.jarrar@ptuk.edu.ps; h.shanak@ptuk.edu.ps

$\bullet$* Department of Applied Mathematics,faculty of Applied Sciences, Palestine Technical University, Kadoorie,Tulkarm,P 305, Palestine. e.mail:r.wannan@ptuk.edu.ps*

$\bf{Abstract:}$ 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. with available results.

**PACS:11.30.Pb;03.65.Ge. **

Correspondence: biswanathrath10@gmail.com ; m.kanan@ubt.ps ; j.asad@ptuk.edu.ps

**I.Introduction **

Real spectra in quantum operators are confined to Hermiticity($H=H^{\dagger}$) as well as PT-symmetry 1.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 properties$TxT^{-1}=x$;$TpT^{-1}=-p$ 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]

[TABLE]

and

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

and reflected energy spectrum of $H_{1}$ only. Further authors reported analytical calculation of energy level using WKB approach[4]

[TABLE]

does not give encouraging results when compared with numerical results. This motivates the present author to calculate spectra of $H_{1,2}$ 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 $V(x)=-x^{2}\log(\frac{i}{x})$ **

The Hamiltonian considered here is

[TABLE]

**Quartic inverted model $V(x)=-x^{4}\log(\frac{i}{x})$ **

Here we consider the Hamiltonian

[TABLE]

Below we present few energy levels

**3.Logarithmic models **

Here we consider the recently prposed models

[TABLE]

[TABLE]

[TABLE]

and present complete spectra in table-2 for $\lambda=1$.

**4.Method of calculation **

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

[TABLE]

where

[TABLE]

Here $|m>$ satisfy the relation

[TABLE]

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 $H_{2}$ and $h_{1}$ are the same. In brief

[TABLE]

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 $\lambda$.

**Authors contribution: **

B.Rath: formulation,computation,writing,finalizing;R.Jarrar: Computtation, finance;H.Shanak: computation and finance; J.Asad: writing,computation and Finance ; R.Wannan-computation,finance.

**Conflict of interest **

Authors declare there is no conflict of interest.

**DATA AVAILABILITY **

No additional data is required . All the datas included in this paper are sufficient.

**Declaration **

Present paper is a modified version of arxiv paper.