Nonlinear T-symmetry Quartic, Sextic, Octic Oscillator Models under Real Spectra

Abstract

We propose nonlinear model T-symmetry operators having quartic, sextic, octic anharmonicity and inverse quadratics under real spectra. In fact, the model operator is non-Hermitian but real in nature. A comparison with the corresponding hermitian counterpart shows higher energy levels (E^T ≫ E^hermitian).

1. Introduction

Since the discovery of $PT$-symmetry, real spectra of quantum operators are now confined to hermiticity, the space–time invariance property [1,2]. However, complex quantum systems can be $PT$ symmetry ($[PT,H]=0$) or T-symmetry ($[T,H]=0$). In the above, P stands for the parity operator having the behavior: $Px{P}^{-1}=-x$ and a time reversal operator (T) transformation nature, i.e., $Ti{T}^{-1}=-i$. The real spectra study under T-symmetry is very limited [3,4], unlike in PT-symmetry [2,5]. In fact, Hatano and Nelson [3] proposed a model theoretical analysis on localization transition in superconductors relating to Meissner’s transverse effect using imaginary vector potential. Later on, T-symmetry analysis involving a mod operator ($\left|x\right|$) and a harmonic oscillator was considered in the literature, e.g., [4].

Till now, all the discussions are pertaining to linear models only. In fact, nonlinear quantum systems are not exactly solvable, so spectral studies are not yet an industry [6]. In this paper, we propose a model of a nonlinear T-symmetry operator using the mean field models reported earlier by Ciosloski [6] and study its spectral nature as follows.

2. Non-Hermitian Models Having Real Eigenvalues

2.1. Operator Model

Here, we present a simple non-hermitian Hamiltonian using a simultaneous coordinate $\left(x\right)$ and momentum $\left(p\right)$ transformation in real and complex space as

$$x=x^{\prime }=x+1$$

(1)

and

$$p=p^{\prime }=p+i$$

(2)

Now applying the same to Harmonic oscillator, we have

$$H^{\prime }={(p+i)}^2+{(x+1)}^2$$

(3)

In a simplified form, this is written as

$$H^{{}^{\prime }}=p^2+x^2+2(x+ip)$$

(4)

In fact, energy levels of this operator $H^{{}^{\prime }}$ are

$$E_n^{{}^{\prime }}=2n+1$$

(5)

which remains the same as a simple harmonic oscillator. In fact, the above operator is linear, and the corresponding nonlinear operator can be written as [6]

$$H_N^{{}^{\prime }}|\psi >=[p^2+\int {\psi }^{*}\left(x\right)x^2\psi \left(x\right)dx+2(x+ip)]|\psi >$$

(6)

In short, this is expressed as [6]

$$H_N^{{}^{\prime }}=p^2+<x^2>+2(x+ip)$$

(7)

here, we use $<x^2>$ as $\int {\psi }^{*}\left(x\right)x^2\psi \left(x\right)dx$. It should be kept in mind that the energy levels of $H_N^{{}^{\prime }}$ are complex, and it is a case of broken T-symmetry. However, energy levels of

$$H_N=p^2+x^2+<x^2>+2(x+ip)$$

(8)

are real. Below, we cite the first four energy levels in

Table 1. First four energy levels of the T-symmetry operator (Equation (8)).

${\mathit{H}}_{\mathit{N}}$
1.0379
3.1139
5.1387
7.1570

.

In this context, we would like to compare the present ground state energy with that of Cioslowski [6] on Hamiltonian

$$H_{Cioslowski}=p^2+x^2+\alpha <x^2>x^2$$

(9)

in

Table 2. Ground state energy comparison.

Energy	$\mathit{\alpha }$	Present	Previous [6]
	0.5	0.59574	0.59574
	1	0.66235	0.66235
$<x^2>$	0.5	0.41964	0.41964
	1	0.37743	0.37743

.

For the interested reader, we present the ground state wave function for $\alpha =1$ in Figure 1.

In the above, we present a simple model of a T-symmetry operator using the coordinate space. However, in terms of a matrix, the above analysis can be reflected as follows.

2.2. Real Non-Hermitian Matrix Having Real Eigenvalues

Here, we present a model (2 × 2) matrix reflecting the non-hermiticity character satisfying the condition as $H\ne H^{†}$. It should be kept in mind that the present non-Hermitian matrix is nothing but a T-symmetry Hamiltonian in a matrix representation [4]. Here, we present a (2 × 2) matrix as

$$A=\left[\begin{array}{cc}60& 11\\ -11& 30\end{array}\right]$$

(10)

with eigenvalues ${\lambda }_1=55.1980$ and ${\lambda }_2=34.8020$. The above-suggested matrix can also be presented pictorially, as in Figure 2.

3. Momentum Transformation in Complex Space: Quartic Operator

Consider the transformation,

$$e^{f\left(x\right)}p{e}^{-f\left(x\right)}=p+i\frac{df\left(x\right)}{dx}$$

(11)

If $f\left(x\right)$ is an odd function of x, then the said Hamiltonian is T-symmetry. However, for an even function, the Hamiltonian is generally $PT$-symmetry in nature [5].

3.1. Invariance in Commutation Relation

Under the above transformation, the commutation relation between the coordinate and momentum remains invariant. Mathematically,

$$[x,p]=i$$

(12)

$$[x,p+i\frac{df}{dx}]=[x,p]+[x,\frac{df}{dx}]=[x,p]=i$$

(13)

Let us consider that $f\left(x\right)=\lambda x^3$ and $\lambda =1/3$. Now, we consider the quartic anharmonic oscillator Hamiltonian as

$$H=p^2+\beta x^4$$

(14)

3.2. Hamiltonian on Momentum Transformation

Here, we transform the momentum as $p\to p+i{x}^2$ and use the same in hamiltonian H. After transformation, the resulting Hamiltonian is written as

$$H=p^2+i{x}^{2}p+ip{x}^2+(\beta -1)x^4$$

(15)

On selecting $\beta =2$, we have

$$H=p^2+i{x}^{2}p+ip{x}^2+x^4$$

(16)

It is seen that in the above equation, $ip$ and $x^2$ are in a mixed form. The term $x^2$ can be separated from $ip$ using the mean field approximation [6] as

$$H=p^2+i<x^2>p+ip<x^2>+x^4$$

(17)

After the mean field approximation, the equation becomes nonlinear in nature [6]. Here, $<x^2>$ means $\int {\psi }^{*}{x}^2\psi dx$.

Instead of a complex space transformation, one can have a real space transformation as $p\to p+x^2$

$$H^{†}=p^2+x^{2}p+p{x}^2+3x^4$$

(18)

However, considering the T-symmetry operator, we present the first 10 energy levels of

$$H^{†}=p^2+<x^2>p+p<x^2>+x^4$$

(19)

and make a possible comparison. Interestingly, we find that the energy levels of the T-symmetry operator are higher compared to its Hermitian counterpart. In fact, both of the two are strongly nonlinear. Using the matrix diagonalization method [2], we present the first ten energy levels in

Table 3. First ten energy levels of T-symmetry and Hermiticity operators.

H	${\mathit{H}}^{†}$
1.2628	0.9293
4.0021	3.6686
7.6581	7.3246
11.8472	11.5136
16.4642	16.1307
21.4408	21.1073
26.7309	26.3974
32.3031	31.9675
38.1254	37.7919
44.1836	43.8500

, as given below.

4. Nonlinear Sextic Hamiltonian with Quadratic Nonlinearity

Now, we consider the Hamiltonian as

$$H=p^2+i<x^2>p+ip<x^2>+x^6-x^4$$

(20)

Using the matrix diagonalization method [2], we present the first ten energy levels in

Table 4. First, the ten energy levels of the T-symmetry sextic nonlinear operator (Equation (19)).

Quantum No.	Sextic Operator
0	1.0681
1	3.6568
2	7.8084
3	13.1266
4	19.3569
5	26.3947
6	34. 1618
7	42. 5987
8	51.6585
9	61.3030

, as given below.

5. Nonlinear Octic Hamiltonian with Quadratic Nonlinearity

Now, we consider the Hamiltonian as

$$H=p^2+i<x^2>p+ip<x^2>+x^8-x^4$$

(21)

Using the matrix diagonalization method [2], we present the first ten energy levels in

Table 5. First ten energy levels of the T-symmetry octic nonlinear operator (Equation (20)).

Quantum No.	Sextic Operator
0	1.1305
1	4.2463
2	9.3934
3	16.2161
4	24.4188
5	33.8533
6	44.4223
7	56.0506
8	68.6778
9	82.2541

as given below.

6. Nonlinear Quartic, Octic Hamiltonian with Quartic Nonlinearity

Now, we consider the Hamiltonian as

$$H=p^2+i<x^4>p+ip<x^4>+x^8$$

(22)

Using the matrix diagonalization method [6], we present the first ten energy levels in

Table 6. Energy levels of T-symmetry octic with a quartic nonlinear operator (Equation (21)).

Quantum No.	Sextic Operator
0	1.2677
1	4.7978
2	10.2869
3	17.3850
4	25.8509
5	35.5398
6	46.3547
7	58.2216
8	71.0812
9	84.8845

, as given below.

7. Nonlinear Singular Model Nonlinearity in Quartic Operator

In the above models, we have considered non-singular models. Here, we focus our attention on a singular model quartic operator as

$$H=p^2+i<\frac{1}{x^2}>p+ip<\frac{1}{x^2}>+x^4$$

(23)

Using the matrix diagonalization method [2], we present the first ten energy levels in

Table 7. Energy levels of T-symmetry singular model quartic operator (Equation (22)).

Quantum No.	Sextic Operator
0	1.0627
1	3.8020
2	7.4580
3	11.6471
4	16.2640
5	21.2407
6	26.5308
7	32.1009
8	37.9253
9	43.9835

, as given below.

8. Method of Calculation

Here, we use the matrix diagonalization method to solve the eigenvalue relation

$$H|\Phi >=E|\Phi >$$

(24)

where

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

(25)

in which $|m>$ satisfies the eigenvalue relation [2]

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

(26)

with $m=0,1,2,3,4\cdots$

Below, we present matrix elements using the above method. The numerical matrix element of (4 × 4) matrix is given below (Equation (17))

$$H(4\times 4)=\left[\begin{array}{cccc}5/4& 1694/1977& 1393/985& 0\\ -1694/1977& 21/4& 1482/1223& 4801/980\\ 1393/985& -1482/1223& 49/4& 981/661\\ 0& 4801/980& -981/661& 69/4\end{array}\right]$$

(27)

Interested readers can check the diagonal element from the relation

$$H_{n,n}=1.25+3.5n+1.5n^2$$

(28)

9. Conclusions

In this paper, we present a model of a nonlinear quartic, sextic and oscillator with singular as well as non-singular models satisfying a T-symmetry condition under a mean field approximation [6]. The models are unbroken in nature. However, if T-symmetry is incorrectly selected, it may give complex spectra (see Equation (7)). In other words, real spectra in quantum operators are not only confined to Hermiticity, PT-symmetry but also to T-symmetry. It should be kept in mind that the suggested models can be extended to fractional operators also. As the topic of T-symmetry is not yet an industry, we found minimal literature on it. We believe one can propose hybrid models as

$$H=p^2+x^2+i<x^2>p+ip<x^2>+x^2<x^2>+x^2<x^4>$$

(29)

and many more.