Shape Invariant Single and Double well Potentials under Spectral Invariance

TL;DR

This paper explores spectral invariance in shape invariant single and double well potentials by employing derivative invariance, providing insights into their structural properties.

Contribution

It introduces a new formulation of spectral invariance based on derivative invariance for shape invariant potentials.

Findings

Spectral invariance characterized in single and double well potentials.

Derivative invariance provides a new perspective on shape invariance.

Potential applications in quantum mechanics and spectral theory.

Abstract

We formulate the structure of spectral invariance in shape invariance single and double well potentials using derivative invariance.

Tables

Matrix A	Matrix $B^{-1}AB$
Eigenvalues of A	Eigenvalues of $B^{-1}AB$
FIG.1. Pictorial view

FIG.1.Non-commutativity nature

FIG.2.Equi-eigenvalue model operators

FIG3: Spectra of $H=p^2+ix$

FIG.2.Spectra of $H=p^2+\frac{i}{x}$

$B_1$	$B_2$	$B_3$
1.156 267	1.156 267	1.156 267
7.562 273	7.562 273	7.562 267
15.291 553	15.191 553	15.291 553
23.766 740	23.766 740	23.766 740
32.789 082	32.766 082	32.789 082
$B_4$	Previous[7]	Previous[4]
1.156 267	1.156 2	1.156 267
7.562 273	7.562 1	7.562 267
15.291 553	15.291 6	15.291 553
23.766 740		23.766 740

$B_1$	$B_2$	$B_3$
1.477 149 7	1.477 149 7	1.477 149 7
11.802 433 5	11.802 433 5	11.802 433 5
25.791 792 3	25.791 792 3	25.791 792 3
42.093 807 7	42.093 807 7	42.093 807 7
60.184 331 2	60.184 331 2	60.184 331 2
$B_4$	exact[5,6]	Previous[4]
1.477 149 7	1.477 149 7	1.477 149 7
11.802 433 5	11.802 433 5	11.802 433 5
25.791 792 3	25.791 792 3	25.791 792 4
42.093 807 7	42.093 807 7	42.093 814 5
60.184 331 2	60.184 331 2	60.185 767 6

Quantum no	$B_1$	$B_2$
0	1.249 08	1.249 08
3	13.738 27	13.738 27
6	31.665 82	31.665 82
9	52.993 79	52.993 79
12	76.976 08	76.976 08
$B_3$	$B_4$	Previous[6]
0	1.249 08	1.249 08
3	13.738 27	13.738 27
6	31.665 82	31.665 82
9	52.993 79	52.993 79
12	76.976 08	76.976 08

Equations

$$AB\ne BA$$

$$A=\left[{\begin{array}[]{ c c }25&20\\ 15&18\\ \end{array}}\right]$$

$$B=\left[{\begin{array}[]{ c c }8&10\\ 13&5\\ \end{array}}\right]$$

$$AB\ne BA$$

$$\frac{AB}{10}=\left[{\begin{array}[]{ c c }46&35\\ 35.4&24\\ \end{array}}\right]$$

$$\frac{BA}{10}=\left[{\begin{array}[]{ c c }35&34\\ 40&35\\ \end{array}}\right]$$

$$B_{1}=\left[{\begin{array}[]{ c c }8&10+3i\\ 10-3i&8\\ \end{array}}\right]$$

$$B_{1}^{-1}AB_{1}=\left[{\begin{array}[]{ c c }-0.8444+18.6667i&-10.5556+16.2667i\\ 31.5556-30.4000i&43.8444-18.6667i\\ \end{array}}\right]$$

$$B_{2}=\left[{\begin{array}[]{ c c }8\pm 2i&10\\ 10&8\mp 2i\\ \end{array}}\right]$$

$$B_{2}^{-1}AB_{2}=\left[{\begin{array}[]{ c c }-9.3750+21.8750i&-8.1250+24.3750i\\ 51.8750-10.6250i&52.3750-21.8750i\\ \end{array}}\right]$$

$$B_{3}=\left[{\begin{array}[]{ c c }8\pm 2i&10+i\\ 10&8\\ \end{array}}\right]$$

$$B_{3}^{-1}AB_{3}=\left[{\begin{array}[]{ c c }-9.3750+21.8750i&-8.1250+24.3750i\\ 51.8750-&8\\ \end{array}}\right]$$

$$B_k=h_{k}H{h}_k^{-1}$$

$$H=\left[{\begin{array}[]{ c c }1+\sqrt{3}i&4\\ 4&1-\sqrt{3}i\\ \end{array}}\right]$$

$$h_{1}=\left[{\begin{array}[]{ c c }1&3+4i\\ 3-4i&1\\ \end{array}}\right]$$

$$B_{1}=\left[{\begin{array}[]{ c c }1-3.2097i&-1.9107+4.4330i\\ -1.9107-4.4330i&1+3.2097i\\ \end{array}}\right]$$

$$h_{2}=\left[{\begin{array}[]{ c c }2&3+4i\\ 3-4i&2\\ \end{array}}\right]$$

$$B_{2}=\left[{\begin{array}[]{ c c }1-5.4395i&-3.4149+5.5612i\\ -3.4149-5.5612i&1=5.4395i\\ \end{array}}\right]$$

$$h_{3}=\left[{\begin{array}[]{ c c }3&3+4i\\ 3-4i&3\\ \end{array}}\right]$$

$$B_{3}=\left[{\begin{array}[]{ c c }1-9.6806i&-6.5981+7.9486i\\ -6.5981-7.9486i&1+9.6806i\\ \end{array}}\right]$$

$$h_{4}=\left[{\begin{array}[]{ c c }4&1+3i\\ 1-3i&4\\ \end{array}}\right]$$

$$B_{4}=\left[{\begin{array}[]{ c c }1-22.1127i&-16.3806+15.2855i\\ -16.3806-15.2855i&1-22.1127i\\ \end{array}}\right]$$

$$h_1=p^2+x^2$$

$$E_n^1=(2n+1)$$

$$h_2=p^2+x^4$$

$$h_3=p^2+ix$$

$$h_4=p^2+\frac{i}{x}$$

$$B_k=h_k{H}_2{h}_k^{-1}$$

$$H_2=p^2+i{x}^3$$

$$B_{k=1,2,3,4}=h_{1,2,3,4}{H}_3{h}_{1,2,3,4}^{-1}$$

Full text

A generalized mathematical model on quantum equivalents of non-commutative operators via commutating eigenvalue relation: PT-symmetry

Biswanath Rath

*Department of Physics, Maharaja Sriram Chandra Bhanj Deo University, Takatpur, Baripada -757003, Odisha, India. *

$\bf{Abstract:}$

Any matrix ”$B$” satisfying the non-commuting relation $[A,B]\neq 0$ with ”$A$”, can be used via $B^{-1}AB$ to reproduce eigenvalues of ”$A$”. This universality relation is also equally valid for any matrix in any branch of physical or social science and also any operator involving co-ordinate$(x)$ or momentum$(p)$. Pictorially this is represented in the following fig.

Many interesting models including logarithmic potential have been considered.

**PACS no-11.30Pb, 03.65.Ge **

**Key words: spectral invariance, PT-symmetry, matrix model, operator,broken PT-symmetry,Un-broken PT-symmetry. **

**I. Introduction **

It is commonly known as any two matrices having the same dimension and non-singular in nature are basically non-commutative in nature[1],

[TABLE]

Now we consider an explicit form of A or B as

**Non-commutativity of matrices[1] **

For example we consider two matrices as $A$

[TABLE]

having eigenvalues $\lambda_{1,2}=39.1706;3.8294$ and $B$

[TABLE]

Here we do not pay importance to eigenvalues of $B$ ( whether reeal or complex) . However, $B$ must be non-singular in nature. Then it is easy to show that .

[TABLE]

For a pictorial representation we consider

[TABLE]

and

[TABLE]

Pictorially this is represented as in Fig-1.

As stated above, we have the freedom to change $B$ as $B_{k}$, where $k=1,2,3,4,.........$ . Below we consider a few different cases as

**Hermiticity **

Here we consider $B_{1}$ as a Hermitian operator as

[TABLE]

The corresponding, transformed matrix is

[TABLE]

whose eigenvalues are the same as $A$.

**PT-symmetry[2] **

Here the model matrix satisfies the condition $[B_{2},PT]=0$.

[TABLE]

.Here $P$ stands for the parity operator having the nature :$PxP^{-1}=-x$; $PpP^{-1}=-p$. Similarly T stands for the time reversal operator having the behaviour: $TxT^{-1}=x$; $TpT^{-1}=-p$ and $Ti=-i$[2].. The model operators, which have been studied for long time by different authors[2-5]

[TABLE]

Here also , the eigenvalues are the same as A

**Arbitrary non-singular matrix **

Here the the matrix considered is very arbitrary in nature.For example

[TABLE]

[TABLE]

In all the cases, we find that eigenvalues remain the same as A.

**2. Universality of eigenvalue invariance (via commutative relation) **

Let us confine to eigenvalues of $A=H$, using the relation

[TABLE]

Here the relation to be read as eigenvalues of LHS = eigenvalues of RHS. Mathematically for eigenvalue determination, it is basically a commutative relation (even though,non-commutative in general).

Below we confine our analysis on complex PT-symmetry operators in matrix as well as operator forms as given below..

**3. Spectral invariance in PT-symmetry operator **

Let us confine our attention to $A=H^{PT}=H$ as a PT-symmetry operator in matrix model.

Here, we consider the Hamiltonian(H) as (2x2) matrix model as

**First model **

[TABLE]

The above model has energy levels $\lambda_{1}=4.6056$ and $\lambda_{2}=-2.6056$. Let us choose different forms of $B_{k}$ as

[TABLE]

The corresponding $B_{1}$ becomes

[TABLE]

**Second model **

Here we consider $h_{2}$ as

[TABLE]

The corresponding $B_{2}$ becomes

[TABLE]

**Third model **

Here we consider $h_{3}$ as

[TABLE]

The corresponding $B_{3}$ becomes

[TABLE]

**Fourth model **

Here we consider $h_{4}$ as

[TABLE]

The corresponding $h_{4}$ becomes

[TABLE]

In all the cases, eigenvalues of $B_{k}$ remain the same as $H$. Pictorially, we have

**4a. Operator model involving $x$ and $p$. **

Here we consider diffent operators as follows.

**Hermitian : $h_{1}+p^{2}+x^{2}$ **

Let us consider a simple harmonic oscillator as $h_{1}$ i.e

[TABLE]

whose energy levels are real and equispaced.

[TABLE]

Now consider another hermitian operator as

[TABLE]

having real spectra [3]. As stated above $h_{k}$ can also ave broken spectra. Accordingly, we consider a broken spectra $h_{4}$ as

[TABLE]

whose spectra is complex see fig-3.

Similarly , we also consider another broken PT-symmetry operator as[5]

[TABLE]

Now use the flowing PT symmetry operator as

The corresponding $B$ operator is

[TABLE]

Let us consider $H_{2}$ as[4,5]

[TABLE]

The energy levels are tabulated in table-1.

**4b. Inverted quartic operator model[5-7] involving $x$ and $p$. **

$H_{3}=p^{2}-x^{4}$

The corresponding $B$ operator is

[TABLE]

**4c.Logarithmic model quartic PT-symmetry[8] involving $x$ and $p$. **

$H_{4}=p^{2}+x^{4}\log(ix)$

The corresponding $B$ operator is

[TABLE]

Here, we present results in table3.

**5.Method of calculation **

Here we solve the eigenvalue relation[9-11]

[TABLE]

where

[TABLE]

where $|m>$ satisfies the eigenvalue relation[8]

[TABLE]

.

**6.Conclusion **

Here, we have focussed attention on PT-symmetry model both matrix and operator involving$(x,p)$ and presented a unique spectral of Hamiltonian under the transformation. In fact, ” intertwining” operators $B_{k};h_{k};H$. It should be borne in mind that the operators$h_{k}$ be non-singular. The eigenvalues of $h_{k}$ are not an important criteria on selection of $B_{k}$. It can have broken spectra also. To justify this, we have considered two potentials $V_{1}=ix$ and $V_{2}=\frac{i}{x}$. Lastly one can have many more operators like this, in which no specific conditions to be imposed on $h_{k}$.

**Conflict of interest: **

Author declares there are no conflicts of interest.

**DATA Requirement : No additional data of any kind is required. **