Rationally Extended Harmonic Oscillator potential, Isospectral Family and the Uncertainity Relations

The idea of Supersymmetric Quantum Mechanics(SQM) [1] is not only useful in solving the quantum mechanical potential problems but has also opened the scope for discovering new exactly solvable potentials. These potentials have applications in diverse areas like inverse scattering [2, 3], soliton theory [4, 5], etc. This sparked a race among researchers to search for a family of isospectral potentials [6, 7, 8, 9]. To accomplish this purpose, several methods were developed like Darboux transformation [10], Darboux Crum Krein Adler Transformation [11], SQM [12, 13], etc. Popular among them was the SQM approach due to its simplicity and it was shown using this approach that for any 1-D potential with atleast one bound state, one can always construct one continuous parameter family of strictly isospectral potentials.

The discovery of the $X_{m}$-exceptional orthogonal polynomials (EOPs) [14, 15, 16] paved the path for discovering new rationally extended shape invariant potentials whose eigenfunctions are in terms of these EOPs. Various properties of these potentials have been studied in detail in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and the references therein. After the discovery of the exceptional Hermite polynomials [30], Fellows and Smith [31] discovered rationally extended one dimensional harmonic oscillator potentials. Their work has been further extended using the SQM approach [32].

The one dimensional harmonic oscillator potential is one of the most important potential having numerous applications. However, so far as we are aware off, there has not been much progress in studying the various properties of the rationally extended family of harmonic oscillator (REHO) potentials. The purpose of this paper is to take a step in that direction. Firstly, we calculate the Heisenberg uncertainty relation $\Delta x\Delta p$ for the REHO potentials. Further, we follow the idea of SQM [1], and generate a one parameter $(\lambda)$ family of rationally extended strictly isospectral potentials including the corresponding Pursey and the Abrahm-Mosses potentials and obtain their eigenfunctions explicitly in terms of the $X_{m}$-Hermite EOPs. We calculate the Heisenberg uncertainty relations for the one parameter family of rationally extended isospectral potentials (including the corresponding Pursey and the Abrahm-Mosses potentials) for different $m$ and $\lambda$.

The plan of the paper is as follows: In Sec. $2$, we briefly discuss the formulation of SQM. In Sec. $3$, we summarise the known important results related to the rationally extended harmonic oscillator potentials. A one parameter $\lambda$ family of isospectral potentials (including the corresponding Pursey and Abrahm-Mosses potentials) are obtained in Sec. $4$ for any even integral $m$. In Sec. $5$, we follow the results discussed in Sec. $3$ and Sec. $4$ and calculate the Heisenberg uncertainty relations for REHO and their isospectral family of potentials (including the corresponding Pursey and Abrahm-Mosses potentials). Finally, we summarize our results and mention some open possible problems in Sec. $6$.

In SQM approach, one considers the Hamiltonian (in the units $\hbar=2m=1$)

$$H^{-}=-\frac{d^{2}}{dx^{2}}+V^{-}(x)-\epsilon\;$$

(1)

where $\epsilon$ is the factorization energy. By assuming the ground state energy of this Hamiltonian $E^{-}_{0}=0$, we factorize $H^{-}$ in terms of $A$ and $A^{\dagger}$ as

$$H^{-}=A^{\dagger}A\;$$

(2)

with

$$A=\frac{d}{dx}+W(x),\qquad A^{\dagger}=-\frac{d}{dx}+W(x).$$

(3)

Here $W(x)$ is known as superpotential which is expressed in term of the ground state eigenfunction as $W(x)=-\ln[\psi^{-}_{0}(x)]^{\prime}$. In this way, another set of Hamiltonian can easily be constructed by reversing the order of the operators $A$ and $A^{\dagger}$ i.e.,

$$H^{+}=AA^{\dagger}=-\frac{d^{2}}{dx^{2}}+V^{+}(x)-\epsilon\;.$$

(4)

Thus, the partner potentials $V^{\mp}(x)$ in term of superpotential are given by

$$V^{\mp}(x)=W(x)^{2}\mp W(x)^{\prime}+\epsilon\;.$$

(5)

Here a prime denotes a derivative with respect to $x$. The precise relationship between the energies and the eigenfunctions of the partner Hamiltonians are

$$E^{-}_{n+1}=E^{+}_{n}\;>0,\qquad n=0,1,2,\cdots,$$

(6)

$$\psi^{-}_{n+1}(x)=\sqrt{\frac{1}{E^{+}_{n}}}A^{\dagger}\psi^{+}_{n}(x),\qquad\psi^{+}_{n}(x)=\sqrt{\frac{1}{E^{-}_{n+1}}}A\psi^{-}_{n+1}(x).$$

(7)

The ground state eigenfunction $\psi^{-}_{0}(x)$ is obtained by solving the differential equation,

$$A\psi^{-}_{0}(x)=0.$$

(8)

One parameter family of isospectral potentials are obtained by redefining the form of the superpotential

$$\hat{W}=W(x)+\phi(x),$$

(9)

and by assuming the uniqueness of the partner potential $V^{+}(x)$ i.e.,

$$V^{+}(x)=\hat{W}(x)^{2}+\hat{W}(x)^{\prime}+\epsilon=W(x)^{2}+W(x)^{\prime}+\epsilon,$$

which gives

$$\phi^{2}(x)+2W\phi+\phi^{\prime}(x)=0.$$

We then find that on substituting $\phi(x)=y^{-1}(x)$, the above equation satisfies the Bernoulli equation

$$y^{\prime}(x)=1+2W(x)y(x),$$

whose solution is

$$\phi(x)=\frac{d}{dx}\ln\left[\mathcal{I}(x)+\lambda\right].$$

Here $\mathcal{I}(x)=\int^{x}_{\infty}\psi^{-}_{0}(x^{\prime})^{2}\;dx^{\prime}$ and $\lambda$ is an integration constant.

Therefore, the potential which is strictly isospectral to $V^{-}(x)$ is given by

	$\displaystyle\hat{V}^{-}(\lambda,x)$	$\displaystyle=\hat{W}(x)^{2}-\hat{W}(x)^{\prime}+\epsilon$
		$\displaystyle=V^{-}(x)-2\frac{d^{2}}{dx^{2}}\ln\left[\mathcal{I}(x)+\lambda\right]\,,$		(10)

where either $\lambda>0$ or $\lambda<-1$ so as to avoid singularity (For details, see[1, 33]). The normalized ground state eigenfunction for the entire family of potentials is given by

$$\hat{\psi}^{-}_{0}(\lambda,x)=\sqrt{\lambda(1+\lambda)}\;\frac{\psi^{-}_{0}(x)}{\mathcal{I}(x)+\lambda}\,.$$

(11)

The normalized excited states eigenfunctions can be easily calculated similar to (7) as

	$\displaystyle\hat{\psi}^{-}_{n+1}(\lambda,\;x)$	$\displaystyle=\sqrt{\frac{1}{E^{+}_{n}}}\hat{A}^{\dagger}\psi^{+}_{n}(x)=\frac{1}{E^{-}_{n+1}}\hat{A}^{\dagger}A\psi^{-}_{n+1}(x),\qquad n=0,1,2,\cdots,$		(12)
	$\displaystyle\hat{A}^{\dagger}$	$\displaystyle=-\frac{d}{dx}-\frac{d}{dx}\ln\left[\frac{\psi^{-}_{0}(x)}{\mathcal{I}(x)+\lambda}\right],\qquad A=\frac{d}{dx}-\frac{d}{dx}\ln\left[\psi^{-}_{0}(x)\right]$

and the Hamiltonian is defined similar to (1) as

	$\displaystyle\hat{H}^{-}$	$\displaystyle=\hat{A}^{\dagger}\hat{A}=-\frac{d^{2}}{dx^{2}}+\hat{V}^{-}(x)-\epsilon$		(13)
	$\displaystyle\hat{A}$	$\displaystyle=\frac{d}{dx}-\frac{d}{dx}\ln\left[\frac{\psi^{-}_{0}(x)}{\mathcal{I}(x)+\lambda}\right]$

It is worth reminding that all these strictly isospectral family of potentials have same partner potential $V^{+}(x)$.

In the limit $\lambda=0$ there is a loss of a bound state and the corresponding potential is called the Pursey potential $\hat{V}^{P}(x)$. An analogous situation occurs in the limit $\lambda=-1$ and the potential is called the Abraham-Moses potential $\hat{V}^{AM}(x)$. The normalized eigenfunctions of $\hat{V}^{P}(x)$ are given by

$$\hat{\psi}^{P}_{n}(x)=\frac{1}{E^{-}_{n+1}}\left(-\frac{d}{dx}-\frac{d}{dx}\ln\left[\frac{\psi^{-}_{0}(x)}{\mathcal{I}(x)}\right]\right)A\psi^{-}_{n+1}(x),\qquad n=0,1,2,\cdots,.$$

(14)

Similarly, the normalized eigenfunctions of $\hat{V}^{AM}(x)$ are given by

$$\hat{\psi}^{AM}_{n}(x)=\frac{1}{E^{-}_{n+1}}\left(-\frac{d}{dx}-\frac{d}{dx}\ln\left[\frac{\psi^{-}_{0}(x)}{\mathcal{I}(x)-1}\right]\right)A\psi^{-}_{n+1}(x),\qquad n=0,1,2,\cdots,.$$

(15)

The energy eigenvalues of the Pursey and the Abraham-Moses Hamiltonians obtained by substituting $\lambda$ equal to $0$ and $-1$ respectively in (13) has the same expressions as that of $H^{+}(x)$, i.e.

$$\hat{E}^{P}_{n}=\hat{E}^{AM}_{n}=E^{+}_{n},\qquad n=0,1,2,3,\cdots$$

(16)

In this section, we briefly review the results obtained in [31, 32] about the REHO potentials. These authors extended the conventional one-dimensional harmonic oscillator potential ($V(x)=x^{2}$) using the idea of SQM and obtained the REHO potentials valid for even co-dimension of $m=0,2,4,....$ and are given by

$$V^{-}_{m}(x)=V(x)-2\left[\frac{\mathcal{H}_{m}^{\prime\prime}}{\mathcal{H}_{m}}-\left(\frac{\mathcal{H}_{m}^{\prime}}{\mathcal{H}_{m}}\right)^{2}+1\right],$$

(17)

where $\mathcal{H}_{m}(x)=(-1)^{m}H_{m}(ix)$ is pseudo-hermite polynomials and factorization energy used in the calculation was $\epsilon=-2m-1$. The normalized ground state eigenfunction $\psi^{-}_{0,m}(x)$ having zero eigenvalue and the excited states eigenfunction$\psi^{-}_{n+1,m}(x)$ for different $m$ are

$$\psi^{-}_{0,m}(x)=\left(\frac{2^{m}m!}{\sqrt{\pi}}\right)^{\frac{1}{2}}\frac{e^{-\frac{x^{2}}{2}}}{\mathcal{H}_{m}(x)},$$

(18)

and

$$\psi^{-}_{n+1,m}(x)=\frac{1}{\sqrt{E^{+}_{n,m}\;2^{n}n!\sqrt{\pi}}}\;\frac{e^{-\frac{x^{2}}{2}}}{\mathcal{H}_{m}(x)}y^{m}_{n+1}(x)\;,\qquad n=0,1,2...,$$

(19)

respectively. Here $y^{m}_{n+1}(x)=\left[\mathcal{H}_{m}(x)H_{n+1}(x)+\mathcal{H}_{m}(x)^{\prime}H_{n}(x)\right]$ and $n=-1,\;0,\;1,\;2,\dots$ is the Exceptional Hermite Polynomial. Note that $y^{m}_{0}(x)=1$. The whole system $\{y^{m}_{n+1}(x)\}$ is the exceptional orthogonal polynomial system, $X_{m}$, of co-dimension $m$ and is orthogonal and complete with respect to the positive-definite measure $\frac{e^{-x^{2}}}{\mathcal{H}_{m}(x)^{2}}$. The Hamiltonian $H^{-}_{m}$ defined similar to (1) has the energy eigenvalues $E^{-}_{n,m}$ given by

$$E^{-}_{n+1,m}=2(n+m+1),\quad n=0,1,2,\cdots,\quad\text{and }E^{-}_{0,m}=0.$$

(20)

It is seen that the difference in energy eigenvalues is $2(m+1)$ units between ground state and first excited state and 2 units between any successive excited states. Therefore energy spectra is equidistant only for $m=0$. Note that $V^{-}_{m}(x)$ is singular at $x=0$ for odd $m$ and therefore $m$ is restricted to positive even integers only.

It is worth noting that the complete set of eigenfunctions for the $m=0$ case, i.e. the one dimensional harmonic oscillator, can be reduced to a single formula containing ground state eigenfunction and excited state eigenfunction given by

$$\Psi_{\nu}(x)=\frac{1}{\sqrt{2^{\nu}\nu!\sqrt{\pi}}}e^{-\frac{x^{2}}{2}}H_{\nu}(x),\qquad\nu=0,\;1,\;2,\cdots\,.$$

The plot of the ground, first and the second excited state eigenfunctions as well as the potential versus position for $m=0,\;m=2\text{ and }m=4$ are given in Fig-1.a ,Fig-1.b and Fig-1.c respectively. It is interesting to note from these figures that as $m$ increases, the eigenfunctions and the potential well becomes sharper. In Table $1$ we have given expressions for the potential $V^{-}_{m}(x)$ as well as the ground and the excited state eigenfunctions in case $m=0,2,4$. Expressions for exceptional Hermite polynomials $y^{m}_{n+1}$ are given in Table $2$ in case $n=0,1,2$ and $m=0,2,4$.

The superpotential $W(x)$ corresponding to the REHO potential is easily obtained from the ground state eigenfunctions $\psi^{-}_{0,m}(x)$ of $V^{-}_{m}(x)$ as

	$\displaystyle W(x)$	$\displaystyle=$	$\displaystyle-\ln[\psi^{-}_{0,m}(x)]^{\prime}$		(21)
		$\displaystyle=$	$\displaystyle x+\frac{\mathcal{H}_{m}(x)^{\prime}}{\mathcal{H}_{m}(x)}.$

The energy eigenvalues of the partner Hamiltonians $H^{\mp}(x)$, defined using (1) and (4), are given by

	$\displaystyle E^{-}_{n+1,m}$	$\displaystyle=2(n+m+1),\quad n=0,1,2,\cdots,\quad\text{with}\quad E^{-}_{0,m}=0.$		(22)
	$\displaystyle E^{+}_{n,m}$	$\displaystyle=2(n+m+1),\quad n=0,1,2,\cdots$		(23)

The One parameter ($\lambda$) family of strictly isospectral potentials corresponding to $V^{-}_{m}(x)$ are easily obtained using (11-12) as

$$\hat{V}^{-}_{m}(\lambda,x)=V^{-}_{m}(x)-2\frac{d^{2}}{dx^{2}}\ln\left[\mathcal{I}_{m}(x)+\lambda\right],$$

(24)

where the integral $\mathcal{I}_{m}(x)$ is calculated using (18) as

$$\mathcal{I}_{m}(x)=\left(\frac{2^{m}m!}{\sqrt{\pi}}\right)\int^{x}_{-\infty}\left[\frac{e^{-\frac{{x^{\prime}}^{2}}{2}}}{\mathcal{H}_{m}(x^{\prime})}\right]^{2}dx^{\prime}.$$

(25)

The expressions for $\mathcal{I}_{m}(x)$ and $\hat{V}^{-}_{m}(\lambda,x)$ for $m$ equal to $0,\;2\text{ and }4$ are given in table-2 and table-3 respectively in terms of the error function $\text{erf}(x)$ defined as

$$\text{erf}(x)=1-\text{erfc}(x),\qquad\text{erfc}(x)=\frac{2}{\sqrt{\pi}}\int^{\infty}_{x}e^{-t^{2}}dt\,.$$

The normalized ground state wave function $\hat{\psi}^{-}_{0,m}(\lambda,x)$ is obtained using (11), (18) and (25) as

$$\hat{\psi}^{-}_{0,m}(\lambda,x)=\sqrt{\lambda(1+\lambda)}\;\frac{\psi^{-}_{0,m}(x)}{\mathcal{I}_{m}(x)+\lambda}\,.$$

(26)

The normalized excited states eigenfunction of $\hat{V}^{-}_{m}(\lambda,x)$, using (12) are given by

	$\displaystyle\hat{\psi}^{-}_{n+1,m}(\lambda,x)$	$\displaystyle=\frac{1}{E^{-}_{n+1,m}}\hat{A}^{\dagger}A\psi^{-}_{n+1,m}(x),\qquad n=0,1,2,\cdots,$		(27)
		$\displaystyle=\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2(n+m+1)2^{n}n!\sqrt{\pi}}}\left(\frac{y^{m}_{n+1}(x)}{\mathcal{H}_{m}(x)}+H_{n}(x)\frac{d}{dx}\ln\left[\mathcal{I}_{m}(x)+\lambda\right]\right)$

and

$$\hat{A}^{\dagger}=-\frac{d}{dx}-\frac{d}{dx}\ln\left[\frac{\psi^{-}_{0,m}(x)}{\mathcal{I}_{m}(x)+\lambda}\right],\qquad A=\frac{d}{dx}-\frac{d}{dx}\ln[\psi^{-}_{0,m}(x)].$$

The expression of $\hat{\psi}^{-}_{n+1,m}(x)$ for $m$ equal to $0,\;2\text{ and }4$ are tabulated in table-4 . As mentioned above, the spectra of $\hat{H}^{-}_{m}$, defined using (13), is strictly isospectral to $H^{-}_{m}$ spectra and is given by Eq. (20).

For $m$ equal to $0,\;2\text{ and }4$, the plots of the strictly isospectral potentials for positive and negative $\lambda$ as a function of $x$ are shown in Fig-(2.a, 3.a, 4.a) and Fig-(2.b, 3.b, 4.b) respectively. Pursey and Abraham-Moses potential plots as a function of $x$ are shown in Fig-(2.c, 3.c, 4.c). The ground-state wavefunction plots as a function of $x$ for various $\lambda$ are shown in Fig-(2.d, 3.d, 4.d). From the figures one observes that the eigen functions and the strictly isospectral potentials become sharper with increasing $m$. In the limit $\lambda$ approaching to $\pm\infty$ the potential $\hat{V}^{-}_{m}(\lambda,\;x)$ approaches to $V^{-}_{m}(x)$. Also notice that the potential starts developing a minimum when $\lambda$ decreases from $\infty$ to zero and the attractive potential well shifts towards $-\infty$ and finally vanishes when $\lambda$ equals zero. There is a loss of bound state and the corresponding potential is called the Pursey potential $V^{P}_{m}(x)$. An analogous situation occurs in the limit $\lambda=-1$ and the potential is called the Abraham-Moses potential $V^{AM}_{m}(x)$.

The Heisenberg Uncertainty relation for position, $x$, and momentum, $p$, is defined as

$$\Delta x\Delta p=\sqrt{\left(<x^{2}>\;-\;<x>^{2}\right)\;\left(<p^{2}>\;-\;<p>^{2}\right)},$$

(30)

where, the angle bracket, $<>$, represents expectation value in a given wavefunction basis.

In this manuscript, we consider the rationally extended one dimensional harmonic oscillator potential associated with exceptional $X_{m}$-Hermite polynomials. Unlike the one dimensional oscillator, for nonzero $m$, while the energy gap between the excited states is $2$ units, the energy gap between the groundstate and first excited state is $m+1$. Using the idea of SQM, we obtained one parameter ($\lambda$) family of strictly isospectral potentials corresponding to the REHO potential as well as the corresponding eigenfunctions. As a special case, as $\lambda$ approaches $0$ or $-1$, we obtained the rationally extended Pursey and the AM potentials respectively as well as their eigenfunctions. Further, we calculated the Heisenberg uncertainty relations $\Delta x\Delta p$ for the REHO and as well as the corresponding strictly isopectral one parameter ($\lambda$) family of potentials. In addition, we also calculated the uncertainity relation for the corresponding Pursey and AM potentials. In the case of the REHO, we showed that the ground state uncertainity increases as $m$ increases. In the case of the strictly isospectral one parameter family of potentials, one finds that the uncertainty relation depends on $m$ as well as $\lambda$. Remarkably, we find that for any $m$, the uncertainty relation is the same for the Pursey and the AM potentials.

There are several open problems. For example, In this paper we have only studied the potentials with even codimension $m$ and obtained the corresponding strictly isospectral one parameter family. Can one similarly obtain the corresponding one parameter family of potentials in case the codimension $m$ is odd. Further, apart from the one dimensional oscillator, are there other rationally extended potentials for which the eigenfunctions can be expressed in terms of exceptional Hermite polynomials? Another obvious question is whether there are exceptional Legendre polynomials and if yes can one discover new potentials whose eigenfunctions can be expressed in terms of exceptional Legendre polynomials? We hope to study some of these problems in coming days.

Acknowledgements
AK is gratefel to Indian National Science Academy (INSA) for awarding INSA Honarary Scientist position at Savitribai Phule Pune University. One of us (RK) is grateful to Ian Marquette for useful comments.

Appendix A. Expectation value of $x$ and $x^{2}$ for one parameter family of REHO
The expectation value of $x$ and $x^{2}$ for various values of positive $\lambda$ are calculated using (26) as:

$$<x>=\begin{cases}\approx-3.0028\|_{m=0}\;\approx-2.2259\|_{m=2}\;\approx-1.8057\|_{m=4}&\text{, when $\lambda=0.00001$}\\ \approx-2.1553\|_{m=0}\;\approx-1.4304\|_{m=2}\;\approx-1.1187\|_{m=4}&\text{, when $\lambda=0.001$}\\ \approx-0.9133\|_{m=0}\;\approx-0.5202\|_{m=2}\;\approx-0.3955\|_{m=4}&\text{, when $\lambda=0.1$}\\ \approx-0.0039\|_{m=0}\;\approx-0.0022\|_{m=2}\;\approx-0.0016\|_{m=4}&\text{, when $\lambda=100$}\\ \approx-0.0004\|_{m=0}\;\approx-0.0002\|_{m=2}\;\approx-0.0002\|_{m=4}&\text{, when $\lambda=1000$}\\ \end{cases}$$

When the values of $\lambda(>0)$ in the above table are replaced by $-|\lambda+1|$ the magnitude of the values of $x$ remain unchanged but the sign reverses.

$$<x^{2}>=\begin{cases}\approx 9.1024\|_{m=0}\;\approx 5.0384\|_{m=2}\;\approx 3.3276\|_{m=4}&\text{, when $\lambda=0.00001$}\\ \approx 4.8071\|_{m=0}\;\approx 2.1621\|_{m=2}\;\approx 1.3309\|_{m=4}&\text{, when $\lambda=0.001$}\\ \approx 1.2338\|_{m=0}\;\approx 0.4198\|_{m=2}\;\approx 0.2446\|_{m=4}&\text{, when $\lambda=0.1$}\\ \approx 0.5000\|_{m=0}\;\approx 0.1556\|_{m=2}\;\approx 0.0896\|_{m=4}&\text{, when $\lambda=100$}\\ \approx 0.5\|_{m=0}\;\approx 0.1556\|_{m=2}\;\approx 0.0896\|_{m=4}&\text{, when $\lambda=1000$}\\ \end{cases}$$

When the values of $\lambda(>0)$ in the above table are replaced by $-|\lambda+1|$ the expectation values of $x^{2}$ remains unchanged. Similarly, the expectation value of $p^{2}$ can be calculated.