A Class of Exactly Solvable Real and Complex $PT$ Symmetric Reflectionless Potentials

Consider a Hamiltonian

$$H^{(1)}(x)=-\frac{d^{2}}{dx^{2}}+V^{(1)}(x),\quad(\hbar=2m=1)$$

(1)

with ground state energy $E_{0}^{(1)}=0$. One can then factorize $H^{(1)}$ in terms of the operators $A$ and $A^{\dagger}$ as

$$H^{(1)}(x)=A^{\dagger}A$$

(2)

with

$$A=\frac{d}{dx}+W(x)\quad\mbox{and}\quad A^{\dagger}=-\frac{d}{dx}+W(x),$$

(3)

where

$$W(x)=-\frac{d}{dx}[\ln\psi^{(1)}_{0}(x)]$$

(4)

is the superpotential, which determines the two partner potentials

$\displaystyle V^{(1)}(x)=W^{2}(x)-W^{\prime}(x)\quad\mbox{and}\quad V^{(2)}(x)=W^{2}(x)+W^{\prime}(x).$

(5)

The eigenvalues and the eigenfunctions of these two potentials (when the SUSY is unbroken) are related by

$$E^{(1)}_{n+1}=E^{(2)}_{n}\qquad E^{(1)}_{0}=0,$$

(6)

and

$$\psi^{(2)}_{n}(x)=\frac{1}{[E^{(2)}_{n}]^{1/2}}A\psi^{(1)}_{n+1}\qquad\psi^{(1)}_{n+1}(x)=\frac{1}{[E^{(2)}_{n}]^{1/2}}A^{\dagger}\psi^{(2)}_{n}$$

(7)

respectively. For the one dimensional case, the transmission $(T^{(1,2)}(k))$ and reflection $(R^{(1,2)}(k))$ amplitudes for the partner potentials $V^{(1,2)}(x)$ are related by

$$R^{(1)}(k)=\bigg{(}\frac{W_{-}+ik}{W_{-}-ik}\bigg{)}R^{(2)}(k)$$

(8)

and

$$T^{(1)}(k)=\bigg{(}\frac{W_{+}-ik^{\prime}}{W_{-}-ik}\bigg{)}T^{(2)}(k)$$

(9)

where

$$k=(E-W^{2}_{-})^{\frac{1}{2}}\quad\mbox{and}\quad k^{\prime}=(E-W^{2}_{+})^{\frac{1}{2}}$$

(10)

with

$$W_{\pm}=W(x\rightarrow\pm\infty).$$

(11)

The one-parameter family of potentials $\hat{V}^{(1)}(\lambda,x)$ which are strictly isospectral to the given potential $V^{(1)}(x)$ are given by

$$\hat{V}^{(1)}(\lambda,x)=V^{(1)}(x)-2\frac{d^{2}}{dx^{2}}\ln(I(x)+\lambda)\,,$$

(12)

where the integral $I(x)$ in term of the normalized ground state wavefunction is given by

$$I(x)=\int^{x}_{-\infty}[\psi^{(1)}_{0}]^{2}(x)\,dx$$

(13)

and $\lambda$ is a constant which is either $>0$ or $<-1$. The corresponding superpotential $\hat{W}(x)$ with the same SUSY partner potential $V^{(2)}(x)$ is given by

$$\hat{W}(x)=W(x)+\frac{d}{dx}\ln[I(x)+\lambda].$$

(14)

The associated normalized ground state wavefunctions to the potential $\hat{V}^{(1)}(\lambda,x)$ is given by

$$\hat{\psi}^{(1)}_{0}(\lambda,x)=\frac{\sqrt{\lambda(1+\lambda)}\psi^{(1)}_{0}(x)}{[I(x)+\lambda]},$$

(15)

while the normalized excited-state ($n=1,2,3...$) eigenfunctions are given by

$$\hat{\psi}^{(1)}_{n+1}(\lambda,x)=\psi^{(1)}_{n+1}(x)+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I^{\prime}(x)}{I(x)+\lambda}\bigg{)}\bigg{(}\frac{d}{dx}+W_{(}x)\bigg{)}\psi^{(1)}_{n+1}(x).$$

(16)

We consider the well known example of real reflectionless potential

$$V^{(1)}(x)=-N(N+1){\rm sech}^{2}(x);\quad-\infty\leq x\leq\infty,$$

(26)

for any positive integer $N>0$. The solutions of the time-independent one-dimensional schrödinger equation corresponding to this potential are well known [32] and given as

$$\psi^{(1)}_{n}(x)=C_{n}^{(N)}{\rm sech}^{N}(x)P^{(-N-\frac{1}{2},-N-\frac{1}{2})}_{n}(i\sinh(x)),$$

(27)

with the energy eigenvalues

$$E^{(1)}_{n}=-(N-n)^{2},\quad n=0,1,2...n_{max}<N$$

(28)

and the normalization constant

$$C_{n}^{(N)}=2^{N}\bigg{[}\frac{n!(N-n)[\Gamma(N-n+\frac{1}{2})]^{2}}{\Gamma(2N-n+1)\pi}\bigg{]}^{1/2}.$$

(29)

Here $P^{(-N-\frac{1}{2},-N-\frac{1}{2})}_{n}(i\sinh(x))$ is the Jacobi polynomial. The corresponding reflection amplitude is zero at all positive energies while the transmission amplitude is given by

$$T^{(1)}(k)=\frac{\Gamma(-N-ik)\Gamma(N-ik+1)}{\Gamma(1-ik)\Gamma(-ik)},$$

(30)

with $k^{2}=E^{(1)}_{n}$ while the transmission probability $|T^{(1)}(k)|^{2}=1$.

The complex $PT$ symmetric Scarf-II potential giving entirely real spectrum is well-known [36, 37] and given by

$$V^{(1)}(x,a,b)=-[b^{2}+a(a+1)]{\rm sech}^{2}(x)+ib(2a+1){\rm sech}(x)\tanh(x);-\infty<x<\infty.$$

(33)

The corresponding bound state energy eigenvalues and the eigenfunctions respectively are

$$E^{(1)}_{n}=-(a-n)^{2},\quad n=0,1,2....n_{max}<a,$$

(34)

and

$$\psi^{(1)}_{n}(x,a,b)=C^{(a,b)}_{n}({\rm sech}x)^{a}\exp(-ib\tan^{-1}(\sinh x))P^{(\alpha,\beta)}_{n}(i\sinh x),$$

(35)

with $\alpha=b-a-\frac{1}{2}$ and $\beta=-b-a-\frac{1}{2}$.

The transmission and the reflection amplitudes of this potential are also well known [37] and are given by

$$T^{(1)}_{scarf}(k,a,b)=\frac{\Gamma(-a-ik)\Gamma(1+a-ik)\Gamma(\frac{1}{2}-b-ik)\Gamma(\frac{1}{2}+b-ik)}{\Gamma(-ik)\Gamma(1+ik)(\Gamma(\frac{1}{2}-ik))^{2}},$$

(36)

and

$$R^{(1)}_{scarf}(k,a,b)=T^{(1)}_{scarf}(k,a,b)\times i\bigg{[}\frac{\cos\pi a\sin\pi b}{\cosh\pi k}+\frac{\sin\pi a\cos\pi b}{\sinh\pi k}\bigg{]}$$

(37)

respectively, where $k^{2}=E^{(1)}_{n}$.

The $PT$ symmetric complex Scarf-II potential $V^{(1)}(x,a,b)$ (given by Eq. (33)) has been extended rationally [19] in terms of classical Jacobi polynomials $P^{(\alpha,\beta)}_{m}(z)$ for any positive integers of $m\geq 0$ given by

	$\displaystyle V^{(1)}_{m,ext}(x,a,b)$	$\displaystyle=$	$\displaystyle V^{(1)}(x,a,b)+2m(2b-m+1)+(2b-m+1)$
		$\displaystyle\times$	$\displaystyle[(-2a-1)+(2b+1)i\sinh x]\bigg{(}\frac{P^{(-\alpha,\beta)}_{m-1}(i\sinh x)}{P^{(-\alpha-1,\beta-1)}_{m}(i\sinh x)}\bigg{)}$
		$\displaystyle-$	$\displaystyle\frac{(2b-m+1)^{2}\cosh^{2}x}{2}\bigg{(}\frac{P^{(-\alpha,\beta)}_{m-1}(i\sinh x)}{P^{(-\alpha-1,\beta-1)}_{m}(i\sinh x)}\bigg{)}^{2}.$

The bound state spectrum of this extended potential is the same (isospectral) as that of the conventional one but the eigenfunctions are different and written in term of exceptional $X_{m}$ Jacobi polynomials $\hat{P}^{(\alpha,\beta)}_{n+m}(i\sinh x)$ as

$$\psi^{(1)}_{ext,n,m}(x,a,b)\propto\frac{{\rm sech}^{a}x\exp[-ib\tan^{-1}(\sinh x)]}{P^{(-\alpha-1,\beta-1)}_{m}(i\sinh x)}\hat{P}^{(\alpha,\beta)}_{n+m}(i\sinh x),$$

(53)

where

	$\displaystyle\hat{P}^{(\alpha,\beta)}_{n+m}(z(x))=(-1)^{m}\bigg{[}\frac{(1+\alpha+\beta+n)}{2(1+\alpha+n)}(z(x)-1)P^{(-\alpha-1,\beta-1)}_{m}(g)P^{(\alpha+2,\beta)}_{n-1}(z(x))$
	$\displaystyle+\frac{(1+\alpha-m)}{(\alpha+1+n)}P^{(-2-\alpha,\beta)}_{m}(z(x))P^{(\alpha+1,\beta-1)}_{n}(z(x))\bigg{]};\quad n,m\geq 0.$		(54)

is the exceptional Jacobi polynomial. Similar to the Scarf-II potential, this extended potential is also SI under the translation of the parameters $a\rightarrow(a-1)$. The transmission and reflection amplitudes for this potential are known [39] and are given by

$$T^{(1)}_{ext,scarf}(k,m,a,b)=T^{(1)}_{scarf}(k,a,b)\zeta(m,a,b)\,,$$

(55)

and

$$R^{(1)}_{ext,scarf}(k,m,a,b)=R^{(1)}_{scarf}(k,a,b)\zeta(m,a,b),$$

(56)

where $\zeta(m,a,b)=\left(\frac{[b^{2}-(ik-\frac{1}{2})^{2}]+(b-ik+\frac{1}{2})(1-m)}{[b^{2}-(ik+\frac{1}{2})^{2}]+(b+ik+\frac{1}{2})(1-m)}\right)$.

Remarkably, it turns out that unlike the conventional Scarf-II potential (33), the corresponding rationally extended Scarf-II potential (3.2) is not invariant under the parametric transformation $b\longleftrightarrow a+\frac{1}{2}$, rather it generates another extended potential [40] given by

$${V}^{(1,p)}_{m,ext}(x,a,b)=V^{(1)}_{m,ext}(x,b\leftrightarrow a+\frac{1}{2}).$$

(57)

The energy eigenvalues of this potential are isospectral to that of the conventional potential obtained after the transformation $b\longleftrightarrow a+\frac{1}{2}$ given by Eq. (39) and the eigenfunction is

$$\psi^{(1,p)}_{ext,n,m}(x,a,b)=\psi^{(1)}_{ext,n,m}(x,a\rightarrow b-1/2,b\rightarrow a+1/2).$$

(58)

This potential (57) is also SI under the translation of parameter $b\longrightarrow b-1$. Since the potentials obtained after parametric transformations are different, hence the scattering amplitudes corresponding to these potentials are also different which are given by

	$\displaystyle T^{(1,p)}_{ext,scarf}(k,m,a,b)$	$\displaystyle=$	$\displaystyle T^{(1)}_{ext,scarf}(k,m,a\rightarrow b-\frac{1}{2},b\rightarrow a+\frac{1}{2})$
	$\displaystyle R^{(1,p)}_{ext,scarf}(k,m,a,b)$	$\displaystyle=$	$\displaystyle R^{(1)}_{ext,scarf}(k,m,a\rightarrow b-\frac{1}{2},b\rightarrow a+\frac{1}{2}).$		(59)

Thus, it turns out that the extended potentials do not respect the parametric symmetry. As a result for a given value of $[a,b]$ (both integers or half integers) one has in fact two different sets of rationally extended potentials for a given $m$ which are reflectionless. The only exceptions to these are the cases when either $[a=(2N-1)/2,b=1/2]$ or $[a=0,b=N]$ where only one rational partner exists. Thus in all, one has $2(2N-1)m+2N$ number of complex PT-invariant reflectionless potentials, with the $2N$ potentials being the nonrational (or $m=0$) ones.

In this case, we fix the value of parameter $N=3$ which gives the potential (26)

$$V^{(1)}(x)=-12{\rm sech}^{2}(x),$$

(60)

with three bound states with the corresponding binding energies being $E^{(1)}_{0}=-9,E^{(1)}_{1}=-4$ and $E^{(1)}_{2}=-1$. The normalized ground state is

$$\psi^{(1)}_{0}(x)=\frac{\sqrt{15}}{4}{\rm sech}^{3}x\,.$$

(61)

Thus the partner potential with its normalized ground state eigenfunction are

$$V^{(2)}(x)=-6{\rm sech}^{2}(x)\,,~{}~{}\psi_{0}^{(2)}(x)=\frac{\sqrt{3}}{2}{\rm sech}^{2}(x)\,.$$

(62)

It is straight forward to calculate the integral (13) using the potential (60) and one obtains

$$I(x)=\frac{1}{16}\bigg{(}8+[8+4{\rm sech}^{2}(x)+3{\rm sech}^{4}(x)]\tanh(x)\bigg{)}\,,$$

(63)

which gives one continuous parameter family of real reflectionless potentials

	$\displaystyle\hat{V}^{(1)}(x,\lambda)$	$\displaystyle=$	$\displaystyle 6{\rm sech}^{2}(x)\bigg{[}-2+\frac{1}{(8+16\lambda+(8+4{\rm sech}^{2}(x)+3{\rm sech}^{4}(x))\tanh(x))^{2}}$		(64)
		$\displaystyle\times$	$\displaystyle[15{\rm sech}^{4}(x)\{(1+3\cosh(2x)+\cosh(4x)){\rm sech}^{6}(x)$
		$\displaystyle+$	$\displaystyle 16\tanh(x)(1+2\lambda+\tanh(x))\}]\bigg{]}.$

The normalized ground state eigenfunction for this potential is obtained as

$\displaystyle\hat{\psi}^{(1)}_{0}(x,\lambda)=\frac{4[15\lambda(\lambda+1)]^{\frac{1}{2}}{\rm sech}^{3}(x)}{(8+16\lambda+(8+4{\rm sech}^{2}(x)+3{\rm sech}^{4}(x))\tanh(x))}.$

(65)

In the limit of $\lambda\rightarrow 0$ and $-1$, we get the reflectionless Pursey and the AM potentials respectively with two bound states. The expressions for these two potentials are given by

$$V^{[P/AM]}(x)=-\frac{24{\rm sech}^{2}(x)[25\cosh(2x)+13\cosh(4x)\mp 3(\mp 5+5\sinh(2x)+4\sinh(4x))]}{(5+11\cosh(2x)\mp 9\sinh^{2}(2x))},$$

(66)

where upper sign corresponds to the Pursey and the lower one for the AM potential. The plots of $\hat{V}^{(1)}(x,\lambda)$ for positive and negative $\lambda$ are shown in Fig. 1(a) and 1(b) respectively. The AM ($V^{[AM]}(x)$), the Pursey ($V^{[P]}(x)$) and the partner potential ($V^{(2)}(x)$) are shown in Fig. 1(c). The normalized ground state wavefunctions for some positive values of $\lambda$ are also shown in Fig. 1(d).

Fig.1(a) One-parameter family of potential $\hat{V}^{(1)}(x,\lambda)$ for positive $\lambda=0.1,0.01,0.0001,0$ and $\infty$. The Pursey potential is shown for $\lambda=0$.

Fig.1(b) One-parameter family of potential $\hat{V}^{(1)}(x,\lambda)$ for negative $\lambda=-1.1,-1.01,-1.0001,-1$ and $-\infty$. The AM potential is shown for $\lambda=-1$.
 
Fig.1(c) The Pursey potential ${V}^{[P]}(x)$, the AM potential ${V}^{[AM]}(x)$ and the partner potential ${V}^{(2)}(x)$.

Fig.1(d) Normalized ground-state wavefunctions $\hat{\psi}^{(1)}_{0}(x,\lambda)$ for some potentials (with positive $\lambda=0.1,0.01,0.001$ and $\infty$.)