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

Suman Banerjeea111e-mail address<EMAIL_ADDRESS>(S.B), Rajesh
Kumar Yadava222e-mail address<EMAIL_ADDRESS>(R.K.Y), Avinash
Khareb333e-mail address<EMAIL_ADDRESS>(A.K) and
Bhabani Prasad Mandalc444e-mail address:bhabani.mandal@gmail.com(BPM)

###### Abstract

We consider the question of the number of exactly solvable complex but PT-
invariant reflectionless potentials with $N$ bound states. By carefully
considering the $X_{m}$ rationally extended reflectionless potentials, we
argue that the total number of exactly solvable complex PT-invariant
reflectionless potentials are $2[(2N-1)m+N]$.

$~{}^{a}$Department of Physics, Sido Kanhu Murmu University, Dumka-814110,
India.
$~{}^{b}$Department of Physics, Savitribai Phule Pune University, Pune-411007,
India.
$~{}^{c}$Department of Physics, Banaras Hindu University, Varanasi-221005,
India.

PACS numbers: 11.30.Pb, 03.65.Ge, 03.65.Nk,

##  1 Introduction

The reflectionless potentials, also known as transparent potentials or black
potentials, are of great importance in physics and engineering. In view of the
numerous applications of the reflectionless potentials, it is very important
to search for new reflectionless potentials. While it is well known that there
are $N$ continuous parameter families of exactly solvable real reflectionless
potentials, to the best of our knowledge, the question of the complex PT-
invariant exactly solvable reflectionless potentials has not been addresses in
the literature.

In the last two decades, after the discovery of the $PT$ (combined parity
($P$) and time reversal ($T$)) symmetric non-hermitian systems [1, 2, 3], it
has been shown that there are non-hermitian complex PT-invariant potentials
which are also reflectionless [4, 5, 6, 7, 8]. After the discovery of the
$X_{m}$ exceptional orthogonal polynomials (EOPs) [9, 10, 11], a group of new
(also known as rationally extended) hermitian as well as $PT$ symmetric non-
hermitian potentials have been constructed [12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] with their solutions in terms
of these EOPs. It is then natural to enquire how many distinct reflectionless
potentials with $N$ bound states can be constructed using both PT symmetry and
$X_{m}$ EOPs. This is the task that we have undertaken in this paper.

Just to set the notation, we first consider a real exactly solvable
reflectionless potential with $N$ bound states and using the method of
supersymmetric quantum mechanics [32, 33, 34, 35] explicitly obtain one
continuous ($\lambda$) parameter family of reflectionless potentials including
the corresponding reflectionless Pursey and Abraham Moses (AM) potentials with
$N-1$ bound states. This can be generalized and one can obtain $N$ continuous
parameter families of real reflectionless potentials with $N$ bound states.

We then consider the case of the non-hermitian $PT$ symmetric Scarf-II
reflectionless potentials with $N$ bound states and discuss the role of the
parametric symmetry. Finally, we consider the rationally extended complex PT-
symmetric scarf-II potential whose eigenfunctions are written in the form of
$X_{m}$ Jacobi EOPs with $m=1,2,3..$, and show that these extended potentials
are also reflectionless and combining all these factors we have in all
$2[(2N-1)m+N]$ number of complex PT-symmetric reflectionless potentials with
$N$ bound states.

The organization of this manuscript is as follows: In Sec. $2$, we briefly
discuss the formalism of supersymmetric quantum mechanics relevant to this
paper [32] and explicitly obtain one continuous parameter family of real
reflectionless potentials with $N$ bound states. In Sec. $3$, we consider the
complex PT-invariant Scarf-II potential with $N$ bound states and obtain
conditions underwhich it is reflectionless. We also discuss the role of
parametric symmetry in counting the number of reflectionless PT-symmetric
complex potentials. We then construct the corresponding $X_{m}$ family of
complex PT-invariant reflectionless potentials with $N$ bound states and argue
that the total number of complex PT-invariant reflectionless potentials is
$2[(2N-1)m+N]$. In Sec. $4$, we consider the case of $N=3$ explicitly. First
we give explicit expression as well as suitable plots for the one parameter
family of real reflectionless potentials and the corresponding eigenfunctions
with three bound states. We then give explicit expression for the real and
imaginary parts of the complex PT-invariant potentials and their
eigenfunctions in the case of three bound states. Finally, in Sec. $5$, we
summarize our results.

##  2 Formalism

In this section, we set the basic notations of supersymmetric quantum
mechanics (SQM) as relevant to the present discussion. We then discuss the
case of the real reflectionless potential with $N$ bound states and obtain one
continuous parameter family of strictly isospectral reflectionless potentials
with $N$ bound states. This can be generalized [32] and one can obtain $N$
continuous parameter family of reflectioless real potentials with $N$ bound
states.

###  2.1 Basic Results of Supersymmetric Quantum Mechanics

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)

####  2.1.1 Pursey potential

The superpotential for this case is defined by putting $\lambda=0$ in Eq. (14)

$W^{[P]}(x)=W(x)+\frac{d}{dx}\ln I(x).$ (17)

and the potential (12) becomes

$V^{[P]}(x)=\hat{V}^{(1)}(\lambda=0,x)=V^{(1)}(x)-2\frac{d^{2}}{dx^{2}}\ln
I(x)\,,$ (18)

while the corresponding eigenvalues are

$E^{[P]}_{n}=E^{(2)}_{n}=E^{(1)}_{n+1}\,,~{}~{}n=0,1,2...\,.$ (19)

The reflection and transmission amplitudes for this case are

$R^{[P]}(k)=\bigg{(}\frac{W_{-}-ik}{W_{-}+ik}\bigg{)}^{2}R^{(1)}(k)$ (20)
$T^{[P]}(k)=-\bigg{(}\frac{W_{-}-ik}{W_{-}+ik}\bigg{)}T^{(1)}(k)\,.$ (21)

####  2.1.2 Abraham-Moses potential

In this case the superpotential and the potential ($\lambda=-1$) are given by

$W^{[AM]}(x)=W(x)+\frac{d}{dx}\ln(I(x)-1),$ (22)

and

$V^{[AM]}(x)=\hat{V}^{(1)}(\lambda=-1,x)=V^{(1)}(x)-2\frac{d^{2}}{dx^{2}}\ln(I(x)-1).$
(23)

The eigenvalues are identical to the Pursey potential and are given by Eq.
(19). The reflection and transmission amplitudes for this case are

$R^{[AM]}(k)=R^{(1)}(k)$ (24)
$T^{[AM]}(k)=-\bigg{(}\frac{W_{+}+ik^{\prime}}{W_{+}-ik^{\prime}}\bigg{)}T^{(1)}(k)\,.$
(25)

###  2.2 Real potentials with $N$-bound states

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
$|T^{(1)}(k)|^{2}=1$.

####  2.2.1 One-parameter family of reflectionless potentials

It is straight forward to obtain the one continuous parameter family of
reflectionless potentials by using Eqs. (12), (13) and (26) and we obtain

$\hat{V}^{(1)}(\lambda,x)=-N(N+1){\rm
sech}^{2}(x)-2\frac{d^{2}}{dx^{2}}\ln(I(x)+\lambda)\,,$ (31)

where

$I(x)=[C_{0}^{N}]^{2}\int_{-\infty}^{x}{\rm sech}^{2N}(y)\,dy\,,$ (32)

and $\lambda>0$ or $\lambda<-1$. Further, it is straight forward to obtain the
corresponding reflectionless Pursey or AM reflectionless potentials with $N-1$
bound states using Eqs. (18) and (23).

This procedure can be iterated $N$ times to find $N$ continuous parameter
family of strictly isospectral reflectionless potentials with $N$ bound states
[32].

##  3 $PT$ Symmetric Complex Reflectionless Potentials

Apart from the $N$ continuous parameter families of real reflectionless
potentials with $N$ bound states, it turns out that there are a vast number of
complex $PT$ symmetric reflectionless potentials with $N$ bound states which
we discuss in this section by starting from the well known complex PT-
invariant Scarf-II potential.

###  3.1 $PT$ symmetric complex Scarf-II potential

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

####  3.1.1 Parametric Symmetry

This potential (33) is invariant under the parametric transformation
$b\leftrightarrow a+\frac{1}{2}$, however the corresponding eigenvalues and
eigenfunctions are different [38] i.e.,

$V^{(1,p)}(x,a,b)=V^{(1)}(x,a\rightarrow b-\frac{1}{2},b\rightarrow
a+\frac{1}{2})=V^{(1)}(x,a,b)$ (38)

but

$E^{(1,p)}_{n}=-(b-n-\frac{1}{2})^{2};\qquad
n=0,1,2....n_{\mbox{max}}<b-\frac{1}{2},$ (39)

and

$\psi^{(1,p)}_{n}(x,a,b)=\psi^{(1)}_{n}(x,a\rightarrow
b-\frac{1}{2},b\rightarrow a+\frac{1}{2}).$ (40)

Here $p$ denotes the quantities obtained after parametric transformation.
However, it is easy to check that the corresponding reflection and
transmission amplitudes as given by Eqs. (36) and (37) are invariant under the
parametric transformation $b\leftrightarrow a+\frac{1}{2}$.

Since the potential (33) has two different sets of eigenvalues and
eigenfunctions $\psi^{(1)}_{n}(x,a,b)$ and $\psi^{(1,p)}_{n}(x,a,b)$, hence
there are two different superpotentials corresponding to the same potential
(33) and are given by

$W(x,a,b)=a\tanh(x)+ib{\rm sech}(x)\,,$ (41)

and

$W^{(p)}(x,a,b)=(b-\frac{1}{2})\tanh(x)+i(a+\frac{1}{2}){\rm sech}(x)\,.$ (42)

This in turn gives two different partner potentials

$\displaystyle V^{(2)}(x,a,b)$ $\displaystyle=$
$\displaystyle(W(x,a,b))^{2}+W(x,a,b)^{\prime}$ (43) $\displaystyle=$
$\displaystyle-(b^{2}+a(a-1)){\rm sech}^{2}(x)+ib(2a-1){\rm sech}(x)\tanh(x)$

and

$\displaystyle V^{(2,p)}(x,a,b)$ $\displaystyle=$
$\displaystyle(W^{(p)}(x,a,b))^{2}+W^{(p)}(x,a,b)^{\prime}$ $\displaystyle=$
$\displaystyle-((b-1)^{2}+a(a+1)){\rm sech}^{2}(x)+i(b-1)(2a+1){\rm
sech}(x)\tanh(x)$

respectively. The first partner potential (43) is SI under translation of
parameter $a\rightarrow a-1$, whereas the second one (3.1.1) is SI under
translation of another parameter $b\rightarrow b-1$. The reflection and
transmission amplitudes of these two partner potentials are related to those
of the potential (33) by

$\displaystyle{R}^{(2)}_{scarf}(k,a,b)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{W_{-}-ik}{W_{-}+ik}\bigg{)}R^{(1)}_{scarf}(k,a,b)$
(45) $\displaystyle=$
$\displaystyle\bigg{(}\frac{a+ik}{a-ik}\bigg{)}R^{(1)}_{scarf}(k,a,b)\,,$
${T}^{(2)}_{scarf}(k,a,b)=-\bigg{(}\frac{a+ik}{a-ik}\bigg{)}{T}^{(1)}_{scarf}(k,a,b)\,.$
(46)

In the parametric case, the reflection and transmission amplitudes are
invariant i.e.,

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

and for the corresponding partner potentials these are related as

$\displaystyle{R}^{(2,p)}_{scarf}(k,a,b)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{W^{(p)}_{-}-ik}{W^{(p)}_{-}+ik}\bigg{)}R^{(1,p)}(k,a,b)$
(48) $\displaystyle=$
$\displaystyle\bigg{(}\frac{b-\frac{1}{2}+ik}{b-\frac{1}{2}-ik}\bigg{)}R^{(1,p)}_{scarf}(k,a,b)\,,$
${T}^{(2,p)}_{scarf}(k,a,b)=-\bigg{(}\frac{b-\frac{1}{2}+ik}{b-\frac{1}{2}-ik}\bigg{)}{T}^{(1,p)}_{scarf}(k,a,b)\,$
(49)

respectively.

####  3.1.2 Conditions for Reflectionless potentials

From the Eq. (37), it follows that all the potential $V^{(1)}(x,a,b)$ and
hence $V^{(2)}(x,a,b)$ and $V^{(2,p)}(x,a,b)$ are reflectionless when the
potential parameters $a$ and $b$ are either both integers or both half
integers. Remarkably, using parametric symmetry it turns out that there are in
fact $2N$ distinct complex $PT$-invariant reflectionless potentials all of
which hold $N$ bound states. Out of these $N$ complex PT-invariant
reflectionless potentials have half-integral values of $a$ and $b$ while the
remaining $N$ complex PT-invariant potentials have integral values of $a$ and
$b$ which we now list one by one.

Case (A): If $a$ and $b$ both half integers

$[a,b]=[(2N-1)/2,1/2],[(2N-3)/2,3/2],...,[3/2,(2N-3)/2],[1/2,(2N-1)/2]\,.$
(50)

On using the fact that the eigenvalues of the complex Scarf-II potential are
given by Eq. (34) while those of the corresponding parametric case are given
by Eq. (39), one can immediately figure out about how many eigenvalues are
coming from the normal scarf-II and how many from the parametric case. For
example, while in the case $[a=(2N-1)/2,b=1/2]$, all the $N$ eigenvalues are
from normal scarf-II, in the case $[a=(2N-3)/2,b=3/2]$, while $N-1$
eigenvalues are coming from normal Scarf-II, one eigenvalue is coming from the
parametric case.

Case (B): If $a$ and $b$ both integers

$[a,b]=[N-1,1],[N-2,2],...,[1,N-1],[0,N]\,.$ (51)

In the case $[a=0,b=N]$, while all the $N$ eigenvalues are from the parametric
case, in the case of $[a=1,b=N-1]$ we have $1$ eigenvalue from the normal
Scarf-II while $N-1$ eigenvalues are coming from the parametric case.

###  3.2 Rationally Extended $PT$ Symmetric Complex Potential

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.

##  4 Illustration For Three Bound States ($N=3$)

In this section, as an illustration, we consider all reflectionless potentials
(both real and complex PT-invariant ones) with three bound states and show the
behavior of these potentials and their corresponding normalized ground state
eigenfunctions through graphical representation.

###  4.1 Real reflectionless potential

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 $\infty$.)

###  4.2 PT symmetric complex Scarf-II potentials

####  4.2.1  The conventional potential

In this case, for three bound states ($N=3$), we have six different possible
combinations of $[a,b]$ given by

$[a,b]=[5/2,1/2],[3/2,3/2],[1/2,5/2],[2,1],[1,2],[0,3]\,,$ (67)

and hence we have six reflectionless potentials. If we plot the potentials for
these combinations of $a$ and $b$, while the potentials $[5/2,1/2]$,
$[3/2,3/2]$ and $[1/2,5/2]$ are same as that of $[0,3]$, $[1,2]$ and $[2,1]$
respectively as can be seen from the plots, the corresponding eigenfunctions
are different for half-integer and integer combinations of both $a$ and $b$.
As mentioned in Eqs. (50) and (51), the integer combination $[0,3]$ is not
acceptable for the potential $V^{(1)}(x,a,b)$, however this is acceptable for
the parametric case $V^{(1,p)}(x,a,b)$. Similarly, the first combination
$[5/2,1/2]$ is well acceptable for the first potential, but not for the
parametric case. The plots of these potentials (real and imaginary parts) are
shown in Fig. $2$. The corresponding eigenfunctions with their parametric
forms are also shown in Fig. $3$ and $4$ receptively. We also compare the
eigenfunctions of conventional $PT$ symmetric potentials with their parametric
counterparts graphically (shown in Fig. $5$).

(a) $a=\frac{5}{2},b=\frac{1}{2}$

(b) $a=\frac{3}{2},b=\frac{3}{2}$

(c) $a=\frac{1}{2},b=\frac{5}{2}$

Fig.2: (a)-(c) Real and imaginary parts of the conventional $PT$ symmetric
Scarf-II potential ($V^{(1)}(x,a,b)$) vs $x$ for different $a$ and $b$.

(a) $a=\frac{5}{2},b=\frac{1}{2}$

(b) $a=\frac{3}{2},b=\frac{3}{2}$

(c) $a=\frac{1}{2},b=\frac{5}{2}$

Fig.3 (i): (a)-(c) Real and imaginary parts of the normalized ground state
eigenfunctions $\psi^{(1)}_{0}(x,a,b)$ vs $x$ for half-integer values of $a$
and $b$.

(a) $a=1,b=2$

(b) $a=2,b=1$

Fig.3 (ii): (a)-(b) Real and imaginary parts of the normalized ground state
eigenfunctions for $\psi^{(1)}_{0}(x,a,b)$ vs $x$ for integer values of $a$
and $b$.

(a) $a=\frac{3}{2},b=\frac{3}{2}$

(b) $a=\frac{1}{2},b=\frac{5}{2}$

Fig.4 (i): (a)-(b) Real and imaginary parts of the normalized ground state
eigenfunctions (for half-integer values of $a$ and $b$) for the parametric
case $\psi^{(1,p)}_{0}(x,a,b)$.

(a) $a=1,b=2$

(b) $a=2,b=1$

(b) $a=0,b=3$

Fig.4 (ii): (a)-(b) Real and imaginary parts of the normalized ground state
eigenfunctions (for integer values of $a$ and $b$) for the parametric case
$\psi^{(1,p)}_{0}(x,a,b)$.

(a)

(b)

Fig.5 (i): (a)-(b) Comparison of real and imaginary parts of the normalized
ground state eigenfunctions for the conventional $\psi^{(1)}_{0}(x,a,b)$ and
parametric cases $\psi^{(1,p)}_{0}(x,a,b)$ for $a=\frac{1}{2},b=\frac{5}{2}$.

(a)

(b)

Fig.5 (ii): (a)-(b) Comparison of real and imaginary parts of the normalized
ground state eigenfunctions for the conventional $\psi^{(1)}_{0}(x,a,b)$ and
parametric cases $\psi^{(1,p)}_{0}(x,a,b)$ for $a=2,b=1$.

####  4.2.2 Rationally extended $PT$ symmetric Scarf-II potential

In this case, unlike the conventional $PT$ symmetric Scarf-II potential, the
extended potentials as well as the corresponding eigenfunctions are completely
different under parametric transformations. For $m=1$, the expression of
potential and the normalizable eigenfunction are given as

$V^{(1)}_{1,ext}(x,a,b)=V^{(1)}(x,a,b)+\frac{(-2(2a+1))}{(-2ib\sinh(x)+2a+1)}+\frac{(2((2a+1)^{2}-4b^{2}))}{(-2ib\sinh(x)+2a+1)^{2}}$
(68)

and

$\psi^{(1)}_{ext,n,m}(x,a,b)\propto\frac{{\rm
sech}^{a}x\exp[-ib\tan^{-1}(\sinh
x)]}{(-2ib\sinh(x)+2a+1)}\hat{P}^{(\alpha,\beta)}_{n+1}(i\sinh x),$ (69)

where $\hat{P}^{(\alpha,\beta)}_{n+1}(i\sinh x)$ is the $X_{1}$ exceptional
Jacobi polynomial.

We consider the same sets of parameters $[a,b]$ (half-integers as well as
integers) as discussed in the above conventional case and show the behaviors
of potentials $V^{(1)}_{1,ext}(x,a,b)$, $V^{(1,p)}_{1,ext}(x,a,b)$ and the
corresponding eigenfunctions $\psi^{(1)}_{1,ext}(x,a,b)$,
$\psi^{(1,p)}_{1,ext}(x,a,b)$ respectively in Figs. $6$, $7$ and $8$.

(a) $a=\frac{1}{2},b=\frac{5}{2}$

(b) $a=\frac{3}{2},b=\frac{3}{2}$

(c) $a=\frac{5}{2},b=\frac{1}{2}$

Fig.6 (i): (a)-(c) Real and imaginary parts of the RE $PT$ symmetric complex
Scarf-II potentials and corresponding eigenfunctions for half-integer values
of $a$ and $b$.

(a) $a=2,b=1$

(b) $a=1,b=2$

Fig.6 (ii): (a)-(c) Real and imaginary parts of the RE $PT$ symmetric complex
Scarf-II potentials and their corresponding eigenfunctions for integer values
of $a$ and $b$.

(a) $a=\frac{3}{2},b=\frac{3}{2}$

(b) $a=\frac{1}{2},b=\frac{5}{2}$

Fig.7 (i): (a)-(b) Real and imaginary parts of the RE $PT$ symmetric complex
Scarf-II potentials and their corresponding eigenfunctions obtained after
parametric transformation for half-integer values of $a$ and $b$.

(a) $a=2,b=1$

(b) $a=1,b=2$

(c) $a=0,b=3$

Fig.7 (ii): (a)-(b) Real and imaginary parts of the RE $PT$ symmetric complex
Scarf-II potentials and their corresponding eigenfunctions obtained after
parametric transformation for integer values of $a$ and $b$.

(a) $a=\frac{1}{2},b=\frac{5}{2}$

(b) $a=\frac{1}{2},b=\frac{5}{2}$

Fig.8 (i): (a)-(b) Comparison between real and imaginary parts of RE $PT$
symmetric complex Scarf-II potential and their corresponding eigenfunctions
obtained after parametric transformation for half integer combination of $a$
and $b$.

(a) $a=2,b=1$

(b) $a=2,b=1$

Fig.8 (ii): (a)-(b) Comparison between real and imaginary parts of RE $PT$
symmetric complex Scarf-II potential and their corresponding eigenfunctions
obtained after parametric transformation for integer combination of $a$ and
$b$.

##  5 Conclusions

In this work, we have made an attempt to obtain all possible exactly solvable
complex PT-invariant reflectionless potentials. As a simple exercise, we first
started with a well-known reflectionless real potential with $N$ bound states
and generated one continuous parameter ($\lambda$) family (which can be easily
generalized to $N$-parameter family) of strictly isospectral reflectionless
potentials. As a special case we have also obtained expressions for the
corresponding reflectionless Pursey and the AM potentials corresponding to
$\lambda=0$ and $-1$ respectively and with $N-1$ bound states.

In the $PT$ symmetric case, we started with the well known complex PT-
invariant Scarf-II potential and showed that it has novel parametric symmetry.
We then showed that there are $N$ number of reflectionless potentials when
both $a$ and $b$ are either integers or half-integers, thereby obtaining $2N$
number of complex PT-invariant reflectionless potentials in total. Further, we
considered the rationally extended $PT$ symmetric reflectionless scarf-II
potential, whose solutions are in terms of $X_{m}$-Jacobi EOPs and shown that
unlike the usual one, this extended potential is not invariant under the
parametric symmetry but instead generates another set of reflectionless
potentials whose solutions are also in terms of $X_{m}$-EOPs. By combining all
these factors we then showed that there are $2[(2N-1)m+N]$ number of complex
PT-invariant reflectionless exactly solvable potentials.

This paper raises few questions. Some of these are, have we really exhausted
the number of complex PT-invariant reflectionless exactly solvable potentials
or are there are still more? While we believe that the answer to the question
is no, one can never be sure. Secondly, since reflectionless potentials have
found wide applications, it would be interesting if one of these complex
reflectionless potential finds some application.

Acknowledgments
AK is grateful to Indian National Science Academy (INSA) for awarding INSA
Honorary Scientist position at Savitribai Phule Pune University. BPM
acknowledges the research grant for faculty under IoE scheme (Number 6031) of
Banaras Hindu University Varanasi.

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#  Edge Intelligence for Autonomous Driving in 6G Wireless System: Design
Challenges and Solutions

Bo Yang, , Xuelin Cao, Kai Xiong, Chau Yuen, , Yong Liang Guan, , Supeng Leng,
, Lijun Qian, , and Zhu Han

###### Abstract

In a level-5 autonomous driving system, the autonomous driving vehicles (AVs)
are expected to sense the surroundings via analyzing a large amount of data
captured by a variety of onboard sensors in near-real-time. As a result,
enormous computing costs will be introduced to the AVs for processing the
tasks with the deployed machine learning (ML) model, while the inference
accuracy may not be guaranteed. In this context, the advent of edge
intelligence (EI) and sixth-generation (6G) wireless networking are expected
to pave the way to more reliable and safer autonomous driving by providing
multi-access edge computing (MEC) together with ML to AVs in close proximity.
To realize this goal, we propose a two-tier EI-empowered autonomous driving
framework. In the autonomous-vehicles tier, the autonomous vehicles are
deployed with the shallow layers by splitting the trained deep neural network
model. In the edge-intelligence tier, an edge server is implemented with the
remaining layers (also deep layers) and an appropriately trained multi-task
learning (MTL) model. In particular, obtaining the optimal offloading strategy
(including the binary offloading decision and the computational resources
allocation) can be formulated as a mixed-integer nonlinear programming (MINLP)
problem, which is solved via MTL in near-real-time with high accuracy. On
another note, an edge-vehicle joint inference is proposed through neural
network segmentation to achieve efficient online inference with data privacy-
preserving and less communication delay. Experiments demonstrate the
effectiveness of the proposed framework, and open research topics are finally
listed.

##  I Introduction

Different from the conventional cellular networks, the next-generation
wireless networks, with various labels such as beyond fifth-generation (B5G)
or sixth-generation (6G), are expected to provide a connection for vehicular
networks with low latency but high reliability ubiquitously via artificial
intelligence (AI) [1]. To prevent accidents caused by human driving errors and
to carry out emissions reduction, automated driving systems have been
developed by tightly integrating with promising technologies, such as surround
sensing, object detection and tracking, control decision-making, and wireless
offloading via multi-access edge computing (MEC) [2, 3]. During the past
decades, machine learning (ML), especially deep learning (DL), has become a
promising technology due to its superiority in extracting inherent features
from large-scale and high-dimensional datasets, which are collected from
embedded sensors such as camera sensors, LiDAR sensors, radar sensors [4]. By
feeding the inferred results to the electronic control unit (ECU) of a
vehicle, the AVs can dynamically adjust the vehicle’s velocity, brake, and
steering according to the surrounding conditions to achieve autonomous or
cooperative adaptive cruise control (CACC). Nevertheless, the inferring
accuracy usually depends on the input data and capability of the deployed DL
model. For example, the object is difficult to be identified via the
convolutional neural network (CNN) model if the input images data is blurry or
if the CNN model has not been sufficiently trained [5].

In recent years, Internet-of-Vehicle (IoV) networks undergo a paradigm shift
from connected vehicles to connected intelligence. The future of vehicular
communications may be a mashup of dedicated short-range communications (DSRC)
and cellular vehicular-to-everything (C-V2X, also LTE Rel.16+). Today the
fight among vehicular communication standards is between “802.11p/DSRC” and
“C-V2X” [6]. 802.11p has the advantage that 5.9 GHz has been approved by the
Department of Transportation and the European Commission, and the technology
is ready for deployment for basic short-range communications. For “Driver-less
V2X”, the situation is different. Due to the massive data anticipated in MEC,
802.11p will likely not be sustainable without significant amendments to the
current standard. In this context, the innovative aspects anticipated via
C-V2X within B5G/6G will play a key role.

As a promising application, the thriving autonomous driving has led to an
upsurge of MEC and AI, which drives the prosperity of edge intelligence (EI)
for greatly facilitating daily life [7, 8, 9]. In this context, EI enables
autonomous driving vehicles (AVs) to sense and react to the surroundings
accurately by offloading the collected data to the powerful edge server co-
located with the base station (BS), as highlighted in Fig. 1. By deploying EI-
based computing service at the edge server, inference computing of AVs can be
achieved with higher accuracy and lower latency. However, many inherent issues
associated with communication and computing under limited bandwidth and
computation resources, data privacy and security, present stumbling blocks
toward the envisioned goal of autonomous driving in 6G era. Motivated by the
above-mentioned appealing characteristics, a flurry of research activities
combining 6G with EI in the autonomous driving system designs has been
sparked.

Figure 1: An illustrative scenario of autonomous driving system.

In this vein, this article aims to explore the EI challenges in autonomous
driving, investigate recent contributions and advances, and then propose
solutions. The major contributions and organization of this article are
summarized as follows. We overview the efforts that have been made on edge
intelligence in vehicular networks and pose the design challenges in Section
II. In Section III, we present the developed two-tier EI-empowered autonomous
driving framework, which outperforms the existing works in the following two
folds: 1) achieving the intelligent offloading strategy via multi-task
learning in near-real-time, and 2) achieving the joint inference with privacy-
preserving and less communication delay through neural network segmentation.
In particular, a multi-task learning model is deployed at the edge server to
infer the optimal offloading decision of the vehicles and the computation
resource allocation of the edge server with high accuracy in near-real-time,
augmented by a neuron network segmentation method to achieve efficient online
inference dynamically with privacy-preserving and less latency raised. A case
study is presented to demonstrate the proposed framework and then evaluated
via experiments in Section IV, followed by listing some critical open research
topics and future guidelines in Section V.

##  II Where Edge Intelligence Meets Autonomous Driving: Applications and
Challenges

In this section, we list some EI applications in autonomous driving and
discuss the technical challenges raised.

###  II-A Applications of EI in Autonomous Driving

There has been lately increasing research interest in incorporating AI
technology (e.g., ML and DL) into edge computing, known as EI, which can help
achieve efficient computing and intelligent decision making to keep the roads
safe. In the following, we list some fundamental computation-intensive
applications aided by EI in autonomous driving, such as object detection,
traffic flow prediction, and path planning.

####  II-A1 Object Detection

Real-time video analytics has been widely applied in autonomous driving (see
Fig. 1), where the data captured by the cameras and other onboard sensors is
analyzed via the AI techniques. To achieve safe and efficient driving (e.g.,
reduce accidents and decrease traffic congestion), autonomous vehicles should
identify the objects timely and accurately according to the surroundings.
However, executing this kind of delay-sensitive tasks via DL generally
requires high computation resources onboard and incurs high bandwidth
consumption by uploading the tasks to the remote cloud centers. To address the
issues and make autonomous driving possible, EI enables the self-driving
vehicles to move some of the video analysis to the edge servers, which are
usually near the data sources and can improve the inference accuracy.

####  II-A2 Traffic Flow Prediction

With the ever-increasing human demands on the intelligent transportation
system and the adoption of autonomous vehicles, timely and accurate
acquisition of traffic flow information will lay out the foundation for many
location-related applications, e.g., navigation. However, massive volumes of
environmental data with various sources will seriously saturate the onboard
storage and become too complex to satisfy the requirements with the
traditional computing techniques and infrastructures [10]. To meet this
challenge, EI represents a trend of “driverless revolution on big data”.
Specifically, by combining AI with advanced edge computing techniques, the
computation resources can be expanded to the edge server to achieve state-of-
the-art performance for hierarchical features learning from the high-
dimensional dataset and satisfy real-time requirements of the time-consuming
tasks.

####  II-A3 Intelligent Decision

Recent years have shown remarkable attention on intelligent decision in
autonomous driving, e.g., online path planning, which aims to improve road
safety by avoiding collision with the surrounding obstacles, vehicles, and
infrastructure. With feasible path planning, AVs can make reasonable decisions
on the critical maneuvers (such as braking, turning, and overtaking.)
according to the surroundings by feeding the inferring results into the
onboard ECU or the CACC system. In practice, intelligent decision can be
formulated as an optimized decision-making problem taking into account the
vehicle dynamics, which, however, makes the decision hardly be achieved in
time [10]. Notably, this becomes a concern increasingly when the B5G/6G
paradigm is taken into account. To meet this concern, EI can be a
revolutionary breakthrough by inferring an optimal decision directly in near-
real-time via a well trained DL model deployed at the edge [11]. Different
from the conventional cloud computing-based methods, e.g., Google Map, with
the aid of AI functioning on the edge, EI can significantly decrease the
wireless communication delay by allowing AVs to upload the collected data
(such as vehicle conditions and surroundings data) to the edge server. As a
result, the AVs may obtain the inferring results within a limited time, and
thus an intelligent, timely, and reliable decision is prone to be achieved at
AVs, especially when the surrounding conditions change dramatically.

###  II-B Technical Challenges

####  II-B1 Infeasible Sensing

In practice, due to the intrinsic features of the embedded sensors, they
usually have restricted perception capacities, which lead to infeasible
sensing of surroundings. Specifically, as a primary sensing method, the main
drawback of visual-based object detection is that the inference performance
(e.g., the accuracy of multi-class classification) usually relies on the
captured images quality. The image quality can be affected by the
surroundings, such as light conditions, weather conditions, image resolution,
and the distance between camera and object. According to National
Transportation Safety Board (NTSB) accident report, a Uber self-driving
testing vehicle struck a pedestrian who was dressed in dark clothing and
walked a bicycle across the road in Arizona at about 9:58 p.m. on March 18,
2018. The bicycle did not have any side reflectors, and the roadway lighting
did not directly illuminate the roadway section. As a result, the pre-trained
deep learning model mistakenly classified the pedestrian as an unknown object
first, then as a vehicle, and finally, as a bicycle with varying expectations
of the future travel paths. It is observed from this accident that the
decision-making by a single kind of sensors (e.g., the cameras in Uber’s
testing case) alone has tended to introduce serious accidents. To avoid this,
a promising candidate method is to fuse alternate sensing modalities for
perception, e.g., fusing from the LiDAR sensors, the sensors implemented at
the intelligent traffic lights, and querying the sensors from other vehicles.
This kind of technology is known as sensor fusion,

####  II-B2 Trade-off between Reliability and Latency

The inference performance generally includes not only the inference accuracy
but also the inference delay. Since the inference accuracy of a pre-trained DL
model is mainly determined by the input data quality and the model capability,
the inference accuracy can be degraded due to the “bad quality” data and
shallow neural networks model. Besides, an additional communication delay is
introduced by offloading the tasks from the mobile device to a more powerful
edge server with a deeper neural network model. Nevertheless, the channel
dynamics may sometimes disable the offloading. Even if not, the introduced
delay could be fatal to delay-sensitive applications. Therefore, there exists
a trade-off between inference accuracy and latency, which makes careful
consideration on the design of edge intelligence inevitable [5]. In order to
achieve a comfortable balance between the reliability and the latency raised,
it becomes necessary to decrease the wireless transmission delay between the
devices and the edge server. For example, the achievable rate of the wireless
link can be improved via some promising technologies in future 6G networks,
such as mmWave transmission, massive MIMO, and holographic reconfigurable
intelligent surfaces. On another note, the data can further be offloaded to
other mobile devices via the device-to-device links with a shorter distance.
In the considered vehicular networks, the vehicles may directly offload the
data to their neighboring vehicles via cellar-V2V or even PC5 interfaces.

####  II-B3 Limited Resources

Compared to a large amount of powerful graphics processing units (GPUs) and
central processing unit (CPUs) integrated at the cloud, the edge servers
usually can hardly bear a massive volume of offloading requests from the
devices due to the constraint of computation, caching, and power resources on
edge servers, and limited communication bandwidth. In this context, joint
optimization of offloading decisions and allocation on the limited resources
available at the edge server plays a particularly important role in edge
intelligence.

####  II-B4 Data Security and Privacy

In many application domains of edge intelligence, data security and privacy
issues are critical because the mobile devices’ data required to be processed
and inferred might carry much private-sensitive information that may not want
to be captured [12, 13]. Take autonomous driving as an example, where the AVs
capture a vast amount of images containing privacy information. Suppose that
this kind of information is directly offloaded to the edge server to be
processed, the user privacy may leak. In general, this issue may occur in both
the model training and inference. To address this issue, performing simple
processing (or training) at the device first and then uploading the
intermediate features to the edge server become promising.

Figure 2: Illustration of the four functionality modules.

##  III Main Functionalities in Autonomous Driving Edge Intelligence

In this section, we discuss how edge intelligence supports the key
functionalities of autonomous driving and list four functionality modules, as
illustrated in Fig. 2.

###  III-A Data Collection Module

As a fundamental building block in the autonomous driving system, the data
collection module collects the data via the onboard sensors. In general, the
collected data can be roughly classified into two categories: sensing data and
vehicle status data. In particular, the sensing data includes the data
collected from the onboard sensors, such as video cameras, LiDAR sensors, and
radar sensors. The vehicle status data usually reveals the vehicles’
computation resources, e.g., the CPU/GPU processing capability, energy power,
and storage capabilities.

###  III-B Distributed Learning Module

Distributed learning (e.g., the federated learning) via the local datasets has
gained considerable attention for providing an EI service for autonomous
driving with privacy-preserving [13]. In particular, distributed learning is
operated at the edge devices based on their collected datasets. Then the
trained local model parameters are uploaded to the edge server for
aggregation. This procedure repeats many times until the model converges. In
this context, AVs can collaboratively train a shared model using real-time
generated mobile data. Although the distributed learning performed by massive
edge devices is sometimes time-consuming due to the limited communication
bandwidth, it is envisioned that distributed learning architectures will still
get a considerable boost in autonomous driving since more advanced
communication technologies are being used under the future B5G/6G networks
[1].

###  III-C Offloading Strategy Decision Module

Offloading strategy decision making within the critical time window is crucial
for safety achievement in autonomous driving networks. Traditionally,
offloading strategy decision making includes two aspects: offloading decision
and computation resources allocation. The offloading strategy decision making
can be formulated as an optimization problem with performance metrics in
latency and energy consumption. For the general self-driving scenarios,
offloading strategy decision making involves solving a mixed integer
programming problem that is usually NP-hard and difficult to solve via
conventional mathematical techniques near-real-time [3]. One promising
approach is to apply deep learning for solving this kind of optimization
problem by training a deep learning model to learn the mapping between the
problem input parameters and the optimal solution.

###  III-D Online Inference Module

As the final step in the edge learning loop (as shown in Fig. 2), the driving
maneuvers are generated by the online inference module based on the new input
data. On the one hand, for the local inference mode (i.e., the inference is
made at the vehicles), the inference performance mainly depends on the
vehicle’s computing capability, which is power limited. On the other hand,
introducing edge inference could provide low-latency services for AVs by
feeding the data into the trained DL model deployed at the edge server.
However, edge inference still faces many challenges, e.g., data privacy and
limited computation capability, which can hardly fully bear and store the
sizeable trained ML model.

Figure 3: An illustrative two-tier edge intelligence structure for autonomous
driving system.

##  IV Two-Tier Edge Intelligence-Empowered Autonomous Driving Framework

Motivated by the preceding considerations, we materialize EI-empowered object
detection for autonomous driving and propose in this article a two-tier
(including the AV-tier and the EI-tier) autonomous driving framework with a
set of design guidelines, as shown in Fig. 3. In the following, we shall
discuss specific components of the proposed framework and provide a concrete
application example to illustrate the paradigm shift of autonomous driving.

Figure 4: A case study on the proposed two-tier autonomous driving framework.

###  IV-A Intelligence Offloading via Multi-task Learning

In the EI-based autonomous driving system, we formulate the binary offloading
decision-making (i.e., offload or not) and the computational resources
allocation (denoting how many computational resources can be allocated to each
vehicle) as a mixed-integer nonlinear programming (MINLP) problem, which is
generally NP-Hard and challenging to solve. In particular, some factors (such
as the number of vehicles, the computational capability of vehicles, and the
channel conditions.) may vary over time, so the conventional relaxation-based
optimization procedure must be executed repeatedly on solving the MINLP each
time the parameters change. Therefore, high computational complexity is
incurred due to numerical iterations, and the solutions are often sub-optimal
and would not scale well [11]. This is increasingly concerned when B5G/6G and
IoT paradigms are taken into account [1]. To meet the coming challenges, one
promising approach is to apply deep learning for solving this kind of NP-hard
optimization problem by training a deep learning model to ‘learn’ the mapping
between the problem input parameters and the optimal solution. In this paper,
we build a multi-task learning (MTL) based framework to infer the solutions
more efficiently with high accuracy, as illustrated in Fig. 3(a). In
particular, a deep neural network model is designed and trained offline at the
BS to obtain the mapping relationship from the input parameters to the output
solutions. Therefore, on receiving the offloading requests from the AVs, the
BS can directly infer the optimal offloading strategy decision in near-real-
time by performing feedforward calculation via the MTL model without
iterations, as shown in Fig. 3(b).

###  IV-B Edge-Vehicle Joint Inference

In the autonomous driving system, the online inference can be classified into
three modes: local-inference mode, edge inference mode, and joint inference
mode. For the local inference, the vehicles perform the inference themselves.
The inference performance mainly depends on the available computational
capability (computation resources and storage space) of the vehicles, which,
however, is usually very limited [5]. For the edge inference, the raw data
must be uploaded to the edge server via the wireless uplink. We note that
there exist two potential hazards associated with this approach. One is data
privacy due to data fully uploading to the edge server. The other is that
considerable communication delay is introduced once many vehicles upload their
raw data with large size and quantity via the bandwidth-limited uplink
channel. Motivated by the challenges in the previous two inference modes, the
edge-vehicle joint inference becomes promising by dividing the trained deep
neural network (DNN) model into two parts: the shallow neural network (SNN)
layers at the vehicle side, and the DNN layers at the edge side. Therefore,
the vehicles only need to upload the intermediate parameters to the edge
server to reduce the end-to-end latency with joint-inference. In practice, the
limited latency gains can be achieved due to the data amplification effect,
which indicates that the size of intermediate output data of the deep neural
networks is larger than that of the input data. In this context, choosing an
appropriate splitting point while avoiding the data amplification effect
becomes critical, and thus the size of uploaded parameters could be far less
than the size of raw data by designing the neural networks model
appropriately.

Figure 5: Inference accuracy of the classification and the MSE of the
regression in the MTL model versus the percentage of training samples is shown
in (a), where the number of AVs is $2$. In (b), the number of AVs versus the
accuracy achieved by the sBB algorithm and the multi-task learning model is
illustrated, respectively, where the percentage of training samples in multi-
task learning model is 1.

###  IV-C Case Study: EI-Assisted Visual Object Detection

In this section, we consider a visual object detection based autonomous
driving system. Due to limited computing resources and a tight energy budget
of self-driving vehicles, they usually need help from the BS to
collaboratively process the delay-sensitive tasks. Since the offloading
strategy decision-making with minimization of AVs’ energy consumption and
latency can be formulated as an MINLP problem, which is NP-Hard, we design an
MTL-based feedforward neural network model by learning the end-to-end mapping
between the problem input parameters and the output solutions. Specifically,
the binary offloading decision is considered as a multi-class classification
problem with the cross-entropy loss, and the edge server computational
resource allocation is considered as a regression problem with the mean square
error (MSE) loss. During the MTL training, the Adam optimizer is used to
minimize the weighted-sum loss, where $\chi_{c}$ and $\chi_{l}$ denote their
weights, respectively. It should be noted that the size of the appropriately
trained MTL model is less than $2$ KB, and thus the edge server has enough
storage space to save the trained MTL model, which can be even cached into the
edge server’s memory in advance to perform the inference more efficiently.
With the offline trained MTL model, the BS can directly infer the offloading
strategy with high accuracy in near-real-time by performing feedforward
calculation without iterations, as illustrated in step 1) - step 2) in Fig. 4.
Therefore, the proposed MTL-based method “moves” the complexity of online
computation to offline training, which can be scaled up across the graphics
processing unit (GPU) clusters.

To achieve privacy-preserving and meet the hardware constraint of AVs, we
investigate it by segmenting a trained CNN model into the lower-level SNN
layers and the higher-level DNN layers. In particular, the SNN layers and DNN
layers usually contain convolutional layers, rectified linear units layer,
polling layer, and fully-connected layers. The SNN layers are deployed on the
resource-constrained vehicles, usually with limited computation resources and
storage space. All the AVs share the same DNN layers, which are deployed at
the edge server and can further improve the inference accuracy by processing
the feature maps generated by the SNN layers. This edge learning setup
fulfills the demand for the timely processing of the video images while taking
into account the practical implementation constraints, e.g., users’ privacy.
As illustrated in step 3) - step 4) in Fig. 4, the lower layers can directly
provide sufficient features for the acceptable object detection performance
without offloading when the captured images quality is sufficiently “good”. On
the contrary, the captured images with “bad” quality would offload the
intermediate feature maps to the higher layers at the edge server to improve
inference accuracy. The edge server then returns the processing results, which
help the vehicles react fast and accurately via the ECUs.

Figure 6: Bad data ratio ($\eta$) versus total weighted-sum cost 0using three
different kinds of inference methods, where the number of AVs is $2$, the
transmission power of AVs is $10$ W, the CPU frequency of the AVs and the edge
server is $1$ GHz and $10$ GHz, respectively.

###  IV-D Illustrative Results

In this subsection, we first evaluate in Fig. 5(a) the impact of the number of
training samples on the inference accuracy of the MTL model, where
$\chi_{c}=\chi_{r}=1$. The inference accuracy is defined as the ratio of the
number of correct predictions to the total number of predictions. We generate
$4\times 10^{4}$ data samples by traversing the combinations of the binary
offloading decision and computational resources allocation ratio with the
exhaustive searching algorithm. In Fig. 5(b), we compare the accuracy of the
proposed MTL model with that of the conventional spatial branch and bound
(sBB) algorithm, where $\chi_{c}=0$ and $\chi_{r}=1$ when the number of AVs is
larger than $5$, and the percentage of training samples in MTL model is $1$.
After, we investigate in Fig. 6 the impact of the “bad” data ratio (denoted as
$\eta$) on the weighted-sum cost (defined as the weighted sum of the delay and
energy consumption of all the AVs) using different inference strategies in the
autonomous driving system.

We observe from Fig. 5(a) that the inference accuracy of binary offloading
decision (i.e., offload or not) is low for the small number of training
samples and increases with the percentage of training samples, whilst the MSE
of regression shows an opposite trend. The reason is that when the training
samples are insufficient, the MTL model can only learn a few distinguishing
features, which can hardly perform inference accurately. As the number of
training samples increases, the MTL model can further extract more abstract
features enabling the MTL model to learn more about the mapping between the
input parameters and the output solutions of the formulated MINLP problem.
From Fig 5(b), we observe that the MTL model always outperforms the sBB
algorithm in the inference accuracy, and the gain is improved as the number of
AVs increases. Compared to the sBB algorithm, almost $120\%$ improvement can
be achieved by MTL model when the number of AVs is $8$. Notably, the MSE of
the MTL model is even less than half of the sBB algorithm. Moreover, by using
multi-task learning, the time consumption solving the MINLP is less than one-
thousandth of the sBB scheme when the number of AVs varies from $2$ to $8$.

Fig. 6 shows that when the “bad” data ratio (denoted as $\eta$) becomes larger
(e.g., more captured images are with low quality), this leads to an increase
of the weighted-sum cost using three inference methods. Specifically, when
$\eta$ is relatively small, e.g., $\eta<0.3$, inference at vehicles becomes
more competitive due to the achieved high inference accuracy without
introducing communication delay. As $\eta$ increases, e.g., $\eta\geq 0.3$,
the inference given by the SNN layers becomes worse, which leads to a sudden
rise in the weighted-sum cost due to the introduced penalty. At this time, the
advantage of edge inference is an explicit benefit of exploiting more powerful
DNN layers to improve inference accuracy via offloading the intermediate
feature maps.

##  V Conclusion and Open Research Topics

In this article, we discussed the potential and challenges of edge
intelligence in connected autonomous driving. We presented a two-tier edge
learning-empowered architecture of wireless computing for AVs and introduced
two functionality designs to make offloading strategy decision efficiently,
ensure data privacy, and improve inference accuracy while meeting the delay
constraint. The effectiveness of the proposed framework was demonstrated via a
case study and experiments.

There are some interesting open research topics for EI-based autonomous
driving architecture design.

  * •

Efficient edge learning with limited labeled training data. Generally, most of
the edge learning tasks in the autonomous driving system are based on
supervised learning, which requires complete labeled training datasets.
However, collecting sufficient labeled data with a correctness guarantee is
still a challenge in practice [14]. To facilitate learning on edge with
limited labeled data, we can intuitively take advantage of the historical data
and combine it with real-time collected data. Also, we can take advantage of
semi-supervised learning, transfer learning, or even autonomous learning [14],
to yield significant training and inference improvements via limited training
data. Furthermore, the sensing deficiency caused by using only video cameras
can be filled up by exploring the potential of sensor fusion, such as LiDAR
sensors, radar sensors, and even by querying the sensor data from other
vehicles.

  * •

Distributed model training with restricted wireless bandwidth. In the
autonomous driving system with edge learning, the training data involved is
generally privacy-sensitive with large size and quantity, e.g., high-
definition video sequences. By introducing distributed learning (e.g.,
federated learning), only the local model parameters are uploaded to the edge
server to preserve data privacy. Interestingly, via the federated learning and
the neuron network segmentation, privacy-preserving can be achieved at the
training side and inference side, respectively. Since each AV usually collects
unique training data separately and the data samples are generally non-
independent and identically distributed among the vehicles, the edge server
prefers to include more AVs’ local FL models to generate a converged global
model. In this context, the model parameters exchange between AVs and servers
may result in high communication costs, and the runtime of each learning
iteration is dominated by the slowest participants, which become a bottleneck
for the autonomous driving system [12]. To mitigate this issue, more AVs can
be allowed to participate in the FL by sharing their FL models with selected
AVs via V2V communications, using the side-link channel in cellular-based V2V
(C-V2V), DSRC, or even millimeter-wave transmission [2, 6].

  * •

Vehicle platoon-aided inference. In practice, the vehicles may sometimes drive
out of the coverage of BS, which will cause considerable influence on the
feasibility of edge inference. To address this challenge, collaborative
caching and computing among vehicles become a competitive candidate solution
[3, 15]. Specifically, the trained DNN model can be divided into multiple
parts (called layers), each of which is pre-deployed at the vehicle belonging
to the same vehicle platoon. Once a vehicle leaves the BS’s coverage, this
vehicle can still achieve reasonable inference by routing the intermediate
parameters to other vehicles within the same vehicle platoon via V2V
communications. In this case, the improvement of inference accuracy is at the
expense of V2V communication delay, and the impact of the platoon dynamics
needs to be further investigated.

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