# Scattering of a twisted electron wavepacket by a finite laser pulse

I. A. Aleksandrov  Department of Physics, St. Petersburg State University, 7/9
Universitetskaya Naberezhnaya, Saint Petersburg 199034, Russia  Ioffe
Institute, Politekhnicheskaya street 26, Saint Petersburg 194021, Russia  D.
A. Tumakov  Department of Physics, St. Petersburg State University, 7/9
Universitetskaya Naberezhnaya, Saint Petersburg 199034, Russia  Ioffe
Institute, Politekhnicheskaya street 26, Saint Petersburg 194021, Russia  A.
Kudlis  ITMO University, Kronverkskiy prospekt 49, Saint Petersburg 197101,
Russia  V. A. Zaytsev  Department of Physics, St. Petersburg State University,
7/9 Universitetskaya Naberezhnaya, Saint Petersburg 199034, Russia  N. N.
Rosanov  Ioffe Institute, Politekhnicheskaya street 26, Saint Petersburg
194021, Russia

###### Abstract

The behavior of a twisted electron colliding with a linearly polarized laser
pulse is investigated within relativistic quantum mechanics. In order to
better fit the real experimental conditions, we introduce a Gaussian spatial
profile for the initial electron state as well as an envelope function for the
laser pulse, so the both interacting objects have a finite size along the
laser propagation direction. For this setup we analyze the dynamics of various
observable quantities regarding the electron state: the probability density,
angular momentum, and mean values of the spatial coordinates. It is shown that
the motion of a twisted wavepacket can be accurately described by averaging
over classical trajectories with various directions of the transverse momentum
component. On the other hand, full quantum simulations demonstrate that the
ring structure of the wavepacket in the transverse plane can be significantly
distorted leading to large uncertainties in the total angular momentum of the
electron. This effect remains after the interaction once the laser pulse has a
nonzero electric-field area.

##  I Introduction

The both massive and massless particles which are capable of carrying nonzero
orbital angular momentum (OAM) are of a great interest for researchers from
different fields of study. The first seminal description of the concept of
such twisted particles was introduced by Allen et al. in the early 90s in Ref.
Allen  _et al._ (1992), where it was discussed in the context of photons. The
wave front of the corresponding beams has a helical spatial structure allowing
the beams to carry a nonzero OAM projection onto the propagation direction.
Soon afterwards, the fact that twisted photons possess discreet OAM was
confirmed experimentally in Ref. Beijersbergen  _et al._ (1993), and two years
later He et al. managed to transfer OAM from light to matter He  _et al._
(1995). In fact, the opportunities related to the information capacity and
other specificities of these optical vortices – twisted photons – have
encouraged numerous studies concerning a huge variety of their applications
Torres and Torner (2011) (see also reviews Allen  _et al._ (1999); Soskin and
Vasnetsov (2001); Padgett  _et al._ (2004); Franke-Arnold  _et al._ (2008);
Knyazev and Serbo (2018); Allen (2016)).

Despite the fact that the development of the field began with the
investigation of photons, it was clear from the very beginning that massive
twisted particles could also be produced. The fundamental possibility that not
only photons can carry OAM but also, for example, electrons was indicated in a
number of studies (see, e.g., Refs. Ouyang  _et al._ (1999); Bagrov and Gitman
(2005); Leary  _et al._ (2008); Bliokh  _et al._ (2011)). For practical
applications, an interesting feature of twisted electrons is that, unlike
photons, they carry a magnetic moment proportional to OAM which can
experimentally reach hundreds of $\hbar$ Mafakheri  _et al._ (2017). It allows
electron vortices to effectively interact with external magnetic fields Bliokh
_et al._ (2007); Verbeeck  _et al._ (2010); Bliokh  _et al._ (2011); Karimi
_et al._ (2012); Lloyd  _et al._ (2012); Gallatin and McMorran (2012); Bliokh
_et al._ (2012), which makes them a versatile tool for investigating magnetic
properties of different materials Rusz and Bhowmick (2013); Beche  _et al._
(2014); Schattschneider  _et al._ (2014); Edström  _et al._ (2016) as well as
for detecting such subtle magnetic effects as, e.g., polarization radiation
Ivanov and Karlovets (2013a, b). The method proposed by Bliokh et al. in Ref.
Bliokh  _et al._ (2007) for producing such electrons was successfully employed
by several groups of experimentalists who have played a pioneering role in
this field of study Uchida and Tonomura (2010); Verbeeck  _et al._ (2010);
McMorran  _et al._ (2011) (see also recent papers Saitoh  _et al._ (2012);
Schattschneider  _et al._ (2012a); Clark  _et al._ (2013); Ryu  _et al._
(2014)). The energy scale in these experiments amounted to $\sim 300$ keV
while the orbital quantum number reached one hundred. This experimental
breakthrough has motivated theorists to deeper analyze the properties of
twisted electrons and also to propose new setups for their generation and
investigation Schattschneider and Verbeeck (2011a); Karimi  _et al._ (2012);
Lloyd  _et al._ (2012); Gallatin and McMorran (2012); Grillo  _et al._ (2014,
2015); Serbo  _et al._ (2015). The positive outcome opened broad prospects not
only for researchers in nuclear and high-energy physics, where this feature
may serve as an efficient tool for a better understanding of processes
occurring in collision and scattering experiments Ivanov (2011); Ivanov and
Serbo (2011); Ivanov (2012); Ivanov  _et al._ (2020), but also for
investigations dealing with magnetic and biological materials Verbeeck  _et
al._ (2011); Schattschneider  _et al._ (2012b); Guzzinati  _et al._ (2013). A
detailed description of theoretical and experimental studies that have emerged
over the last decade was presented by Bliokh et al. in a comprehensive review
Bliokh  _et al._ (2017). One should note here that the relativistic and
nonrelativistic descriptions lead to substantially different results for
observable quantities. In particular, due to the nontrivial coupling of the
electron spin and its angular momentum according to the Dirac equation, the
latter is not conserved in contrast to the case of the nonrelativistic
treatment. The detailed analyses of Schrödinger equation for the case of
twisted electrons can be found, e.g., in Ref. Schattschneider and Verbeeck
(2011b).

In order to describe the behavior of twisted electrons within relativistic
theory, one has to construct the corresponding solutions of the Dirac equation
which differ from the usual plane-wave states. For instance, the solution in
the form of the so-called Bessel beams can be considered Bliokh  _et al._
(2011). However, one should also find out whether such solutions properly
mimic the real experimental setups. For example, the Bessel beams themselves
are not localized in space. To eliminate this shortcoming, one can introduce
wavepackets (WP) or consider other solutions of the Dirac equation, e.g., the
so-called exponential WP proposed in Refs. Bialynicki-Birula and Bialynicka-
Birula (2017, 2019a). Alternatively, relativistic electron vortices can be
described by using the Foldy-Wouthuysen transformation of the Dirac equation
followed by a series of quite natural approximations Barnett (2017). It allows
one to derive the equations that can be solved much easier in contrast to the
Dirac one. However, the legitimacy of the Foldy-Wouthuysen representation in
this context is not that clear. In particular, the determination of the
correct operators for physical observables and the possibility of a
probabilistic interpretation of the wavefunction have given rise to heated
debates Silenko  _et al._ (2018); Bialynicki-Birula and Bialynicka-Birula
(2019b); Silenko  _et al._ (2019). Recently, the number of approaches has been
published which offer alternative methods for obtaining exact solutions of the
Dirac equation on the basis of Bateman-Hillion ansatz Ducharme  _et al._
(2021) and by means of newly developed relativistic dynamic inversion approach
Campos  _et al._ (2021).

The list of theoretical studies of physical processes involving twisted
electrons is very extensive (see, e.g., Refs. Seipt  _et al._ (2014);
Bandyopadhyay  _et al._ (2015); Matula  _et al._ (2014); Karlovets  _et al._
(2017); Zaytsev  _et al._ (2017); Kosheleva  _et al._ (2018); Groshev  _et
al._ (2020); Karlovets (2012); Hayrapetyan  _et al._ (2014)). Let us note that
the formalism describing the phenomena associated with the scattering
processes of electrons in matter is relatively well developed; the recent
studies have covered a number of important issues playing a vital role for the
completeness of the theoretical basis Matula  _et al._ (2014); Karlovets  _et
al._ (2017); Zaytsev  _et al._ (2017); Kosheleva  _et al._ (2018); Groshev
_et al._ (2020). This contrasts, however, with the study of the behavior of
twisted electrons colliding with a laser pulse. Although the problem and
related issues have already been considered in a number of works Karlovets
(2012); Hayrapetyan  _et al._ (2014), we aim to obtain more accurate results
by taking into account the finite size of both the electron WP and the laser
pulse. For this purpose, within relativistic quantum mechanics, we follow the
theoretical approach employed in Ref. Aleksandrov  _et al._ (2020) for a
plane-wave electron, where the spatial envelope of the laser pulse along its
propagation direction was introduced and the initial electron WP had a
Gaussian profile. Following this scheme, one explores a more realistic setup
from the experimental viewpoint.

Our computations are based on the following procedure. First, a localized
electron WP is constructed from the Bessel beams while the laser pulse is
assumed to be a linearly polarized plane wave. The WP is then expanded in
terms of the Volkov solutions Wolkow (1935). Having obtained the corresponding
expansion coefficients, one can construct the solution of the Dirac equation
at arbitrary time. We analyze various observable quantities such as the
probability density, angular momentum and its dispersion, and mean values of
the coordinates. These quantities are computed for various parameters of the
external laser field and the initial electron state.

The paper is organized as follows. In Sec. II we describe the setup under
consideration. In Sec. III we briefly outline the approach based on the
expansion of the wavefunction in terms of the Volkov solutions in the case of
a plane-wave electron state. Section IV is devoted to a detailed description
of the analogous method for treating twisted electrons. We discuss our
numerical results in Sec. V.

Throughout the paper, we use atomic units $\hbar=m_{e}=-e=1$.

##  II Setup. Electron wavepacket colliding with a laser pulse

We consider a twisted electron wavepacket propagating along the $z$ axis and a
linearly polarized plane-wave laser pulse traveling along the opposite
direction (see Fig. 1). The latter is described by the following expression
for the electric component:

$\displaystyle E_{x}(t,z)=\mathcal{E}(ct+z),$ (1)
$\displaystyle\mathcal{E}(\xi)=E_{*}F(\omega\xi/c)\sin(\omega\xi/c+\phi).$ (2)

Here $E_{*}$ and $\omega$ are the field amplitude and frequency, respectively,
the function $F$ is a smooth Gaussian-shape envelope,
$F(\eta)=\mathrm{exp}(-\eta^{2}/a^{2})$, where $a$ is a dimensionless
parameter governing the pulse duration, and $\phi$ is a carrier-envelope phase
(CEP). It is convinient to introduce the field peak intensity

$I=\frac{1}{8\pi\alpha}E_{*}^{2},$ (3)

where $\alpha$ is the finite structure constant. The only nonzero component of
the potential $A^{\mu}$ is $A^{1}=A_{x}(t,z)=\mathcal{A}(ct+z)$ with

$\mathcal{A}(\xi)=-c\int\limits_{-\infty}^{\xi}\mathcal{E}(\xi^{\prime})d\xi^{\prime}.$
(4)

We underline here that the vector potential does not necessarily vanish for
$\xi\to+\infty$. We will denote this limit by
$\mathcal{A}_{0}=\mathcal{A}(+\infty)$. Nonzero values of $\mathcal{A}_{0}$
correspond to a nonzero unipolarity of the laser pulse, i.e., nonzero momentum
transferred to the electron along the electric field direction:

$\mathcal{A}_{0}=-c\int\limits_{-\infty}^{+\infty}\mathcal{E}(\xi^{\prime})d\xi^{\prime}=-\frac{\sqrt{\pi}caE_{*}}{\omega}\,\mathrm{e}^{-a^{2}/4}\sin\phi.$
(5)

Figure 1: Initial state of the counterpropagating twisted wavepacket (left)
and laser pulse (right). The momentum projection of the former is $p_{z}\equiv
p_{\parallel}$. The laser field is polarized along the $x$ direction according
to Eq. (1) and has amplitude $E_{*}$. The center of the initial WP corresponds
to $z=0$ (the $x$ axis in the figure does not cross the origin).

In what follows, we will exploit the fact that the envelope $F(\eta)$ has an
almost finite support, i.e., the function $\mathcal{A}(\xi)$ almost vanishes
for $\xi<-\xi_{\text{max}}$ and does not differ much from $\mathcal{A}_{0}$
for $\xi>\xi_{\text{max}}$. This means that at the initial time instant
$t_{\text{in}}=-(L+\xi_{\text{max}})/c$, the laser pulse is localized within
the region $z\in(L,\ L+2\xi_{\text{max}})$, where $L$ should be sufficiently
large, so that the pulse and the electron WP do not overlap.

The final state after the interaction is considered at
$t=t_{\text{out}}=(\tilde{L}+\xi_{\text{max}})/c$, where $-\tilde{L}$ is the
position of the right edge of the laser pulse. One has to make sure that
$\tilde{L}$ is sufficiently large, so that the external field and the electron
WP no longer overlap.

In what follows, we will discuss how the initial state of the electron WP can
be evolved in time by means of the exact (Volkov) solutions of the Dirac
equation in the external plane-wave background.

##  III Plane-wave electron state

First, we will examine a plane-wave electron state and briefly recap the
approach which we previously employed in Ref. Aleksandrov  _et al._ (2020).
Since the external field does not depend on $x$ and $y$, the corresponding
components of the generalized momentum are conserved. With respect to the $z$
direction, we construct a WP of the following form:

$\psi^{\text{(0)}}_{\bm{p},s}(\bm{x})=\frac{1}{(2\pi)^{3/2}}\,\mathrm{e}^{i\bm{p}\bm{x}}\\!\int\limits_{-\infty}^{+\infty}\\!dq\,f(q)\,\mathrm{e}^{iqz}u(\bm{p}+q\bm{n},s),$
(6)

where $\bm{n}$ is a unit vector along $z$ and the smearing function $f(q)$
determines the spectral content of the initial WP. It is normalized to ensure
$\langle\psi^{\text{(0)}}_{\bm{p},s}|\psi^{\text{(0)}}_{\bm{p},s^{\prime}}\rangle=\delta_{ss^{\prime}}$.
The initial spin state of the electron is governed by the constant bispinors
$u(\bm{p},s)$ corresponding to the positive-energy solutions of the Dirac
equation ($s=\pm$). Together with bispinors $v(\bm{p},s)$, they form an
orthonormal and complete set. These bispinors obey

$\displaystyle\big{(}c\,\bm{\alpha}\cdot\bm{p}+\beta c^{2}\big{)}u(\bm{p},s)$
$\displaystyle=$ $\displaystyle\varepsilon u(\bm{p},s),$ (7)
$\displaystyle\big{(}c\,\bm{\alpha}\cdot\bm{p}+\beta c^{2}\big{)}v(\bm{p},s)$
$\displaystyle=$ $\displaystyle-\varepsilon v(\bm{p},s),$ (8)

where $\alpha_{i}$ and $\beta$ are the standard Dirac matrices, and
$\varepsilon=c\sqrt{c^{2}+\bm{p}^{2}}$.

The initial condition is
$\psi_{\bm{p},s}(t_{\text{in}},\bm{x})=\psi^{\text{(0)}}_{\bm{p},s}(\bm{x})$,
and our goal is to evolve the wavefunction in time. The main idea is to expand
the initial state in terms of the Volkov solutions Wolkow (1935) and then
combine them at some given $t$ using the same coefficients as they do not
depend on time. The Volkov states are given by

$\displaystyle\varphi^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t,\bm{x})$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\,\mathrm{e}^{i\zeta\bm{p}^{\prime}\bm{x}}f^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t,z),$
(9) $\displaystyle f^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t,z)$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-i\zeta\varepsilon^{\prime}t}\,\mathrm{exp}\Bigg{\\{}i\int\limits_{-\infty}^{n\cdot
x}\\!d\xi\,\frac{1}{2(n\cdot
p^{\prime})}\,\bigg{[}\frac{2}{c}\,(p^{\prime}\cdot
A(\xi))+\frac{\zeta}{c^{2}}\,A^{2}(\xi)\bigg{]}\Bigg{\\}}$ (10)
$\displaystyle\times$ $\displaystyle\bigg{[}1-\frac{\zeta}{2c(n\cdot
p^{\prime})}\,(\gamma\cdot n)(\gamma\cdot
A)\bigg{]}w_{\zeta}(\bm{p}^{\prime},s^{\prime}),$

where $\zeta=\pm$ denotes the sign of the energy,
$w_{+}(\bm{p}^{\prime},s^{\prime})=u(\bm{p}^{\prime},s^{\prime})$, and
$w_{-}(\bm{p}^{\prime},s^{\prime})=v(-\bm{p}^{\prime},s^{\prime})$. In our
case, it yields

$\displaystyle f^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t,z)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{-i\zeta\varepsilon^{\prime}t}\,\mathrm{exp}\Bigg{\\{}-\frac{i}{\varepsilon^{\prime}+cp_{z}^{\prime}}\Bigg{[}p_{x}^{\prime}\int\limits_{-\infty}^{\xi}\\!d\xi^{\prime}\,\mathcal{A}(\xi^{\prime})+\frac{\zeta}{2c}\int\limits_{-\infty}^{\xi}\\!d\xi^{\prime}\,\mathcal{A}^{2}(\xi^{\prime})\Bigg{]}\Bigg{\\}}$
(11) $\displaystyle\times$
$\displaystyle\bigg{[}1+\frac{\zeta}{2(\varepsilon^{\prime}+cp_{z}^{\prime})}\,\mathcal{A}(\xi)(\gamma^{0}+\gamma^{3})\gamma^{1}\bigg{]}w_{\zeta}(\bm{p}^{\prime},s^{\prime}),$

where $\xi=n\cdot x=ct+z$.

To construct the time-dependent solution of the Dirac equation, we make use of
the fact that the Volkov states form an orthonormal and complete set Boca and
Florescu (2010); Di Piazza (2018), so the wavefunction can be expanded as
follows:

$\psi_{\bm{p},s}(t,\bm{x})=\sum_{\zeta}\sum_{s^{\prime}}\int\\!d\bm{p}^{\prime}\,C^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}\varphi^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t,\bm{x}),$
(12)

where the coefficients $C^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}$ do not
depend on time because the Dirac Hamiltonian is Hermitian. They can be
evaluated at the time instant $t=t_{\text{in}}$ as a usual inner product:

$\displaystyle C^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}$ $\displaystyle=$
$\displaystyle\langle\varphi^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t_{\text{in}})|\psi^{\text{(0)}}_{\bm{p},s}\rangle$
(13) $\displaystyle\equiv$
$\displaystyle\int\\!d\bm{x}\,\big{[}\varphi^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}(t_{\text{in}},\bm{x})\big{]}^{\dagger}\psi^{\text{(0)}}_{\bm{p},s}(\bm{x}).$

It turns out that in the plane-wave background, these coefficients are
“diagonal” with respect to $p_{x}$ and $p_{y}$:
$C^{(\zeta)}_{\bm{p}^{\prime},s^{\prime}}=\delta(p_{x}^{\prime}-\zeta
p_{x})\delta(p_{y}^{\prime}-\zeta
p_{y})c^{(\zeta)}_{p_{z}^{\prime},s^{\prime}}$. One can explicitly verify that
$-i\partial_{x}\varphi^{(+)}_{\bm{p}^{\prime},s^{\prime}}(t,\bm{x})=p_{x}^{\prime}\varphi^{(+)}_{\bm{p}^{\prime},s^{\prime}}(t,\bm{x})$,
so the index $p_{x}^{\prime}$ corresponds to the generalized momentum. Since
the initial wavefunction (6) corresponds to the positive-energy subspace of
solutions, the coefficients $c^{(-)}_{\bm{p}^{\prime},s^{\prime}}$ vanish, so
one can use only the positive-energy Volkov solutions ($\zeta=+$). Note that
the Volkov functions possess a well-defined sign of the energy which does not
depend on time (this is in accordance with the fact that a plane-wave
background does not produce $e^{+}e^{-}$ pairs). We obtain

$\displaystyle c^{(+)}_{p_{z}^{\prime},s^{\prime}}$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{+\infty}\\!\frac{dz}{2\pi}\int\limits_{-\infty}^{+\infty}\\!dq\,f(q)\,\mathrm{e}^{i(p_{z}-p_{z}^{\prime})z}\mathrm{e}^{iqz}$
(14) $\displaystyle\times$
$\displaystyle\big{[}f^{(+)}_{\bm{p}^{\prime},s^{\prime}}(t_{\text{in}},z)\big{]}^{\dagger}u(\bm{p}+q\bm{n},s),$

where $p^{\prime}_{x}=p_{x}$ and $p_{y}^{\prime}=p_{y}$. The wavefunction can
be then evaluated at arbitrary time instant $t$ via

$\psi_{\bm{p},s}(t,\bm{x})=\sum_{s^{\prime}}\int\\!dp_{z}^{\prime}\,c^{(+)}_{p_{z}^{\prime},s^{\prime}}\varphi^{(+)}_{p_{x},p_{y},p_{z}^{\prime},s^{\prime}}(t,\bm{x}).$
(15)

##  IV Twisted electron

In this section, we will discuss how the approach described above should alter
if one replaces a plane-wave electron with a twisted one. First, we discuss
the case of Bessel beams, which are spatially infinite, and then turn to the
analysis of localized WPs.

###  IV.1 Bessel beam

Let us consider the so-called Bessel beam of twisted electrons traveling along
the $z$ axis ($p_{\parallel}\equiv p_{z}$) Bliokh  _et al._ (2011):

$\psi^{(\text{B})}_{l,p_{\parallel},p_{\perp},s}(\bm{x})=\frac{\sqrt{p_{\perp}}}{2^{3/2}\pi}\,\mathrm{e}^{ip_{\parallel}z}g_{l,p_{\parallel},p_{\perp},s}(x,y),$
(16)

where

$g_{l,p_{\parallel},p_{\perp},s}(x,y)\equiv
g_{l,p_{\parallel},p_{\perp},s}(\rho,\varphi)=\sum_{k=-1}^{+1}i^{k}a_{k}\mathrm{e}^{i(l+k)\varphi}J_{l+k}(p_{\perp}\rho).$
(17)

Here

$a_{0}=\left(\begin{tabular}[]{c}{$\sqrt{1+c^{2}/\varepsilon}\ \vec{w}$}\\\
\\\ {$\sqrt{1-c^{2}/\varepsilon}\ \sigma_{z}\cos\theta_{0}\,\vec{w}$}\\\ \\\
\end{tabular}\right),\qquad a_{-1}=\begin{pmatrix}0\\\ 0\\\
\bar{\beta}\sqrt{\Delta}\\\ 0\end{pmatrix},\qquad a_{1}=\begin{pmatrix}0\\\
0\\\ 0\\\ \bar{\alpha}\sqrt{\Delta}\end{pmatrix},$ (18)

$\vec{w}=(\bar{\alpha},\bar{\beta})^{\text{t}}$ is either $(1,0)^{\text{t}}$
or $(0,1)^{\text{t}}$ for $s=1/2$ and $s=-1/2$, respectively,
$\Delta=(1-c^{2}/\varepsilon)\sin^{2}\theta_{0}$,
$\varepsilon=c\sqrt{c^{2}+p^{2}_{\perp}+p^{2}_{\parallel}}$,
$p_{\parallel}=|\bm{p}|\cos{\theta_{0}}$, and
$p_{\perp}=|\bm{p}|\sin\theta_{0}$. The functions $a_{k}$ depend on
$p_{\parallel}$, $p_{\perp}$, and $s$ but are independent of the spatial
coordinates. The transverse coordinates can either be $x$ and $y$ or $\rho$
and $\varphi$. The wavefunction is determined by the following quantum
numbers: $l$, $s$, $p_{\parallel}$, and $p_{\perp}$. Note that the orbital
angular momentum and the spin projection are not well defined and only the
projection of the total angular momentum $m=l+s$ is conserved. Note also that
the Bessel beams with given $s$ have a nonzero projection onto the Volkov
solutions (9) with the other sign of $s^{\prime}$. The Bessel beams (16) are
normalized according to

$\langle\psi^{(\text{B})}_{l,p_{\parallel},p_{\perp},s}|\psi^{(\text{B})}_{l,p_{\parallel}^{\prime},p_{\perp}^{\prime},s^{\prime}}\rangle=\delta_{ss^{\prime}}\delta(p_{\parallel}-p_{\parallel}^{\prime})\delta(p_{\perp}-p_{\perp}^{\prime}).$
(19)

This inner product is evaluated in Appendix A.

In what follows, we will calculate the coefficients
$C_{\bm{p}^{\prime},s^{\prime}}^{(\zeta)}$ [see Eq. (13)] using the explicit
form of Volkov states [see Eqs. (9)–(11)]. One obtains

$C_{\bm{p}^{\prime},s^{\prime}}^{(\zeta)}=\frac{\sqrt{p_{\perp}}}{8\pi^{5/2}}\int\limits_{-\infty}^{\infty}\\!dz\,\mathrm{e}^{i(p_{\parallel}-\zeta
p^{\prime}_{\parallel})z}\big{[}f_{\bm{p^{\prime}},s^{\prime}}^{(\zeta)}(t_{\rm
in},z)\big{]}^{\dagger}\int\limits_{-\infty}^{\infty}\\!dx\\!\int\limits_{-\infty}^{\infty}\\!dy\,\mathrm{e}^{-i\zeta(p^{\prime}_{x}x+p^{\prime}_{y}y)}g_{l,p_{\parallel},p_{\perp},s}(x,y).$
(20)

The double integral over $x$ and $y$ can be calculated analytically (see
Appendix A). We arrive at

$C_{\bm{p}^{\prime},s^{\prime}}^{(\zeta)}=\delta(p_{\perp}-p^{\prime}_{\perp})c_{p_{\parallel}^{\prime},\varphi_{p^{\prime}},s^{\prime}}^{(\zeta)},$
(21)

where
$\bm{p}^{\prime}=(p_{\perp}^{\prime}\cos\varphi_{p^{\prime}},p_{\perp}^{\prime}\sin\varphi_{p^{\prime}},p_{\parallel}^{\prime})$
and

$c_{p_{\parallel}^{\prime},\varphi_{p^{\prime}},s^{\prime}}^{(\zeta)}=\frac{1}{4\pi^{3/2}}\frac{1}{\sqrt{p_{\perp}}}\,i^{-\zeta
l}\mathrm{e}^{il\varphi_{p^{\prime}}}\int\limits_{-\infty}^{\infty}\\!dz\,\mathrm{e}^{i(p_{\parallel}-\zeta
p^{\prime}_{\parallel})z}\big{[}f_{\bm{p}^{\prime},s^{\prime}}^{(\zeta)}(t_{\rm
in},z)\big{]}^{\dagger}\sum_{k=-1}^{+1}(-1)^{k\delta_{\zeta,-1}}a_{k}\mathrm{e}^{ik\varphi_{p^{\prime}}}.$
(22)

Here $\delta_{\zeta,-1}$ reflects that the Bessel beam contains only the
positive-energy components. Accordingly, we construct a decomposition of the
Bessel beam in terms of plane waves with given $p_{\perp}$ which are
determined by the momentum projection $p_{\parallel}^{\prime}$, direction of
the transverse momentum component $\varphi_{p^{\prime}}$, and spin quantum
number $s$. In what follows, we will discuss the analogous expansion in the
case of a localized twisted WP.

###  IV.2 Twisted wavepacket

To make the wavefunction finite in the $z$ direction, one has to combine
solutions with different longitudinal momenta. In the case of a plane-wave
electron, these solutions usually correspond to given values of the transverse
projections $p_{x}$ and $p_{y}$. However, in the case of vortex states, these
are not well defined. In this study, we will fix the magnitude of the
transverse momentum component $p_{\perp}$, so the expression for the twisted
WP reads

$\psi^{(0)}_{l,p_{\parallel},p_{\perp},s}(\bm{x})=\int\limits_{-\infty}^{+\infty}\\!dq\,f(q)\psi^{(\text{B})}_{l,p_{\parallel}+q,p_{\perp},s}(\bm{x}),$
(23)

where the smearing function $f(q)$ satisfies

$\int\limits_{-\infty}^{+\infty}\\!dq\,|f(q)|^{2}=1.$ (24)

We choose a Gaussian profile

$f(q)=\frac{1}{(\pi\sigma^{2})^{1/4}}\exp\left(-q^{2}/2\sigma^{2}\right),$
(25)

where $\sigma$ determines the WP width in momentum space, i.e. the uncertainty
in $p_{\parallel}$. In all our calculations we set $\sigma=$ 10 a.u.

###  IV.3 Plane-wave decomposition of a twisted wavepacket

Assuming that at $t=t_{\text{in}}=-(L+\xi_{\text{max}})/c$ the WP has the form
(23), we will now evaluate the positive-energy coefficients with the aid of
Eqs. (21) and (22). Since the laser field almost vanishes for
$\xi<-\xi_{\text{max}}$, its left edge is initially located to the right of
$z=L$. We assume that the electron WP is localized within the interval $[-L,\
L]$. It means that in Eq. (22), where $p_{\parallel}$ should be replaced with
$p_{\parallel}+q$ according to Eq. (23), one can integrate only over this
region. The condition $z\in[-L,\ L]$ corresponds to
$\xi\in[-2L-\xi_{\text{max}},\ -\xi_{\text{max}}]$, so the external field and
its potential involved in $f_{\bm{p}^{\prime},s^{\prime}}^{(\zeta)}(t_{\rm
in},z)$ vanish. It allows us to perform integration over $z$:

$\int\limits_{-L}^{L}\\!dz\,\mathrm{e}^{i(p_{\parallel}+q-p^{\prime}_{\parallel})z}=\frac{2\sin(p_{\parallel}+q-p_{\parallel}^{\prime})L}{p_{\parallel}+q-p_{\parallel}^{\prime}}.$
(26)

The results proved to be independent of $L$ and $\xi_{\text{max}}$, provided
they are sufficiently large. One obtains

$C_{\bm{p}^{\prime},s^{\prime}}^{(+)}=\frac{1}{2\pi^{3/2}}\,i^{-l}\mathrm{e}^{il\varphi_{p^{\prime}}}\int\limits_{-\infty}^{+\infty}dq\,f(q)\,\mathrm{e}^{i\varepsilon^{\prime}t_{\text{in}}}(p_{\perp})^{-1/2}\delta(p_{\perp}-p_{\perp}^{\prime})\,\frac{\sin(p_{\parallel}+q-p_{\parallel}^{\prime})L}{p_{\parallel}+q-p_{\parallel}^{\prime}}\,\sum_{k=-1}^{+1}\mathrm{e}^{ik\varphi_{p^{\prime}}}u^{\dagger}(\bm{p}^{\prime},s^{\prime})a_{k}.$
(27)

Here $A_{k}$ involve $p_{\parallel}+q$. The wavefunction at arbitrary time
instant $t$ can then be evaluated via

$\psi_{\bm{p},s}(t,\bm{x})=\frac{i^{-l}\sqrt{p_{\perp}}}{2\pi^{3/2}}\sum_{s^{\prime}}\int\limits_{-\infty}^{+\infty}\\!dp_{\parallel}^{\prime}\int\limits_{0}^{2\pi}\\!d\varphi_{p^{\prime}}\,\mathrm{e}^{il\varphi_{p^{\prime}}}\int\limits_{-\infty}^{+\infty}dqf(q)\,\mathrm{e}^{i\varepsilon^{\prime}t_{\text{in}}}\,\frac{\sin(p_{\parallel}+q-p_{\parallel}^{\prime})L}{p_{\parallel}+q-p_{\parallel}^{\prime}}\sum_{k=-1}^{+1}\mathrm{e}^{ik\varphi_{p^{\prime}}}[u^{\dagger}(\bm{p}^{\prime},s^{\prime})a_{k}]\varphi^{(+)}_{\bm{p}^{\prime},s^{\prime}}(t,\bm{x}).$
(28)

In this expression,
$\bm{p}^{\prime}=(p_{\perp}\cos\varphi_{p^{\prime}},p_{\perp}\sin\varphi_{p^{\prime}},p_{\parallel}^{\prime})$.

##  V Results

Here we implement the procedure described in Sec. IV and examine various
observable quantities concerning the structure and dynamics of twisted WPs
interacting with finite laser pulses. We start our analysis from the following
set of parameters. The laser field parameters are $I\approx 1.3\times 10^{13}$
W/cm2, $a=9$, and $\omega=0.242$ a.u. The electron state corresponds to $l=3$,
$\theta_{0}=\pi/4$, and kinetic energy $E_{\rm kin}=817.4$ keV. These
parameters are chosen as in Ref. Hayrapetyan  _et al._ (2014). The WP width in
momentum space is $\sigma=$ 10 a.u. Note that in Ref. Hayrapetyan  _et al._
(2014) the Gaussian envelope function is involved in the vector potential,
while here we use it in the electric field itself [see Eq. (2)], which allows
us to construct unipolar laser pulses, $\mathcal{A}_{0}\neq 0$.

###  V.1 Overall dynamics

In order to analyze the twisted WP dynamics, we integrate the electron
probability density $|\psi_{\bm{p},s}(t,\bm{x})|^{2}$ over $z$ at various time
instants $t$ and plot the corresponding snapshots in the $xy$ plane. As the
external field does not exert a force along the $y$ axis, we observe
displacements only in the $x$ direction. Several snapshots are displayed in
Fig. 2. Although the WP moves as a whole along the $x$ axis, its ring
structure remains the same Hayrapetyan  _et al._ (2014). However, this is not
always the case as demonstrated in the following example. In Fig. 3 we present
the analogous results for higher frequency and amplitude of the laser pulse:
$\omega=4.84$ a.u. and $I\approx 2.1\times 10^{18}$ W/cm2. In this case, the
spatial extent of the twisted WP exceeds the laser wavelength, so different
“slices” perpendicular to the $z$ axis interact with different parts of the
laser pulse and thus have different $x$ displacements. Accordingly, the
overall structure of the WP blurs or even disintegrates. As will be discussed
later, this effect leads also to large uncertainties in the angular momentum
of the twisted electron. Note that the parameters chosen in Figs. 2 and 3 lead
to the same value of the classical nonrelativistic oscillation amplitude, so
the $x$ displacements do not differ much for these two field configurations.

Figure 2: Snapshots of the twisted WP density integrated over $z$ at different
$t$. The $x$ coordinate corresponds to the position of the WP center. The
laser field and WP parameters are $I\approx 1.3\times 10^{13}$ W/cm2, $a=9$,
$\omega=0.242$ a.u., $l=3$, $\theta_{0}=\pi/4$, and $E_{\rm kin}=817.4$ keV.
Figure 3: Snapshots of the twisted WP density integrated over $z$ at different
$t$. The $x$ coordinate corresponds to the position of the WP center. The
laser frequency is $\omega=4.84$ a.u. and intensity is $I\approx 2.1\times
10^{18}$ W/cm2. The rest parameters are the same as in Fig. 2.

We have not yet explored the WP dynamics with respect to the $z$ axis, nor
have we specified the $x(t)$ dependence. It turns out that great insights can
be obtained from the analysis of classical trajectories. This will be
discussed next.

###  V.2 Comparison with classical motion

Twisted electron states do not possess well-defined momentum projections along
the $x$ and $y$ axes as they are characterized only by the magnitude
$p_{\perp}$ of the transverse momentum component. It suggests that the
dynamics of a twisted electron cannot be traced by solving the classical
equations of motion. Here we will evolve classical trajectories with various
initial directions of the transverse momentum component and compare them with
the results of our quantum simulations based on solving the Dirac equation. As
will be seen, the crucial parameter here is the electric-field area of the
laser pulse:

$S_{E}=\int\limits_{-\infty}^{+\infty}E_{x}(t,z)dt=-\frac{1}{c}\,\mathcal{A}_{0}.$
(29)

The classical equations of motion can be solved by following the same
procedure as was outlined in Ref. Aleksandrov  _et al._ (2020), where the
external field was assumed to be monochromatic. Let $\bm{p}_{0}$ be the
initial momentum of the electron. Then the final momentum projections read

$\displaystyle p_{x}$ $\displaystyle=$ $\displaystyle p_{0x}-S_{E},$ (30)
$\displaystyle p_{y}$ $\displaystyle=$ $\displaystyle p_{0y},$ (31)
$\displaystyle p_{z}$ $\displaystyle=$ $\displaystyle
p_{0z}-\frac{S_{E}}{\varepsilon_{0}/c-p_{0z}}\,\Big{(}p_{0x}+\frac{1}{2}\,S_{E}\Big{)},$
(32)

where $\varepsilon_{0}=c\sqrt{c^{2}+\bm{p}_{0}^{2}}$. Although the $z$
component of the momentum of the twisted WP (23) approximately corresponds to
$p_{\parallel}$, the $x$ and $y$ projections are completely unknown and only
obey $p_{0x}^{2}+p_{0y}^{2}=p_{\perp}^{2}$. Let us introduce angle
$\varphi_{p_{0}}$:
$\bm{p}_{0}=(p_{\perp}\cos\varphi_{p_{0}},p_{\perp}\sin\varphi_{p_{0}},p_{\parallel})$.
Changing $\varphi_{p_{0}}$ we obtain a set of classical trajectories, none of
which is supposed to follow the mean values of the momentum and coordinate
operators within the quantum computations. However, we will also average the
results over $\varphi_{p_{0}}$ in order to approximately predict the quantum
behavior. Our goal is to examine the accuracy of this approach although it
completely neglects the effects of quantum interference.

In Fig. 4 we display the mean values of the $x$ coordinate of the electron WP
as a function of $t$ for the parameters used in Fig. 2. One observes that by
averaging the classical predictions over $\varphi_{p_{0}}$, one obtains a very
accurate approximation of the results of quantum simulations. Although the
classical trajectories are extremely sensitive to the value of
$\varphi_{p_{0}}$, for $\varphi_{p_{0}}=\pi/2$ the results are also close to
the quantum mean. Note, however, that in the case $\varphi_{p_{0}}=\pi/2$, the
$y$ coordinate tends to infinity with increasing $t$, while the actual
position of the electron WP always corresponds to $y=0$. This suggests that
one has indeed to average the classical predictions. Moreover, the classical
trajectories turn out to be extremely sensitive to the value of
$\varphi_{p_{0}}$ (see the results for $\varphi_{p_{0}}=\pi/2+\delta$ in Fig.
4).

Figure 4: Comparison between the mean value of the $x$ coordinate of the
twisted WP and the classical predictions for various directions
$\varphi_{p_{0}}$ of the initial transverse momentum. Here $\delta=-4\times
10^{-6}$. The electron and laser parameters are the same as in Fig. 2.

In Fig. 5 we show that the mean classical trajectory is also very close to the
quantum mean of the $z$ coordinate. Here we use different parameters to make
the curve nontrivial and thus more evidently demonstrate the efficiency of the
classical approach. Finally, we note that classical trajectories are also used
to guess a proper position of the spatial box within quantum calculations.

Figure 5: Comparison between the mean value of the $z$ coordinate of the
twisted WP and the classical predictions averaged over the direction
$\varphi_{p_{0}}$ of the initial transverse momentum. The laser field and WP
parameters are $I\approx 3.5\times 10^{18}$ W/cm2, $a=9$, $\omega=0.15$ a.u.,
$l=3$, $\theta_{0}=11.3^{\circ}$, and $E_{\rm kin}=0.014$ keV. Figure 6: Local
current density of a twisted electron integrated over the polar angle
$\varphi$ and $z$ coordinate at $t=t_{\text{in}}$. The electron state with
$m=l+s=-1/2$ is constructed from the Bessel beams (16) [labeled as (B), solid
lines] and from the “rotated” functions (34) and (35) [labeled as (R), dashed
lines]. In the latter case, the current density can have negative values. The
momentum components are $p_{\perp}=p_{\parallel}$ = 10 a.u.

###  V.3 Current density

Now we turn to the analysis of the local current density along the $z$ axis in
the absence of the external field. As was demonstrated in Refs. Kosheleva  _et
al._ (2018); Groshev  _et al._ (2020), in the case of Bessel beams, this
quantity may be negative at certain positions in space, while the wavepacket
itself travels with a positive speed in the $z$ direction (previously, the
analogous finding in the case of twisted light was reported in Ref. Surzhykov
_et al._ (2016)). However, the local current values in the states (16) are
given by

$\displaystyle j_{z}(\bm{x})$ $\displaystyle=$
$\displaystyle\psi^{(\text{B})\dagger}_{l,p_{\parallel},p_{\perp},s}(\bm{x})\alpha_{z}\psi^{(\text{B})}_{l,p_{\parallel},p_{\perp},s}(\bm{x})$
(33) $\displaystyle=$
$\displaystyle\frac{p_{\perp}}{4\pi^{2}}\frac{pc}{\varepsilon}\cos\theta_{0}J^{2}_{l}(p_{\perp}\rho),$

where $p=\sqrt{p_{\perp}^{2}+p_{\parallel}^{2}}$, so the current is always
positive. It turns out that one may indeed obtain negative values of $j_{z}$
if one constructs a linear combination of two Bessel beams (16) with given
$m=l+s$ and $s=\pm 1/2$. The functions used in Refs. Kosheleva  _et al._
(2018); Groshev  _et al._ (2020) are exactly those combinations (see Appendix
B).

Let us construct wavefunctions
$\psi^{(\text{R})}_{m,p_{\parallel},p_{\perp},\mu}$ according to

$\displaystyle\psi^{(\text{R})}_{m,p_{\parallel},p_{\perp},1/2}(\bm{x})$
$\displaystyle=$
$\displaystyle\cos\frac{\theta_{0}}{2}\psi^{(\text{B})}_{m-1/2,p_{\parallel},p_{\perp},1/2}(\bm{x})$
(34) $\displaystyle+$ $\displaystyle
i\sin\frac{\theta_{0}}{2}\psi^{(\text{B})}_{m+1/2,p_{\parallel},p_{\perp},-1/2}(\bm{x}),$
$\displaystyle\psi^{(\text{R})}_{m,p_{\parallel},p_{\perp},-1/2}(\bm{x})$
$\displaystyle=$ $\displaystyle
i\sin\frac{\theta_{0}}{2}\psi^{(\text{B})}_{m-1/2,p_{\parallel},p_{\perp},1/2}(\bm{x})$
(35) $\displaystyle+$
$\displaystyle\cos\frac{\theta_{0}}{2}\psi^{(\text{B})}_{m+1/2,p_{\parallel},p_{\perp},-1/2}(\bm{x}).$

We distinguish between the functions (34) and (35) by means of the quantum
number $\mu$, which determines the helicity of the electron beam Serbo  _et
al._ (2015). Instead of Eq. (33), it yields now

$\displaystyle\tilde{j}_{z}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{p_{\perp}}{4\pi^{2}}\frac{pc}{\varepsilon}\bigg{\\{}\big{[}\cos(\theta_{0}/2)J_{m-\mu}(p_{\perp}\rho)\big{]}^{2}$
(36) $\displaystyle-$
$\displaystyle\big{[}\sin(\theta_{0}/2)J_{m+\mu}(p_{\perp}\rho)\big{]}^{2}\bigg{\\}},$

which coincides, e.g., with Eq. (13) from Ref. Kosheleva  _et al._ (2018) up
to a common normalization factor $1/(2\pi p_{\perp})$. We perform computations
with the twisted WP (23) consisting of the Bessel beams (16) and with a
similar combination of the “rotated” beams (34) and (35). The results for
$p_{\perp}=p_{\parallel}$ = 10 a.u., and $m=-1/2$ at $t=t_{\text{in}}$ are
given in Fig. 6, where we present the local current density integrated over
the polar angle $\varphi$ and over the $z$ coordinate. We observe that the
local current density can indeed be negative in the case of the “rotated”
wavefunctions (dashed lines). Finally, we note that the sum of two currents in
Eq. (36) with different signs of $\mu$ precisely corresponds to the sum of
currents (33) for given $m$ and $l=m\pm 1/2$, i.e., the sum of the solid lines
in Fig. 6 equals the sum of the dashed ones.

Figure 7: Mean values of the $z$ projection of the total angular momentum as a
function of the field area $S_{E}$ (left) and the dispersion $DJ_{z}$ (right).
The parameters are: $I\approx 3.5\times 10^{16}$ W/cm2, $a=0.9$, $\omega=0.15$
a.u., $l=3$, $\theta_{0}=11.3^{\circ}$, and $E_{\rm kin}=1.41$ keV.

###  V.4 Total angular momentum and its dispersion

The twisted WP (23) involves Bessel beams with given numbers $l$ and $s$,
which means that the electron wavefunction has initially a well-defined value
of the $z$ projection of the total angular momentum $m=l+s$. Here we will
evaluate the uncertainty of this number which arises due to the interaction
with the external laser field. To this end, we will compute a mean value of
the operator $J_{z}=L_{z}+S_{z}$ and its dispersion (standard deviation),

$DJ_{z}=\sqrt{\langle J^{2}_{z}\rangle-\langle J_{z}\rangle^{2}}.$ (37)

In the case of an individual Bessel beam
$\psi^{(\text{B})}_{l,p_{\parallel},p_{\perp},s}$, one obtains

$\displaystyle\langle L_{z}\rangle$ $\displaystyle=$ $\displaystyle
l+s(1-c^{2}/\varepsilon)\sin^{2}\theta_{0},$ (38) $\displaystyle\langle
S_{z}\rangle$ $\displaystyle=$ $\displaystyle
s-s(1-c^{2}/\varepsilon)\sin^{2}\theta_{0},$ (39) $\displaystyle\langle
J_{z}\rangle$ $\displaystyle=$ $\displaystyle l+s=m,$ (40) $\displaystyle
DJ_{z}$ $\displaystyle=$ $\displaystyle 0.$ (41)

Combining Bessel beams according to Eq. (23) and evolving the WP in time, we
find out that the interaction with the laser pulse can lead to large values of
$DJ_{z}$. In Fig. 7 we present $\langle J_{z}\rangle$ and $DJ_{z}$ at the
final time instant $t=t_{\text{out}}$ as a function of the electric-field area
$S_{E}$ for the following parameters: $I\approx 3.5\times 10^{16}$ W/cm2,
$a=0.9$, $\omega=0.15$ a.u., $l=3$, $\theta_{0}=11.3^{\circ}$, and $E_{\rm
kin}=1.41$ keV. We change the field area by means of the CEP parameter $\phi$
[see Eq. (5)]. The mean value of the total angular momentum remains almost the
same even for large $S_{E}$ but there is still a certain deviation from $3.5$
a.u. due to the changes in the spin projection $\langle S_{z}\rangle$
Aleksandrov  _et al._ (2020). More important, the dispersion turns out to be
significant. This effect can be explained if one recalls that the $z$ slices
of the electron wavefunction can have very different displacements along the
$x$ direction. While it does not alter the mean value $\langle J_{z}\rangle$
since $\langle y\rangle=0$ and $\langle p_{y}\rangle=0$, the dispersion
$DJ_{z}$ increases with the relative displacement among different $z$ slices,
which, in turn, is proportional to $S_{E}$. This suggests that a highly
unipolar finite laser pulse can substantially damage the vortex structure of
the electron state.

##  VI Conclusion

In this study, the interaction of a twisted wavepacket with a linearly
polarized laser pulse was investigated within relativistic quantum mechanics.
The process was simulated taking into account the finite size of the both
objects. A special focus was placed on the dynamics of the electron state and
evolution of its various characteristics. It was demonstrated that the
coordinates of the center of the wavepacket can be accurately determined by
averaging over the classical trajectories corresponding to different
orientations of the initial momentum of the particle. This observation is
expected to be highly beneficial as it allows one to obtain valuable
predictions without performing quantum calculations, i.e., without solving the
Dirac equation. The analysis of classical trajectories was also extremely
helpful in interpreting other effects revealed in our study. Namely, it was
shown that the vortex structure of the electronic state can be destroyed by
highly unipolar laser pulses. This was found by a direct inspection of the
ring structure of the wavepacket and also by computing the dispersion of the
total angular momentum, which proved to rapidly increase with the pulse area.
Besides, it was demonstrated that the local current density can have negative
values if the Bessel beams are properly combined within the electron
wavefunction.

##  Appendix A Normalization of Bessel beams and integration over transverse
coordinates

Here we will show that the Bessel beams (16) are normalized according to Eq.
(19). Although the presence of $\delta(p_{\parallel}-p^{\prime}_{\parallel})$
in Eq. (19) is obvious, let us briefly discuss how the second delta-function
should appear. According to Eq. (18), the wavefunction contains three terms
with Bessel functions of different orders. First, we note that the integration
over $\varphi$ makes all of the terms with two different orders vanish, so one
has to consider only “diagonal terms”. The corresponding integral over $\rho$
reads

$\int\limits_{0}^{\infty}\\!d\rho\rho
J_{l}(p_{\perp}\rho)J_{l}(p^{\prime}_{\perp}\rho).$ (42)

Calculating this integral with a finite upper limit, one can show that

$\int\limits_{0}^{R}\\!d\rho\rho
J_{l}(p_{\perp}\rho)J_{l}(p^{\prime}_{\perp}\rho)\underset{R\rightarrow\infty}{\longrightarrow}\frac{1}{p_{\perp}}\delta(p_{\perp}-p^{\prime}_{\perp}),$
(43)

provided $p_{\perp},p^{\prime}_{\perp}>0$. Combining all the terms, we obtain
Eq. (19).

The relation (43) is also useful in calculating the integral over $x$ and $y$
in Eq. (20). According to Eq. (17), there are three terms. Let us consider
that containing $a_{k}$:

$\displaystyle\mathcal{I}_{k}$ $\displaystyle\equiv$ $\displaystyle
a_{k}i^{k}\int\limits_{-\infty}^{\infty}\\!dx\\!\int\limits_{-\infty}^{\infty}\\!dy\,\mathrm{e}^{-i\zeta(p^{\prime}_{x}x+p^{\prime}_{y}y)}\mathrm{e}^{i(l+k)\varphi}J_{l+k}(p_{\perp}\rho)$
(44) $\displaystyle=$ $\displaystyle
a_{k}i^{k}\int\limits_{0}^{\infty}\\!d\rho\rho\int\limits_{0}^{2\pi}\\!d\varphi\,\mathrm{e}^{[-i\zeta
p^{\prime}_{\perp}\rho\cos{(\varphi_{p^{\prime}}-\varphi)}]}$
$\displaystyle\times$
$\displaystyle\mathrm{e}^{i(l+k)\varphi}J_{l+k}(p_{\perp}\rho).$

To perform integration over $\varphi$, we utilize the following identity
Abramowitz and Stegun (1965):

$\int\limits_{0}^{2\pi}\\!d\varphi\,\mathrm{e}^{iz\cos\varphi}\,\mathrm{e}^{in\varphi}=2\pi
i^{n}J_{n}(z).$ (45)

Thus, one obtains

$\displaystyle\mathcal{I}_{k}$ $\displaystyle=$ $\displaystyle 2\pi
a_{k}i^{2k+l}\mathrm{e}^{i(l+k)\varphi_{p^{\prime}}}\int\limits_{0}^{\infty}\\!d\rho\rho
J_{l+k}(p_{\perp}\rho)J_{l+k}(-\zeta p^{\prime}_{\perp}\rho)$ (46)
$\displaystyle=$ $\displaystyle\frac{2\pi
a_{k}}{p_{\perp}}\,(-1)^{k\delta_{\zeta,-1}}i^{-\zeta
l}\mathrm{e}^{i(l+k)\varphi_{p^{\prime}}}\,\delta(p_{\perp}-p^{\prime}_{\perp}).$

Here, we have used $J_{l}(-z)=(-1)^{l}J_{l}(z)$. It immediately brings us to
Eq. (22).

##  Appendix B Connection with the wavefunctions from Refs. Serbo  _et al._
(2015); Kosheleva  _et al._ (2018); Groshev  _et al._ (2020)

In Refs. Kosheleva  _et al._ (2018); Groshev  _et al._ (2020) the relativistic
wavefunctions of twisted electrons were chosen in the form
$\tilde{\psi}_{m,p_{\parallel},p_{\perp},\mu}$,

$\displaystyle\tilde{\psi}_{m,p_{\parallel},p_{\perp},1/2}(\bm{x})$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\frac{\mathrm{e}^{ip_{\parallel}z}}{\sqrt{2\varepsilon}}\begin{pmatrix}\sqrt{\varepsilon+c^{2}}\cos(\theta_{0}/2)\,\mathrm{e}^{i(m-1/2)\varphi}J_{m-1/2}(p_{\perp}\rho)\\\
i\sqrt{\varepsilon+c^{2}}\sin(\theta_{0}/2)\,\mathrm{e}^{i(m+1/2)\varphi}J_{m+1/2}(p_{\perp}\rho)\\\
\sqrt{\varepsilon-c^{2}}\cos(\theta_{0}/2)\,\mathrm{e}^{i(m-1/2)\varphi}J_{m-1/2}(p_{\perp}\rho)\\\
i\sqrt{\varepsilon-c^{2}}\sin(\theta_{0}/2)\,\mathrm{e}^{i(m+1/2)\varphi}J_{m+1/2}(p_{\perp}\rho)\end{pmatrix},$
(47) $\displaystyle\tilde{\psi}_{m,p_{\parallel},p_{\perp},-1/2}(\bm{x})$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\frac{\mathrm{e}^{ip_{\parallel}z}}{\sqrt{2\varepsilon}}\begin{pmatrix}i\sqrt{\varepsilon+c^{2}}\sin(\theta_{0}/2)\,\mathrm{e}^{i(m-1/2)\varphi}J_{m-1/2}(p_{\perp}\rho)\\\
\sqrt{\varepsilon+c^{2}}\cos(\theta_{0}/2)\,\mathrm{e}^{i(m+1/2)\varphi}J_{m+1/2}(p_{\perp}\rho)\\\
-i\sqrt{\varepsilon-c^{2}}\sin(\theta_{0}/2)\,\mathrm{e}^{i(m-1/2)\varphi}J_{m-1/2}(p_{\perp}\rho)\\\
-\sqrt{\varepsilon-c^{2}}\cos(\theta_{0}/2)\,\mathrm{e}^{i(m+1/2)\varphi}J_{m+1/2}(p_{\perp}\rho)\end{pmatrix},$
(48)

where $m$ is the projection of the total angular momentum as used in the main
text and $\mu$ is an additional quantum number governing the helicity of the
beam Serbo  _et al._ (2015). These solutions are normalized according to

$\langle\tilde{\psi}_{m,p_{\parallel},p_{\perp},\mu}|\tilde{\psi}_{m,p_{\parallel}^{\prime},p_{\perp}^{\prime},\mu^{\prime}}\rangle=\frac{1}{2\pi
p_{\perp}}\,\delta_{\mu\mu^{\prime}}\delta(p_{\parallel}-p_{\parallel}^{\prime})\delta(p_{\perp}-p_{\perp}^{\prime}).$
(49)

For a given value of $m$ and given momentum, Eq. (16) yields two functions
with $s=\pm 1/2$ and $l=m-s$. Each of these functions can be represented as a
combination of the two solutions (47) and (48):

$\begin{pmatrix}\psi^{(\text{B})}_{m-1/2,p_{\parallel},p_{\perp},1/2}\\\
\psi^{(\text{B})}_{m+1/2,p_{\parallel},p_{\perp},-1/2}\end{pmatrix}=\sqrt{2\pi
p_{\perp}}\begin{pmatrix}\cos(\theta_{0}/2)&-i\sin(\theta_{0}/2)\\\
-i\sin(\theta_{0}/2)&\cos(\theta_{0}/2)\end{pmatrix}\begin{pmatrix}\tilde{\psi}_{m,p_{\parallel},p_{\perp},1/2}\\\
\tilde{\psi}_{m,p_{\parallel},p_{\perp},-1/2}\end{pmatrix}.$ (50)

Accordingly, the wavefunctions (47) and (48) coincide with the functions (34)
and (35) up to a factor of $\sqrt{2\pi p_{\perp}}$:

$\tilde{\psi}_{m,p_{\parallel},p_{\perp},\mu}(\bm{x})=\sqrt{2\pi
p_{\perp}}\psi^{(\text{R})}_{m,p_{\parallel},p_{\perp},\mu}(\bm{x}).$ (51)

Finally, we note that in Ref. Serbo  _et al._ (2015) twisted electron states
were described by the wavefunctions which differ from those in Eqs. (47) and
(48) only in an overall numerical factor.

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# Stability of Certain Higher Degree Polynomials

Shanta Laishram, Ritumoni Sarma  Stat Math Unit, Indian Statistical Institute,
New Delhi 110016, India<EMAIL_ADDRESS>Department of Mathematics, Indian
Institute of Technology Delhi, New Delhi-110016, India<EMAIL_ADDRESS>and  Himanshu Sharma  Department of Mathematics, Indian Institute of
Technology Delhi, New Delhi-110016, India<EMAIL_ADDRESS>
###### Abstract.

One of the interesting problems in arithmetic dynamics is to study the
stability of polynomials over a field. In this paper, we study the stability
of $f(z)=z^{d}+\frac{1}{c}$ for $d\geq 2$, $c\in{\mathbb{Z}\setminus\\{0\\}}$.
We show that for infinite families of $d\geq 3$, whenever $f(z)$ is
irreducible, all its iterates are irreducible, that is, $f(z)$ is stable. For
$c\equiv 1\pmod{4}$, we show that all the iterates of $z^{2}+\frac{1}{c}$ are
irreducible. Also we show that for $d=3$, if $f(z)$ is reducible, then the
number of irreducible factors of each iterate of $f(z)$ is exactly $2$ for
$|c|\leq{10^{12}}$.

###### Key words and phrases:

Stable, Eventually stable, non-Archimedean valuation

##  1\. Introduction

An important question in the field of arithmetic dynamics is to study the
recurrence sequences satisfying $t_{n}=f(t_{n-1})$, where
$t_{0}\in{\mathbb{Q}}$ and $f(z)\in{\mathbb{Q}[z]}$ with $\deg(f(z))\geq{2}$.
One can ask how many primes are there in the sequence $(t_{n})$ or which
primes are dividing at least one element of the sequence $(t_{n}).$ Many
authors have investigated these questions in [[19],[15],[12],[10],[9]]. One
interesting problem in this direction is about the stability and eventual
stability of polynomials over a field. In fact stability and eventual
stability have been recently used in proving finite index results for some
arboreal representations in [4] and [5].

If each iterate of $f(z)\in{\mathbb{Q}[z]}$ is irreducible over $\mathbb{Q}$
then we say that $f(z)$ is stable over $\mathbb{Q}$. More generally if the
number of irreducible factors of iterates of $f(z)$ is bounded by a constant,
that is, there exist $n_{0}\in{\mathbb{N}}$ such that the number of
irreducible factors of $f^{n}(z)$ remains constant for $n\geq{n_{0}}$ then we
say that $f(z)$ is eventually stable.

We consider the stability and eventual stability of $z^{d}+b$ with
$b\in\mbox{$\mathbb{Q}$},b\neq 0$. Put $b=\frac{a}{c}$ with
$a,c\in\mbox{$\mathbb{Z}$},c\neq 0$. It was shown in [11, Theorem $1.6$] that
$f(z)=z^{d}+\frac{a}{c}$ is eventually stable when $a\neq{1}$. The stability
and eventual stability of $z^{d}+\frac{1}{c}$ is not known completely. Even
for the quadratic polynomial $z^{2}+\frac{1}{c}$ over $\mathbb{Q}$, it is not
completely known though some partial results are available in [8]. We refer to
[2] for a more detailed survey. In this paper, we consider the stability and
eventual stability of the polynomial
$z^{d}+\frac{1}{c},c\in\mathbb{Z}\setminus\\{0\\}$. We prove the following
results.

######  Theorem 1.

Let $d\geq 2$ be a positive integer and $f(z)=z^{d}+\frac{1}{c}$ where
$c\neq{0}$ is an integer. We have

  1. (a)

Let $d=2$. If $c\equiv 1\pmod{4}$ then each iterate of $f(z)$ is irreducible
over $\mathbb{Q}$.

  2. (b)

Let $d>2$. Assume that $f(z)$ is irreducible. Then $f(z)$ is stable over
$\mathbb{Q}$ when

    1. $1.$

$d\geq 3$ is odd.

    2. $2.$

$d=2^{r}$ with $r\geq 2.$

    3. $3.$

$d=2^{r}\cdot 3^{s}$ with $r,s\geq{1}.$

    4. $4.$

$d=2^{r}\cdot 5^{s}\cdot 7^{t}$ with $r\geq 1,s,t\geq 0$ and $d\equiv
1\pmod{3}$

    5. $5.$

$d\equiv 4\pmod{12}$.

We note that when $d=2$ and $c=-m^{2}$, then $c\equiv 0,3\pmod{4}$ and
$f(z)=z^{2}+\frac{1}{c}$ is reducible. We refer to [8, Theorem 1.3] for other
values of $c$ for which each iterate of $z^{2}+\frac{1}{c}$ is irreducible
over $\mathbb{Q}$. The case $d>3$ odd and not divisible by $3$ follows from
[7, Theorem 7]. From Theorem 1 and the remarks made in [7] and , we believe
that the following conjecture is true.

######  Conjecture 1.

If $f(z)=z^{d}+\frac{1}{c},c\in{\mathbb{Z}\setminus\\{0\\}},d\geq{3}$, is
irreducible over $\mathbb{Q}$ then $f(z)$ is stable over $\mathbb{Q}$.

Though we are not able to completely prove this conjecture unconditionally, by
using an explicit version of $abc$-conjecture due to Baker [1] (see Conjecture
4), we are able to show that the Conjecture 1 is true.

######  Theorem 2.

The explicit $abc-$Conjecture implies Conjecture 1. That is
$f(z)=z^{d}+\frac{1}{c}$ with $c\in{\mathbb{Z}\setminus\\{0\\}}$ and $d\geq 3$
is stable over $\mathbb{Q}$ whenever $f(z)$ is irreducible over $\mathbb{Q}$.

When $f(z)=z^{d}+\frac{1}{c}$ reducible over $\mathbb{Q}$, nothing much is
known about the irreducible factors of iterates of $f(z)$ for $d\geq 3$. In
this paper, we prove the following result about the eventual stability of
$f(z)=z^{3}+\frac{1}{c}$ when it is reducible.

######  Theorem 3.

Let $f(z)=z^{3}+\frac{1}{c},c\in{\mathbb{Z}\setminus\\{0\\}}$. If $f(z)$ is
reducible over $\mathbb{Q}$ then $f^{n}(z)$ has exactly two irreducible
factors over $\mathbb{Q}$ for each $n\in{\mathbb{N}}$ and for $c$ with
$|c|\leq{10^{12}}.$

We believe that above result should be true for all $c$ and we propose the
following conjecture.

######  Conjecture 2.

Let $f(z)=z^{3}+\frac{1}{c},c\in{\mathbb{Z}\setminus\\{0\\}}$. If $f(z)$ is
reducible over $\mathbb{Q}$ then $f^{n}(z)$ has exactly two irreducible
factors over $\mathbb{Q}$ for each $n\in{\mathbb{N}}$ and for all $c$.

We prove Theorems 1 and 2 in Section 4. Theorem 3 is proved in Section 3. In
Section 2, we give the preliminaries.

##  2\. Preliminaries

Let $K$ be a number field, $\phi(z)\in K(z)$ and let
$\alpha\in\mathbb{P}^{1}(K).$ Assume that $\phi(z)=\frac{f(z)}{g(z)}$, where
$f(z)$ and $g(z)$ are coprime in $K[z]$. The degree of a rational function
$\phi(z)=\frac{f(z)}{g(z)}$ is defined as
$\deg{\phi(z)}=\max\\{\deg{f(z)},\deg{g(z)}\\}$. Let
$\phi^{n}(z)=\phi(\phi^{n-1}(z))$, the $n$-th iterate of $\phi(z)$ and put
$\phi^{n}(z)=\frac{f_{n}(z)}{g_{n}(z)}$, where $f_{n}(z),g_{n}(z)\in K[z]$ are
coprime polynomials.

######  Definition 4.

The pair $(\phi,\alpha)$ is said to be stable over $K$ if $f_{n}(z)-\alpha
g_{n}(z)$ is irreducible over $K$ for each $n\in{\mathbb{N}}$.

However stability is not preserved under field extensions. A weaker condition
called eventual stability behaves well with respect to finite field
extensions.

######  Definition 5.

If there exist a constant $C(\phi,\alpha)$ such that the number of irreducible
factors of $f_{n}(z)-\alpha g_{n}(z)$, for all $n\geq{1}$, is bounded by
$C(\phi,\alpha)$ then we say that ($\phi$, $\alpha$) is eventually stable over
$K$. If the number of irreducible factors of $g_{n}(z)$ is similarly bounded
then we say that $(\phi,\infty)$ is eventually stable.

If $(\phi,0)$ is eventually stable then we say that $\phi(z)$ is eventually
stable. Eventual stability of a rational function over $K$ is preserved over
finite extensions of $K$.

A map $\nu:K\to\mathbb{Z}\cup\\{\infty\\}$ is called a discrete valuation on
$K$ if it satisfies the following properties:

  * (1)

$\nu(x)=\infty\Leftrightarrow x=0$,

  * (2)

$\nu(xy)=\nu(x)+\nu(y)~{}~{}\forall\,x,~{}y\in K$ and

  * (3)

$\nu(x+y)\geq\inf\\{\nu(x),\nu(y)\\}$.

Suppose $p$ is a rational prime. Then every rational number $x$ can be written
as $p^{t}x_{0}$ where $t\in\mathbb{Z}$ and $p$ divides neither the numerator
nor the denominator of $x_{0}$. Then,
$\nu_{p}:\mathbb{Q}\setminus\\{0\\}\to\mathbb{Z}$ defined by $\nu_{p}(x)=t$ is
a discrete valuation which is called the $p$-adic valuation on $\mathbb{Q}$.

Given a discrete valuation $\nu$ on the number field $K$, $R$ denote the
following ring $\\{x\in K:\nu(x)\geq{0}\\}$. Then $\mathfrak{p}=\\{x\in
K:\nu(x)>0\\}$ is the unique maximal ideal of the ring $R$. The field
$k=R/\mathfrak{p}$ is called the residue field of $\nu$. Denote by
$\tilde{x}\in{\mathbb{P}^{1}(\kappa)}$ the reduction modulo $\mathfrak{p}$ of
$x\in\mathbb{P}^{1}(K)$. Further we denote by $\tilde{f}(z)$ the polynomial
obtained from $f(z)\in R[z]$ by reducing each coefficient modulo
$\mathfrak{p}$.

Let $\phi(z)=\frac{f(z)}{g(z)}\in{K(z)}$ where $f(z),g(z)\in{R[z]}$. Then we
say that $\frac{f(z)}{g(z)}$ is normalized if $f(z),g(z)\in{R[z]}$ are coprime
and atleast one of the coefficients of $f(z)$ or $g(z)$ is a unit in $R$.
Define $\tilde{\phi}(z)=\frac{\tilde{f}(z)}{\tilde{g}(z)}$. Suppose $\phi(z)$
is non constant. Then $\phi(z)\in{K(z)}$ is said to have good reduction at
$\nu$ if $\deg(\tilde{\phi}(z))=\deg(\phi(z))$. We say that $\phi(z)$ is
bijective on residue extensions for the discrete valuation $\nu$ on $K$ if
$\tilde{\phi}(z)$ defines a bijection on $\mathbb{P}^{1}(E)$ for every finite
extension $E$ of the residue field $\kappa$ of $\nu$.

######  Proposition 2.1.

[13] Let $K$ be a number field with discrete valuation $\nu$,
$\phi(z)\in{K(z)}$ be such that $\deg{\phi(z)}\geq{2}$ and let
$\alpha\in{\mathbb{P}^{1}(K)}$ be non-periodic with respect to $\phi(z)$.
Assume $\phi(z)$ is bijective on residue extensions for $\nu$ and has good
reduction at $\nu$. Assume further $\phi^{n}(z)=\frac{f_{n}(z)}{g_{n}(z)}$ is
normalized and $\alpha\neq{\infty}$. Then the number of irreducible factors of
$f_{n}(z)-\alpha g_{n}(z)$ over $K$ is at most

  * 1.

$\nu(\phi(\alpha)^{-1}-\alpha^{-1})$ if $\nu(\alpha)<0$,

  * 2.

$\nu(\phi^{i}(\alpha)-\alpha)$ if $\nu(\alpha)\geq{0}$ for
$i=\min\\{{n\geq{1}:\tilde{\phi}^{n}(\tilde{\alpha})=\tilde{\alpha}}\\}.$

######  Lemma 2.1.

[8] Suppose $g(x)\in{K[x]}$ is a monic irreducible polynomial with degree
$d\geq{1}$ and $char(K)\neq 2$. Let $f(x)\in K[x]$ be a monic quadratic
polynomial and $\gamma$ be such that $f^{\prime}(\gamma)=0$. If no element of

$\\{(-1)^{d}g(f(\gamma))\\}\cup\\{g(f^{n}(\gamma)):n\geq{2}\\}$

is a square in K, then $g(f^{n}(x))$ is irreducible over $K$ for each
$n\in{\mathbb{N}}$.

We will use the following result for $d=3$ in the proof of Theorem 3 which
deals with the irreducible factors of the iterates of $z^{3}+\frac{1}{c}$ when
it is reducible.

######  Lemma 2.2 (Capelli’s Lemma, [14]).

Suppose $f(x),g(x)$ are polynomials over the field $K$ such that $g(x)$ is
irreducible. Then $g(f(x))$ is irreducible over $K$ if and only if
$f(x)-\beta$ is irreducible over $K(\beta)$ for every root
$\beta\in{\overline{K}}$ of $g(x)$.

######  Proposition 2.2.

[7] Let $n\geq{3}$ be an odd integer and let
$f(x)=x^{n}-b\in{\mbox{$\mathbb{Q}$}[x]}$. For each $m\geq{1}$, let
$S(n,m)=\\{b\in{\mbox{$\mathbb{Q}$}}:f^{m}\,\text{is irreducible
but}\,f^{m+1}\,\text{is reducible}\,\text{over}\,\,{\mbox{$\mathbb{Q}$}}\\}.$
Then $S(n)=\cup_{m=1}^{\infty}S(n,m)$ is finite and is empty if $3\nmid n$.

The above result was proved by using Lemma 2.2 and the non-existence of
primitive solutions of the generalized Fermat equation. Let $p,q,r$ be
integers $\geq 2$ and consider the generalized Fermat equation

(2.1) $x^{p}+y^{q}={z^{r}}.$

Given a triple $(a,b,c)\in{\mathbb{Z}}^{3}$, we say that $(a,b,c)$ is a
solution of equation $(2.1)$ if $a^{p}+b^{q}=c^{r}$. If $a,b,c$ are pairwise
coprime then this solution is called ${\it proper}$. A proper solution
$(a,b,c)$ is _primitive_ if $abc\neq 0$. A well-known conjecture regarding the
generalized Fermat equations is due to Tijdeman and Zagier (see [16]), also
known as Beal’s conjecture.

######  Conjecture 3.

There are no primitive solutions of the diophantine equation

$x^{p}+y^{q}=z^{r}$

in ${\mathbb{Z}}$ for $p,q,r\geq 3.$

This is open. However there are a number of partial results on this
conjecture. For the proof of our theorems, we need the following result on the
non-existence of primitive solutions of the following equations, see [3].

######  Lemma 2.3.

There are no primitive solutions for the equation $x^{p}+y^{q}=z^{r}$ when

$(p,q,r)\in\\{(2,3,7),(2,3,10),(2,n,4),(2,n,6),(3,3,2n),(3,n,6),(n,n,2),(n,n,3)\\}$

where $n\geq 2$ and further $n\geq 3$ when $(p,q,r)\in\\{(3,n,6),(n,n,3)\\}$;
$n\geq{4}$ when $(p,q,r)\in\\{(2,n,6),(n,n,2)\\}$ and $n\geq{6}$ when
$(p,q,r)=(2,n,4)$.

The next result is a Catalan’s conjecture, now a theorem of
Mih$\breve{a}$ilescu [18].

######  Lemma 2.4.

The only solution of $x^{m}-y^{n}=1$ in integers $x,y,m>1,n>1$ with $xy\neq 0$
is given $3^{2}-2^{3}=1$.

We end with this section with the explicit abc-conjecture due to Baker [1]. If
$m$ is a positive integer, then its radical $N(m)$ is the product of distinct
prime divisors of $m$ and $\omega(m)$ denotes the number of distinct primes
dividing $m$.

######  Conjecture 4.

[Explicit abc-conjecture] Let $a,b$ and $c$ be pairwise coprime integers
satisfying $a+b=c.$ Then

$c<\frac{6}{5}N\frac{(\log N)^{\omega}}{\omega!}$

where $N=N(abc)$ and $\omega=\omega(N).$

An easily applicable formulation was given by Laishram and Shorey [16, Theorem
1]. The next result is contained in [16, Theorem 1].

######  Proposition 2.3.

Assume Conjecture 4. Let $a,b$ and $c$ be pairwise coprime integers such that
$a+b=c.$ Then

$c<N^{1+\frac{3}{4}}\qquad{\rm where}\quad N=N(abc).$

Further for $0<\epsilon\leq{\frac{3}{4}}$, there exist $N_{\epsilon}$,
depending only on $\epsilon$, such that whenever $N\geq{N_{\epsilon}}$, we
have $c<N^{1+\epsilon}$. In particular, for $\epsilon=\frac{7}{12}$,
$N_{\epsilon}=\exp(204.75)$.

Explicit $abc-$Conjecture can be used to give a general result on the
Conjecture 3, see [16, Theorem 3].

######  Proposition 2.4.

Assume Conjecture 4. Then there are no primitive solutions of the equation
$x^{p}+y^{q}=z^{r}$ with $p\geq 3,q\geq 3,r\geq 3$ and

$[p,q,r]\notin\\{[3,5,\ell]:7\leq\ell\leq 23,\ell\ {\rm
prime}\\}\cup\\{[3,4,\ell]:\ell\ {\rm prime}\\}$

where $[p,q,r]$ denote all the permutations of ordered triples $(p,q,r)$.

##  3\. Proof of Theorem 3

We start this section with a lemma which is analogous to Lemma 2.1.

######  Lemma 3.1.

Let $g(z)$ be a monic irreducible polynomial over a number field $K$ and let
$f(z)=z^{3}+\frac{1}{c}\in{K[z]}$. If none of
$\Big{\\{}g(f^{n}(0))\Big{\\}}_{n\geq{1}}$ is a cube in $K$ then $g(f^{n}(z))$
is irreducible for each $n\in{\mbox{$\mathbb{N}$}}$.

###### Proof.

We prove it by induction on $n$. For $n=0,~{}~{}g(z)$ is given to be
irreducible. Assume now irreducibility of $g(f^{n-1}(z))$. Then by Capelli’s
Lemma 2.2, $g(f^{n}(z))$ is irreducible over $K$ if and only if
$z^{3}+1/c-\gamma$ is irreducible over $K(\gamma)$, that is, $1/c-\gamma$ is
not a cube in $K(\gamma)$, for every root $\gamma$ of $g(f^{n-1})(z)$. Note
that

$\displaystyle N_{K(\gamma)/K}\left(\frac{1}{c}-\gamma\right)$
$\displaystyle=\prod_{\alpha~{}~{}\text{is root
of}~{}~{}g(f^{n-1}(z))}\left(\frac{1}{c}-\alpha\right)=g\left(f^{n-1}\left(\frac{1}{c}\right)\right)=g(f^{(n)}(0)).$

If $N_{K(\gamma)/K}(1/c-\gamma)$ is not a cube in $K$ then $1/c-\gamma$ is not
a cube in $K(\gamma)$. Thus, if $g(f^{(n)}(0))$ is not a cube in $K$ then
$z^{3}+1/c-\gamma$ is irreducible over $K(\gamma)$ so that $g(f^{n}(z))$ is
irreducible over $K$ for each $n\in{\mbox{$\mathbb{N}$}}.$ ∎

######  Lemma 3.2.

Let $f(z)=z^{d}+\frac{1}{c}$ with $d$ an odd integer. For any prime $p$
dividing $d$, if $f^{n}(0)$ is not a $p$-th power in $\mathbb{Q}$ for $c>0$
then it is also not a $p$-th power for $c<0$ in $\mathbb{Q}$. Additionally, If
$d=3$ then similar result holds for $g(f^{n}(0))$ for any divisor $g(z)$ of
$f(z)$.

###### Proof.

Let $f_{c}(z)=z^{d}+\frac{1}{c}$ with $d$ an odd integer. By induction, we
have $f^{n}_{-c}(0)=-f^{n}_{c}(0)$ for all $n\geq{1}$. Therefore
$f^{n}_{-c}(0)$ is a $p$-th power if and only if $f^{n}_{c}(0)$ is a $p$-th
power in $\mathbb{Q}$. Suppose $d=3$, $f_{c}(z)$ is reducible and $g(z)|f(z)$.
It is also easy to observe that
$g_{-c}(f^{n}_{-c}(0))=\pm{g_{c}(f^{n}_{c}(0))}$. Hence the proposition
follows. ∎

Let $f(z)=z^{3}+\frac{1}{c}$. We know that $f(z)$ is reducible if and only if
$c=m^{3}$ for some $m\in{\mathbb{Z}}.$ In that case
$f(z)=(z+\frac{1}{m})(z^{2}-\frac{z}{m}+\frac{1}{m^{2}})$. Set

(3.1) $g_{1}(z)=z+\frac{1}{m}\quad{\rm and}\quad
g_{2}(z)=z^{2}-\frac{z}{m}+\frac{1}{m^{2}}.$

Since $g_{1}(f(z))=z^{3}+1/m^{3}+1/m=z^{3}+\frac{m^{2}+1}{m^{3}}$,
$g_{1}(f(z))$ is reducible if and only if $m^{2}+1$ is a cube in $\mathbb{Z}.$
By Lemma 2.4, $m^{2}+1$ is not a cube unless $m=0$. Hence $g_{1}(f(z))$ is
irreducible for all $m\neq{0}.$ For considering the irreducible factors of
iterates of $f(z)$, we require the following lemma for which the proof is
similar to that of [14, Proposition 5.4].

######  Lemma 3.3.

Let $f(z)=z^{3}+\frac{1}{c}$ be reducible with the irreducible factors
$g_{1}(z)$ and $g_{2}(z)$ as in (3.1). Then $(g_{i}(f^{n}(0))),n\geq{1},$ is a
rigid divisibility sequence for $i=1,2$.

###  3.1. Proof of Theorem 3:

Let $f(z)=z^{3}+\frac{1}{c}$ be reducible. Then $c=m^{3}$ for some $0\neq
m\in{\mathbb{Z}}$ and write $f(z)=g_{1}(z)g_{2}(z)$, where $g_{1}(z),g_{2}(z)$
are given by (3.1). Let $w_{n}(m)$ be the numerator of $g_{1}(f^{n-1}(0))$, so
that $w_{2}(m)$ is the numerator of $g_{1}(f(0))$. We observe that
$w_{2}(m)=m^{2}+1$ is not a cube in $\mathbb{Z}$ by Lemma 2.4. By Lemma 3.3,
it follows that $({w_{n}(m)})$ is a rigid divisibility sequence. Hence
$w_{2n}(m)$ is not a cube in $\mathbb{Z}$ for each $n\geq{1}$.

For a given $k\in{\mathbb{N}}$ and $m\not\equiv 0\pmod{k}$, the sequence
$(g_{1}(f^{n}(0))\pmod{k}$ eventually becomes a repeating cycle and we search
for values of $k$ and congruence classes of $m$ modulo $k$ such that
$(g_{1}(f^{n}(0))\textrm{mod}\ k)$ is not a cube for each
$n\in{\mbox{$\mathbb{N}$}}$. Since we have shown above that $w_{2j}$ is not a
cube in $\mathbb{Z}$ for each $j\geq{1}$, that is, $g_{1}(f^{2j-1}(0)$ is not
a cube in $\mathbb{Q}$, it is enough to check that
$(g_{1}(f^{2j}(0))\textrm{mod}\ k)$ is not a cube for $j\geq{1}$. In fact,
with the help of SAGE we verify for each $j\in{\mbox{$\mathbb{N}$}}$ that
$(g_{1}(f^{2j}(0))$ is not a cube for $m$ belonging to the congruences classes
modulo $k\in\\{7,13,19,31,37,43\\}$ that are listed in Table 1.

$k$ | $m\pmod{k}$
---|---
$7$ | $m\equiv\pm 1,\pm 3$
$13$ | $m\equiv\pm 1,\pm 2,\pm 3,\pm 6$
$19$ | $m\equiv\pm 2,\pm 4$
$31$ | $m\equiv\pm 1,\pm 3,\pm 4,\pm 6,\pm 8,\pm 9,\pm 10,\pm 11,\pm 12$
$37$ | $m\equiv\pm 3,\pm 9,\pm 17$
$43$ | $m\equiv\pm 2,\pm 5,\pm 8,\pm 10,\pm 12,\pm 13,\pm 14,\pm 15,\pm 20$
Table 1.

One can verify that the congruence classes in Table 1 cover all integers
belonging to the interval $[1,10^{4}]$ except for 267 of them. For each of the
remaining integers, we could find a prime $p<150$ such that
$g_{1}(f^{n}(0)\pmod{p}$ is not a cube for every even integer $n$. For
example, $p=73$ works for $m=4342$. Hence by Lemma 3.1, $g_{1}(f^{n}(z)$ is
irreducible for all $n\geq{1}$ when $1\leq{m}\leq{10^{4}}$.

Let $x_{n}(m)$ be the numerator of $g_{2}(f^{n-1}(0))$. In particular
$x_{2}(m)$ is the numerator of $g_{2}(f(0)).$ It again follows from Lemma 3.3
that $(x_{n}(m))$ is a rigid divisibility sequence. Since
$g_{2}(f(0))=\frac{m^{4}-m^{2}+1}{m^{6}}$, we have $x_{2}(m)=m^{4}-m^{2}+1$.
Now $x_{2}(m)$ is a cube if and only if the elliptic curve $y^{2}-y+1=x^{3}$
has integral point with $y=m^{2}$. It follows from the curve $243.a1$ in LMFDB
[6] that $y^{2}-y+1=x^{3}$ has only integral points
$(1,1),(7,19),(1,0),(7,-18)$. Since $m\neq{0}$, $m^{4}-m^{2}+1$ is a cube in
$\mathbb{Q}$ for $m=\pm{1}$ only, that is, $x_{2}(m)$ is not a cube in
$\mathbb{Z}$ for $|m|\geq{2}$. Let $m=1$. In this case $f(z)=z^{3}+1$ and
$g_{2}(z)=z^{2}-z+1$, we verify that $z^{6}+z^{3}+1=g_{2}(f(z))$ is
irreducible over $\mathbb{Q}$. Further we checked that
$g_{2}(f^{n}(0))\pmod{7}$ is not a cube for each $n\geq{2}$. Hence by Lemma
3.1, $g_{2}(f^{n}(z))$ is irreducible for all $n\in{\mbox{$\mathbb{N}$}}.$
Similarly we verified that $g_{2}(f^{n}(z))$ is irreducible for each
$n\in{\mbox{$\mathbb{N}$}}$ when $m=-1$. Hence we take $|m|\geq 2$. Since
$(x_{n}(m))$ is a rigid divisibility sequence and $x_{2}(m)$ is not a cube,
for $n\geq 1$, $x_{2n}(m)$ is also not a cube in $\mathbb{Z}.$ Again with the
help of SAGE we verify for each $j\in{\mbox{$\mathbb{N}$}}$ that
$(g_{1}(f^{2j}(0))$ is not a cube for $m$ belonging to the congruences classes
modulo $k\in\\{7,13,19,31,37\\}$ that are listed in Table 2.

$k$ | $m\pmod{k}$
---|---
$7$ | $m\equiv\pm 1,\pm 2,\pm 3$
$13$ | $m\equiv\pm 1,\pm 2,\pm 3,\pm 4,\pm 6$
$19$ | $m\equiv\pm 3,\pm 5$
$31$ | $m\equiv\pm 1,\pm 4,\pm 7,\pm 8,\pm 9,\pm 11,\pm 14$
$37$ | $m\equiv\pm 4,\pm 7,\pm 9,\pm 12,\pm 16,\pm 17,\pm 18$
Table 2.

One can verify that the congruence classes in Table 2. cover all integers
belonging to the interval $[1,10^{4}]$ except for $88$ of them. For each of
the remaining integers, we could find a prime $p<150$ such that
$g_{1}(f^{n}(0)\pmod{p}$ is not a cube for every even integer $n$. For
example, $p=67$ works for $m=2730$. Hence $(g_{2}(f^{n}(z))$ is irreducible
for each $n\in{\mbox{$\mathbb{N}$}}$ by Lemma 3.1. This proves Theorem 3.

##  4\. Proof of Theorems 1 and 2

For the proof of Theorems 1 and 2, we need the following lemma.

######  Lemma 4.1.

Let $g(z)$ be a monic irreducible polynomial over a number field $K$ and let
$f(z)=z^{d}+\frac{1}{c}\in{K[z]}$. Then for $m\geq 1$, $g(f^{m}(z))$ is
irreducible if $g(f^{m-1}(z))$ is irreducible and ${g(f^{m}(0))}$ is not a
$p$-th power in $K$ for each prime $p$ dividing $d$. Hence for $m\geq 1$,
$g(f^{m}(z))$ is irreducible if ${g(f^{j}(0))}$ is not a $p$-th power in $K$
for each prime $p$ dividing $d$ and for each $1\leq j\leq m$.

###### Proof.

Assume $g(f^{m-1}(z))$ be irreducible. By Lemma 2.2 we have $g(f^{m}(z))$ is
irreducible if and only if for every root $\beta$ of $g(f^{m-1}(z))$,
$f(z)-\beta=z^{d}+1/c-\beta$ is irreducible over $K({\beta)}$. By [17, Theorem
$9.1$], $f(z)-\beta$ is irreducible if for every prime $p$ dividing $d$ we
have $\beta-1/c\notin{K(\beta)}^{p}$ and if $4|d$ then
$\beta-1/c\notin{-4K(\beta)}^{4}$. We know that
$\beta-\frac{1}{c}\notin{K(\beta)}^{p}$ for a prime $p$ if
$N_{K(\beta)/K}(\beta-\frac{1}{c})\notin{K}^{p}$. Note that

$\displaystyle N_{K(\beta)/K}\left(\beta-\frac{1}{c}\right)$
$\displaystyle=\prod_{\text{roots}~{}\alpha\ {\rm of}\
g(f^{m-1}(z))}\left(\alpha-\frac{1}{c}\right)=(-1)^{t}g(\left(f^{m-1}\left(\frac{1}{c}\right)\right)=(-1)^{t}g(f^{(m)}(0))$

where $t=\deg\,(g(f^{m-1}(z)))$. Hence if $g(f^{(m)}(0))$ is not a $p$-th
power in $K$ then $\beta-1/c\notin{K(\beta)}^{p}$. Now let $4|d.$ Since
$g(f^{(m)}(0))$ is not a square in $K$ and $4|t$, we have from
$N_{K(\beta)/K}(1/c-\beta)=g(f^{(m)}(0))$ that $1/c-\beta$ is not a square and
hence $1/c-\beta\notin{4K(\beta)}^{4}$. Hence we conclude that if
$(g(f^{m}))(0)$ is not a $p$-th power in $K$ for each prime $p$ dividing $d$
and $g(f^{m-1}(z))$ is irreducible, then $g(f^{m}(z))$ is irreducible. This
proves the first assertion. The latter follows inductively from the first
assertion. ∎

Apply Lemma 4.1 to $K=\mbox{$\mathbb{Q}$}$ and $f(z)=g(z)=z^{d}+\frac{1}{c}$
whenever it is irreducible over $\mathbb{Q}$. Let $a_{n}$ denote the numerator
of $f^{n}(0)$. Then $a_{n}$ satisfies $a_{1}=1$ and for $n>1$,

$a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}\qquad{\rm and}\qquad
f^{n}(0)=\frac{a_{n}}{c^{d^{n-1}}}.$

By induction, $a_{n},a_{n+1}$ and $c$ are pairwise coprime for
$n\in{\mathbb{N}}$. Note that for any prime $p$ dividing $d$, the denominator
of $f^{n}(0)$ is always of the form $(c^{e})^{p}$ for some positive integer
$e$. So, for a prime $p$ that divides $d$, $f^{n}(0)$ is not a $p$-th power in
$\mathbb{Q}$ unless $a_{n}$ is a $p$-th power in $\mathbb{Z}$. Let $n=2$. Then
$a_{1}=1$ implying $a_{2}=1+c^{d-1}$. By Lemma 2.4, $a_{2}$ is a power only
when $c=2,d=4$ in which case $a_{2}=3^{2}$. By Lemma 4.1, $f^{2}(z)$ is
irreducible for all $(d,c)\neq(4,2)$. For $(d,c)=(4,2)$, we check that
$f^{2}(z)=(x^{4}+\frac{1}{2})^{4}+\frac{1}{2}$ is irreducible over
$\mathbb{Q}$. Thus to prove Theorem 1 it is enough to consider $n\geq 3$.

###  4.1. Proof of Theorem 1:

Let $f(z)=z^{d}+b$. As in the statement of Proposition 2.2, let $S(d,m)$ be
the set of $b\in{\mathbb{Q}}$ for which $f^{m}(z)$ is irreducible but
$f^{m+1}(z)$ is not and set $S(d)=\cup_{m=1}^{\infty}S(d,m)$. By Proposition
2.2, $S(d)=\emptyset$ for all $d$ not divisible by $2$ or $3$. In other words,
for $d$ not divisible by $2$ or $3$, if $f(z)=z^{d}+\frac{1}{c}$ is
irreducible then all its iterates are also irreducible. Hence for the proof of
Theorem 1, we may suppose that $2|d$ or $3|d$.

Let $d=2$. We apply Proposition 2.1 to $K=\mathbb{Q}$,
$\phi(z)=f(z)=z^{2}+\dfrac{1}{c}$ and $\nu=\nu_{2},$ the $2$-adic valuation on
$\mathbb{Q}$. Here the corresponding residue field is $\mathbb{F}_{2}$. $f$
has good reduction when $c$ is odd in which case $\tilde{f}(z)=z^{2}+1$. Let
$c$ be odd. Then $\tilde{f}(0)=1,\tilde{f}(1)=0$ so that the period $i=2$ by
Proposition 2.1. That is $f^{n}(z)$ has at most $\nu_{2}(f^{2}(0))$
irreducible factors. Now $f(0)=\frac{1}{c},f^{2}(0)=\frac{1+c}{c^{2}}$ so that
$\nu_{2}(f^{2}(0))=\nu_{2}(1+c)$. When $c\equiv 1\pmod{4}$, we have
$\nu_{2}(1+c)=1$ so that all the iterates $f^{n}(z)$ are irreducible over
$\mathbb{Q}$. This proves $(a)$.

Now suppose $d\geq 3$. As stated after the proof of Lemma 4.1, we need to show
for $n\geq 3$ that for no prime $p$ dividing $d$ $a_{n}$ is a $p$-th power in
$\mathbb{Z}$ . We consider various cases of (b) one by one.

Let $d$ be an odd integer divisible by $3$. Then $d^{n-1}-1$ is even and
$a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}=x_{1}^{3}+y_{1}^{2k}$ and if $p\neq 3$ is a
prime dividing $d$ then we also have
$a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}=x_{2}^{p}+y_{2}^{2}$ for some integers
$x_{1},x_{2},y_{1},y_{2},k$ with $k\geq 3$. For each $n\geq{3}$, $a_{n}$ is
not a $p$-th power for any prime divisor of $d$ because if $a_{n}$ is a cube
then $a_{n}+(-x_{1})^{3}=y_{1}^{2k}$ and if $a_{n}$ is a $p$-th power for any
odd prime $p\neq{3}$ dividing $d$ then $a_{n}+(-x_{2})^{p}=y_{2}^{2}$ which
are not possible by Lemma 2.3.

Let $d=2^{r}$ with $r\geq 2$. Then $f(z)=z^{2^{r}}+\frac{1}{c}$ and
$a_{n}=a_{n-1}^{2^{r}}+c^{(2^{r})^{n-1}-1},\,n\geq{3}$. We need to show for
each $n\geq 3$ that $a_{n}$ is not a square in $\mathbb{Z}$. If $a_{n}=y^{2}$
for some $y$, then taking $k=(2^{r})^{n-1}-1>6$ odd, we get an equation
$y^{2}+(-c)^{k}=(a_{n}^{2^{r-2}})^{4}$ which has no primitive solution by
Lemma 2.3.

Let $d=2^{r}\cdot 3^{s}$ with $r,s\geq 1$. Then $6|d$ and hence
$a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}=x^{6}+y^{k}$ for some integers $x,y,k$ with
odd $k\geq 5$. Considering equations $a_{n}+(-y)^{k}=x^{6}$ if $a_{n}$ is a
square and $a_{n}+(-y)^{k}=x^{6}$ if $a_{n}$ is a cube, we get a contradiction
by Lemma 2.3. Hence $a_{n}$ is neither a square nor a cube for $n\geq 3$.

Let $d=2^{r}\cdot 5^{s}\cdot 7^{t}$ with $r\geq 1,s,t\geq{0}$ and $d\equiv
1\pmod{3}$. We assume that either $s>0$ or $t>0$ since $d=2^{r}$ is already
considered. If $a_{n}=z^{p}$ for a prime $p|d$, we have

$a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}=\begin{cases}x_{1}^{10}+y^{3}\implies
z^{2}+(-y)^{3}=x^{10}_{1}&{\rm if}\ s>0\ {\rm and}\ a_{n}=z^{2}\\\
x_{3}^{7}+y^{3}\implies z^{2}+(-y)^{3}=x_{3}^{7}&{\rm if}\ s=0\ {\rm and}\
a_{n}=z^{2}\\\ x_{2}^{5}+y^{3}\implies z^{5}+(-x_{2})^{5}=y^{3}&{\rm if}\ s>0\
{\rm and}\ a_{n}=z^{5}\\\ x_{3}^{7}+y^{3}\implies
x^{7}_{3}+(-z)^{7}=x_{3}^{7}&{\rm if}\ t>0\ {\rm and}\
a_{n}=z^{7}.\end{cases}$

We get a contradiction by Lemma 2.3.

Let $d\equiv 4\pmod{12}$ with $d>4$. Then $3|(d-1)$ and $4|d$ and we have

$a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}=x^{4}_{1}+y^{k}_{1}=x^{p}_{2}+y^{3}_{2}$

some integers $x_{1},x_{2},y_{1},y_{2},k,p$ with odd $k\geq 6$ and for any odd
prime $p|d$. Considering equations $a_{n}+(-y_{1})^{k}=x^{4}_{1}$ if $a_{n}$
is a square and $a_{n}+(-x)^{p}=y^{3}_{2}$ if $a_{n}$ is a $p-$th power for an
odd prime $p|d$, we get a contradiction by Lemma 2.3. Hence $a_{n}$ is not a
$p-$the power for any prime divisor of $d$. This proves Theorem 1.

###  4.2. Proof of Theorem 2:

We may assume, by Theorem 1, that $d$ is an even integer which is not
considered in Theorem 1. As stated after the proof of Lemma 4.1 we need aWe
already noted , that it is enough to show that for $n\geq 3$, $a_{n}$ is not a
$p$-th power in $\mathbb{Z}$ for any prime $p$ dividing $d$. Let $p|d$ and
suppose that $a_{n}=z^{p}$ for some integer $z$. Then

$z^{p}=a_{n}=a_{n-1}^{d}+c^{d^{n-1}-1}=x^{p}+y^{k}$

for some integers $x,y,k$ with odd $k\geq 5$.

From now on, we assume explict abc Conjecture 4 . By Proposition 2.4,
$z^{p}+(-x)^{p}=y^{k}$ has no primitive solutions when $p$ is odd. Hence we
assume that $p=2$ and therefore we have $z^{2}=a^{d}_{n-1}+c^{d^{n-1}-1}$. It
suffices to show that this equation is not possible. We have
$c<z^{\frac{2}{d^{n-1}-1}}$ and $a_{n-1}<z^{\frac{2}{d}}$ so that

$N=N(z^{2}\cdot a^{d}_{n-1}\cdot c^{d^{n-1}-1})=N(z\cdot a_{n-1}\cdot
c)<z^{1+\frac{2}{d}+\frac{2}{d^{(n-1)}-1}}$

where $N(r)$ is the radical of $r$. By Theorem 2.3 applied to
$a^{d}_{n-1}+c^{d^{n-1}-1}=z^{2}$, we obtain $z^{2}<N^{\frac{7}{4}}$ implying

$\displaystyle(z^{2})^{\frac{4}{7}}<z^{1+\frac{2}{d}+\frac{2}{d^{n-1}-1}}\implies\frac{8}{7}<1+\frac{2}{d}+\frac{2}{d^{n-1}-1}\implies\frac{1}{14}<\frac{1}{d}+\frac{1}{d^{2}-1}.$

since $n\geq 3$. Clearly $d>14$ so that $d^{2}-1>2d$ and hence

$\frac{1}{14}<\frac{1}{d}+\frac{1}{2d}=\frac{3}{2d}\implies d<21.$

By Theorem 1, we need to consider $d=14$ where we have the equation
$x^{2}=a_{n}=a^{14}_{n-1}+c^{14^{n-1}-1}$. For $n$ odd, considering the
equation $x^{2}+(-y)^{3}=(a^{2}_{n-1})^{7}$ where $y^{3}=c^{14^{n-1}-1}$, we
get a contradiction by Lemma 2.3. Hence we assume that $n$ is even. In
particular $n\geq 4$. Also $c=\pm 1$ is not possible by Lemma 2.4 since
$a_{n-1}\neq 0,\pm 1$. Thus $|c|\geq 2$ and $n\geq 4$. We apply Theorem 2.3 to
$a^{14}_{n-1}+c^{14^{n-1}-1}=z^{2}$. If

$N=N(z^{2}\cdot a^{14}_{n-1}\cdot c^{14^{n-1}-1})<\exp(204.75),$

we obtain

$2^{14^{3}-1}\leq|c|^{14^{n-1}-1}\leq
N^{\frac{7}{4}}\leq\exp(204.75\cdot\frac{7}{4})$

which is a contradiction. Hence $N\geq\exp(204.75)$. Taking
$\epsilon=\frac{7}{12}$ in Theorem 2.3 again, we obtain
$z^{2}<N^{1+\frac{7}{12}}$ implying

$\displaystyle(z^{2})^{\frac{12}{19}}<N\leq
z^{1+\frac{2}{14}+\frac{2}{14^{n-1}-1}}\implies\frac{24}{19}<1+\frac{1}{7}+\frac{2}{14^{3}-1}<1+\frac{1}{7}+\frac{2}{19}$

since $n\geq 4$. This is a contradiction again. This proves Theorem 2.

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