# Vacuum instability in QED with an asymmetric $x$-step. New example of
exactly solvable case

A. I. Breev<EMAIL_ADDRESS>Department of Theoretical Physics, Tomsk State
University Novosobornaya Sq. 1, 634050 Tomsk, Russia  S. P. Gavrilov
<EMAIL_ADDRESS><EMAIL_ADDRESS>Herzen State Pedagogical
University of Russia, Moyka embankment 48, 191186, St. Petersburg, Russia  D.
M. Gitman<EMAIL_ADDRESS>P.N. Lebedev Physical Institute, 53
Leninskiy ave., 119991 Moscow, Russia;  Institute of Physics, University of
São Paulo, Rua do Matão, 1371, CEP 05508-090, São Paulo, SP, Brazil.

###### Abstract

We present a new exactly solvable case in strong-field $QED$ with one-
dimensional step potential ($x$-step). The corresponding $x$-step is given by
an analytic asymmetric with respect to the axis $x$ reflection function. The
step can be considered as a certain analytic "deformation" of the symmetric
Sauter field. Moreover, it can be treated as a new regularization of the Klein
step field. We study the vacuum instability caused by this $x$-step in the
framework of a nonperturbative approach to strong-field $QED$. Exact solutions
of the Dirac equation used in the corresponding nonperturbative calculations,
are represented in the form of stationary plane waves with special left and
right asymptotics and identified as components of initial and final wave
packets of particles. We show that in spite of the fact that the symmetry with
respect to positive and negative bands of energies is broken, distribution of
created pairs and other physical quantities can be expressed via elementary
functions. We consider the processes of transmission and reflection in the
ranges of the stable vacuum and study physical quantities specifying the
vacuum instability. We find the differential mean numbers of electron-positron
pairs created from the vacuum, the components of current density and energy-
momentum tensor of the created electrons and positrons leaving the area of the
strong field under consideration. Besides, we study the particular case of the
particle creation due to a weakly inhomogeneous electric field and obtain
explicitly the total number, the current density and energy-momentum tensor of
created particles. Unlike the symmetric case of the Sauter field the
asymmetric form of the field under consideration causes the energy density and
longitudinal pressure of created electrons to be not equal to the energy
density and longitudinal pressure of created positrons.

Pair creation, Schwinger effect, Dirac equation, exact solutions
Mathematics Subject Classification 2010: 81Q05, 81V10, 33C50

###### pacs:

03.65.Pm, 12.20.-m, 23.20.Ra

##  I Introduction

The Schwinger effect, that is, creation of charged particles from the vacuum
by strong external electric-like and gravitational fields (the vacuum
instability) has been attracting attention already for a long time; see, e.g.,
monographs BirDav82 ; GMR85 ; GriMaM ; FGS . This is a nonperturbative effect
of quantum field theory ($QFT$), which has not yet received a convincing
experimental confirmation. However, recent progress in laser physics allows
one to hope that the vacuum instability will be experimentally observed in the
near future even in laboratory conditions. Recently, this, as well as the real
possibility of observing an analogue of the Schwinger effect in condensed
matter physics (in the graphene, topological insulators, 3D Dirac and Weyl
semimetals, antiferromagnets, etc.) has increased theoretical interest in the
problem and led to the development of various analytical and numeric
approaches, see recent reviews VafVis14 ; GelTan16 ; Adv-QED22 ; Progr23 .
From general quantum theory point of view, the most clear formulation of the
problem of particle production from the vacuum by external fields is
formulated for time-dependent external electric fields that are switched on
and off at infinitely remote times $t\rightarrow\pm\infty$, respectively. The
idealized problem statement described above was considered for uniform time-
dependent external electric fields. Such kind of external fields are called
the $t$-electric potential steps ($t$-steps). A complete nonperturbative with
respect to the external background formulation of strong-field $QED$ with such
external fields was developed in Refs. FGS ; Gitma77 ; it is based on the
existence of exact solutions of the Dirac equation with time dependent
external field (more exactly, complete sets of exact solutions). However,
there exist many physically interesting situations in high-energy physics,
astrophysics, and condensed matter where external backgrounds formally are
time-independent. In our works; see Refs. GavGi16 ; GavGi20 ; BGavGi23 , a
nonperturbative approach in $QED$ with the so-called $x$-potential steps, or
simply $x$-steps, was developed. The $x$-steps represent time-independent
inhomogeneous electric-like external fields of a constant direction. The
latter approach is based on the existence of special exact solutions of the
Dirac or Klein-Gordon equations with corresponding $x$-steps. In cases when
such solutions can be found and all the calculations can be done analytically,
we refer to these cases as to exactly solvable ones. Sauter potential and the
Klein step, considered in the pioneer works Klein27 ; Sauter31a ; Sauter-pot ,
belong to the class of exactly solvable cases. Initially they were considered
in the framework of the relativistic quantum mechanics, which gave rise to a
rather long-lasting discussion about the Klein paradox (a detailed historical
review can be found in Refs. DomCal99 ; HansRavn81 ). In the work GavGi16  it
was pointed out that this paradox and other misunderstandings in considering
quantum effects in fields of strong $x$-steps can be consistently solved as
many particle effects of the $QFT$ ($QED$) with unstable vacuum. Recently, a
number of new exactly solvable cases were presented and studied in detail in
the framework of general approach GavGi16 ; GavGi20 . Particularly,
interesting are the cases of a constant electric field between two capacitor
plates ($L$-constant electric field) GavGit16b , a field of a piecewise form
of continuous exponential functions GavGitSh17 , and a piecewise and a
continuous configuration of an inverse-square step AdoGavGit20 . Exactly
solvable cases are interesting not only in themselves, but also due to the
fact that they allow you to develop and test new approximate and numerical
methods for calculating quantum effects in strong-field $QFT$. One can find a
number of application of these exactly solvable cases in high-energy physics
and condensed matter physics; see, e.g., GavGit13 ; AbrZub16 ; AdoHeGG21 ;
AdoGavGit23 .

In this article, we present a new exactly solvable case for strong-field $QED$
with $x$-step. For the generality, the field is considered in $d=D+1$ \-
dimensional Minkowski space-time, parametrized by the coordinates
$X=\left(t,\mathbf{r}\right)$,
$\mathbf{r}=\left(x^{1}=x,\mathbf{r}_{\perp}\right)$,
$\mathbf{r}_{\perp}=x^{2},\ldots,x^{D}$. The electric field is constant and
has only one component along the $x$-axis,
$\mathbf{E}\left(X\right)=\left(E^{1}\left(x\right)=E\left(x\right),0,...,0\right)$.
The field is given by a step potential $A_{0}(x)$, so that
$E\left(x\right)=-A_{0}^{\prime}\left(x\right)$.

We note that among the above exactly solvable cases only the Sauter electric
field is given by an analytic function,

$\displaystyle
A_{0}^{(\mathrm{Sauter})}(x;L,E_{S})=-LE_{S}\tanh\left(x/L\right),$
$\displaystyle
E_{(\mathrm{Sauter})}(x;L,E_{S})=E_{S}\cosh^{-2}\left(x/L\right),\;E_{S}>0,\;L>0.$
(1)

This field reaches its maximum value at $x=0$ and is symmetric with respect to
the origin. Unlike the above mentioned cases given by piecewise smooth
x-steps, physical quantities calculated for the analytic Sauter field are
presented by elementary functions, which makes this case especially convenient
for physical interpretations. Here we present a new example of exactly
solvable case in which the external field is given by the following analytic
function:

$\displaystyle A_{0}(x)=\frac{\sigma
E_{0}}{\sqrt{1+\exp\left(\frac{x}{\sigma}\right)}},\quad E_{0}>0,\ \sigma>0,\
$ $\displaystyle
E(x)=\frac{E_{0}}{8}\sqrt{1+\exp\left(\frac{x}{\sigma}\right)}\cosh^{-2}\left(\frac{x}{2\sigma}\right)\
.$ (2)

The potential energy of an electron (with the charge $q=-e,$ $e>0$) is
$U\left(x\right)=-eA_{0}\left(x\right)$. It tends to different in the general
case constants values $U\left(-\infty\right)$ and $U\left(+\infty\right)$ as
$x\rightarrow-\infty$ and $x\rightarrow+\infty$, respectively,

$U\left(-\infty\right)\equiv U_{\mathrm{L}}=-eE_{0}\sigma,\ \
U\left(+\infty\right)\equiv U_{\mathrm{R}}=0\,.$ (3)

The magnitude $\delta U$ of the potential step is given by the difference

$\delta U=U_{\mathrm{R}}-U_{\mathrm{L}}=eE_{0}\sigma$ (4)

Note $\delta U$ is equal to the increment of kinetic energy, if the particle
retains the direction of motion and moves in the direction of acceleration,
and if toward the opposite, then this increment changes sign. Depending on the
magnitude $\delta U$, the step is called noncritical or critical one, see
GavGi16 ,

$\begin{array}[]{l}\delta U<\delta U_{c}=2m\,,\text{ {noncritical step}}\\\
\delta U>\delta U_{c},\text{ \ \ \ \ \ \ \ \ \ {critical step}}\end{array}\,.$
(5)

If the magnitude $\delta U$ is large enough, the particle production from the
vacuum could be essential.

The electric field (2) differs from the Sauter field (1) at $L=2\sigma$ by the
presence of an additional term in Eq. (2),

$E(x)=\sqrt{1+\exp\left(\frac{x}{\sigma}\right)}E_{(\mathrm{Sauter})}\left(x;2\sigma,\frac{E_{0}}{8}\right).$
(6)

There is no symmetry of the field $E(x)$ with respect to the point
$x_{\mathrm{M}}=\sigma\ln 2$, in which the field has the maximum value
$E_{\max}=E_{0}/(3\sqrt{3})$. While the Sauter field exhibits this symmetry
with respect to the point of its maximum. Moreover, they generally increase in
a similar way, but the Sauter field decreases faster. One can say that field
(2) for a given value of the parameter $\sigma$ is a certain "deformation" of
the Sauter field, which turns on at $x>0$ and turns off at
$x\rightarrow+\infty$. Finally, we note that distributions of created pairs by
the Sauter field are symmetric with respect of the energy $p_{0}$. The latter
symmetry is not inherent in realistic asymmetric fields. We demonstrate that
in the case of the asymmetric analytic field (2), with broken symmetry with
respect to $p_{0}$, the distributions of created pairs and other physical
quantities can be still expressed in terms of elementary functions.

The article is organized as follows: In Sect. II, we construct exact solutions
of the Dirac equation with a new example of $x$-step given by a analytic
asymmetric function. These solutions are presented in the form of stationary
plane waves with special left and right asymptotics and identified as
components of initial and final wave packets of particles and antiparticles.
We find coefficients of mutual decompositions of the initial and final
solutions. In Sec. III, we consider the processes of transmission and
reflection in ranges of the stable vacuum. In Sec. IV, we calculate physical
quantities specifying the vacuum instability. We find differential mean
numbers of electron-positron pairs created from the vacuum, as well as
components of current density and energy-momentum tensor of the created
electrons and positrons leaving the area of the strong external field. In Sec.
V, we consider a particular case of the particle creation due to a weakly
inhomogeneous electric field and obtain explicitly the total number, current
density and energy-momentum tensor of the created particles. A new
regularization of the Klein step is considered in Sec. VI, which is used then
in calculating the corresponding vacuum instability. Section VII contains some
concluding remarks. In Appendix A, we describe briefly basic elements of a
nonperturbative approach to $QED$ with $x$-steps. In Appendix B, we show that
the density of created pairs and the probability of the vacuum to remain a
vacuum obtained from exact formulae for the slowly varying field in the
leading-term approximation are in agreement with results following in the
framework of a locally constant field approximation (LCFA). In Appendix C, we
list some useful properties of hypergeometric functions. We use the system of
units, where $c=\hbar=1$.

##  II Solutions of Dirac equation with asymmetric potential $x$-step

###  II.1 General solution

Let us consider the Dirac equation with a $x$-step in the Hamiltonian form:

$\displaystyle i\partial_{0}\psi\left(X\right)=\hat{H}\psi\left(X\right)\,,$
$\displaystyle\hat{H}=\gamma^{0}\left(-i\gamma^{j}\partial_{j}+m\right)+U\left(x\right)\,,\
\ j=1,\ldots,D\,.$ (7)

The Dirac spinor $\psi\left(X\right)$ has $2^{\left[d/2\right]}$ components,
$\left[d/2\right]$ denotes the integer part of $d/2$, and $\gamma^{\mu}$ are
$2^{\left[d/2\right]}\times 2^{\left[d/2\right]}$ Dirac matrices in $d$
dimensions, $\left[\gamma^{\mu},\gamma^{\nu}\right]_{+}=2\eta^{\mu\nu}\,\,$,
and $U\left(x\right)=-eA_{0}\left(x\right),$ where $A_{0}(x)$ is given by Eq.
(2).

There exist solutions of Eq. (7) in the form of stationary plane waves
propagating along the space-time directions $t$ and $\mathbf{r}_{\perp}$. In
this case the Dirac spinors labeled by quantum numbers $n$ have the form:

$\displaystyle\psi_{n}\left(X\right)=\exp\left(-ip_{0}t+i\mathbf{p}_{\perp}\mathbf{r}_{\perp}\right)\psi_{n}\left(x\right)\,,\
\ n=\left(p_{0},\mathbf{p}_{\perp},\sigma\right)\,,$
$\displaystyle\psi_{n}\left(x\right)=\left\\{\gamma^{0}\left[p_{0}-U\left(x\right)\right]+i\gamma^{1}\partial_{x}-\bm{\gamma}_{\perp}\mathbf{p}_{\perp}+m\right\\}\varphi_{n}\left(x\right)v_{\chi,\sigma}\,,$
(8)

where the spinors $\psi_{n}\left(x\right)$ and the scalar functions
$\varphi_{n}\left(x\right)$ depend exclusively on $x$ while $v_{\chi,\sigma}$
is a set of constant orthonormalized spinors, satisfying the following
conditions:

$\displaystyle\gamma^{0}\gamma^{1}v_{\chi,\sigma}$ $\displaystyle=$
$\displaystyle\chi v_{\chi,\sigma},\ \
v_{\chi,\sigma}^{{\dagger}}v_{\chi^{\prime},\sigma^{\prime}}=\delta_{\chi,\chi^{\prime}}\delta_{\sigma,\sigma^{\prime}}\
,$ $\displaystyle\chi$ $\displaystyle=$ $\displaystyle\pm 1,\ \
\sigma=\left(\sigma_{s}=\pm 1\,,\ s=1,2,...,\left[d/2\right]-1\right)$ (9)

Quantum numbers $s$ and $\chi$ describe the spin polarization (if $d\leq 3$
there are no spin degrees of freedom that are described by the quantum numbers
$s$). Solutions $\psi_{n}^{\left(\chi\right)}\left(X\right)$ and
$\psi_{n}^{\left(\chi^{\prime}\right)}\left(X\right)$ given by Eqs. (8) that
differ only by values of $\chi$ are linearly dependent __ if __ $d>3$_._
Therefore, it suffices to work with solutions corresponding to one of possible
values of $\chi$, and sometimes we omit the subscript $\chi$, supposing that
the spin quantum number $\chi$ is fixed in a certain way. Due to the same
reason, there exists, in fact, only $J_{(d)}=2^{[d/2]-1}$ different spin
states (labeled by quantum numbers $\sigma$) for a given set of
$p_{0},\mathbf{p}_{\bot}$. Substituting Eq. (8) into Eq. (7), one finds that
scalar functions $\varphi_{n}\left(x\right)$ obey the following second-order
ordinary differential equation:

$\left\\{\frac{d^{2}}{dx^{2}}+\left[p_{0}-U\left(x\right)\right]^{2}-\pi_{\perp}^{2}+i\chi\partial_{x}U\left(x\right)\right\\}\varphi_{n}\left(x\right)=0\,,\
\pi_{\perp}=\sqrt{\mathbf{p}_{\perp}^{2}+m^{2}}.$ (10)

Solutions of a similar type of equation

$\left\\{\frac{d^{2}}{dt^{2}}+\left[p_{x}-\tilde{U}\left(t\right)\right]^{2}+\pi_{\perp}^{2}-i\tilde{\chi}\partial_{t}\tilde{U}\left(t\right)\right\\}\tilde{\varphi}_{n}\left(t\right)=0,$
(11)

where $\tilde{U}\left(t\right)=-A_{x}\left(t\right)$, were recently found in
Ref. BrGavGitSh21  for the case of a nonperturbative treatment of the vacuum
instability due to __ a time-dependent electric field $E\left(t\right)$, given
by the potential:

$A_{x}\left(t\right)=\frac{\sigma E_{0}}{\sqrt{1+\exp\left(t/\sigma\right)}}\
.$

It is quite obvious that equation (10) can be obtained from equation (11) by a
substitution

$t\rightarrow x,\;p_{x}\rightarrow
p_{0},\;\pi_{\perp}^{2}\rightarrow-\pi_{\perp}^{2},\;\tilde{\chi}\rightarrow-\chi.$

Therefore, solutions of equation (10) can be obtained from solutions of
equation (11) using the same substitution. As the result, the general solution
of Eq. (10) can be represented as a linear combination of the functions
$\varphi_{n,i}\left(x\right)$,

$\displaystyle\varphi_{n,i}\left(x\right)=\
\left.\left(1+z\right)^{i\alpha_{1}}\left(1-z\right)^{i\alpha_{2}}\hat{M}_{n}w_{n,i}\left(\frac{z+1}{2}\right)\right|_{z=z(x)}\
,$ $\displaystyle\hat{M}_{n}=\frac{bz-(\alpha_{1}-\alpha_{2}+2\chi
eE_{0}\sigma^{2})}{\left(ia-1\right)b}\frac{d}{dz}+1\ ,\
z(x)=\sqrt{1+\exp\left(\frac{x}{\sigma}\right)}\ ,$ $\displaystyle
a=\alpha_{1}+\alpha_{2}-\sqrt{2\left(\alpha_{1}^{2}+\alpha_{2}^{2}\right)-(2eE_{0}\sigma^{2})^{2}},\
\
b=\alpha_{1}+\alpha_{2}+\sqrt{2\left(\alpha_{1}^{2}+\alpha_{2}^{2}\right)-(2eE_{0}\sigma^{2})^{2}}\
,$
$\displaystyle\alpha_{1}=\sigma\sqrt{\pi_{\perp}^{2}-(p_{0}-eE_{0}\sigma)^{2}},\
\alpha_{2}=\sigma\sqrt{\pi_{\perp}^{2}-(p_{0}+eE_{0}\sigma)^{2}}\ .$ (12)

In the above combination, we use two pairs of linearly independent solutions
$w_{n,i}\left(\xi\right)$ with additional indices $i=1,\ldots,4$. The first
pair reads:

$\displaystyle
w_{n,1}\left(\xi\right)=\xi^{-(ia-1)}F\left(ia-1,i(a-2\alpha_{1});2i\alpha_{2};1-\xi^{-1}\right)\
,$ $\displaystyle
w_{n,2}\left(\xi\right)=\xi^{i(a-2\alpha_{1})-1}(1-\xi)^{1-2i\alpha_{2}}F\left(i(2\alpha_{1}-a),-ia;2(1-i\alpha_{2});1-\xi^{-1}\right)\
,$ (13)

where $F\left(\alpha,\beta;\gamma;\xi\right)$ are Gaussian hypergeometric
functions Bateman . Solutions (13) are well-defined in a vicinity of the
singular point $\xi=1,$ which corresponds to $x\rightarrow-\infty$. The second
pair reads:

$\displaystyle
w_{n,3}\left(\xi\right)=\left(-\xi\right)^{-(ia-1)}F\left(ia-1,i(a-2\alpha_{1})-1;i(a-b);\xi^{-1}\right)\
,$ $\displaystyle
w_{n,4}\left(\xi\right)=\left(-\xi\right)^{-ib}F\left(ib,i(b-2\alpha_{1})+1;i(b-a)+2;\xi^{-1}\right)\
.$ (14)

Solutions (14) are well-defined in a vicinity of the singular point
$\xi=\infty$, which corresponds to $x\rightarrow+\infty$. Using functions (13)
and (14) one can construct four complete sets $\varphi_{n,i}\left(x\right)$,
$i=1,2,3,4$, of solutions of equation (10).

###  II.2 Solutions with special left and right asymptotics

Unlike the explicitly time-dependent solutions of Eq. (11) the solutions given
by Eqs. (8) and (12) are stationary plane waves. In the treatment of the
vacuum instability in the $x$-case under consideration, they describe
qualitatively different cases depending on the ranges of quantum numbers. By
this reason their role in calculations and interpretation of physics of the
vacuum instability in the $x$-case is quite different compared to similar in
appearance the above cited time-dependent solutions. Further, we carry out a
nonperturbative study of the vacuum instability using solutions (12) within
the framework of the approach formulated in Refs. GavGi16 ; GavGi20 .

Due to local properties of equation (10) at $x\rightarrow\mp\infty$ (where the
electric field is zero), the scalar functions $\varphi_{n}\left(x\right)$ have
definite left “L” and right “R” asymptotics:

$\ {}_{\zeta}\varphi_{n}\left(x\right)=\
_{\zeta}\mathcal{N}e^{i\zeta\left|p^{\mathrm{L}}\right|x}\ \text{{as}}\
x\rightarrow-\infty\,,$ $\ {}^{\zeta}\varphi_{n}\left(x\right)=\
^{\zeta}\mathcal{N}e^{i\zeta\left|p^{\mathrm{R}}\right|x}\text{ {as}}\ \
x\rightarrow+\infty\,.$ (15)

Here ${}_{\zeta}\mathcal{N}$ and$\ {}^{\zeta}\mathcal{N}$ are some
normalization constants, and
$p^{\mathrm{L/R}}=\zeta\left|p^{\mathrm{L/R}}\right|$,
$\zeta=\pm=\mathrm{sgn}\left(p^{\mathrm{L}}\right)=\mathrm{sgn}\left(p^{\mathrm{R}}\right)$,
denotes real asymptotic momenta along the $x$-axis,

$\left|p^{\mathrm{L/R}}\right|=\sqrt{\pi_{0}\left(\mathrm{L/R}\right)^{2}-\pi_{\perp}^{2}},\
\ \pi_{0}\left(\mathrm{L/R}\right)=p_{0}-U_{\mathrm{L/R}}\,,$ (16)

where $U_{\mathrm{L/R}}$ is given by Eq. (3). Then for the corresponding Dirac
spinors the following relations hold:

$\displaystyle\hat{p}_{x}\ {}_{\zeta}\psi_{n}\left(X\right)$ $\displaystyle=$
$\displaystyle\zeta\left|p^{\mathrm{L}}\right|\
_{\zeta}\psi_{n}\left(X\right)\ \text{{as}}\ x\rightarrow-\infty\,,$
$\displaystyle\hat{p}_{x}\ ^{\zeta}\psi_{n}\left(X\right)$ $\displaystyle=$
$\displaystyle\zeta\left|p^{\mathrm{R}}\right|\
^{\zeta}\psi_{n}\left(X\right)\ \text{{as}}\ x\rightarrow+\infty\,.$ (17)

Note that the electric field under consideration can be neglected at
sufficiently big $\left|x\right|$, let say, in the macroscopic regions
$S_{\mathrm{L}}$ on the left of $x=x_{\mathrm{L}}<0$ and $S_{\mathrm{R}}$ on
the right of $x=x_{\mathrm{R}}>0$.__ To this end one can choose finite
$x_{\mathrm{L/R}}$ such that

$1-\frac{U\left(x_{\mathrm{R}}\right)-U\left(x_{\mathrm{L}}\right)}{\delta
U}\ll 1.$ (18)

__ We assume that the asymptotic behavior (17) is sufficiently good
approximation for all the functions $\ {}_{\zeta}\psi_{n}\left(X\right)$ and
${}^{\zeta}\psi_{n}\left(X\right)$ if $x<x_{\mathrm{L}}$ and
$x>x_{\mathrm{R}}$, which means that particles are free in the regions __
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$.

Nontrivial sets of Dirac spinors $\left\\{\
{}_{\zeta}\psi_{n}\left(X\right)\right\\}$ and $\left\\{\
{}^{\zeta}\psi_{n}\left(X\right)\right\\}$, that are key elements of the above
mentioned approach, do exist for the quantum numbers $n$ satisfying the
conditions:

$\pi_{0}\left(\mathrm{L/R}\right)^{2}>\pi_{\perp}^{2}\Leftrightarrow\left\\{\begin{array}[]{l}\pi_{0}\left(\mathrm{L/R}\right)>\pi_{\perp}\\\
\pi_{0}\left(\mathrm{L/R}\right)<-\pi_{\perp}\end{array}\right.\,.$ (19)

Note that $\pi_{0}\left(\mathrm{L}\right)>\pi_{0}\left(\mathrm{R}\right)$. As
a result of these inequalities, the complete set of the quantum numbers $n$
can be divided in some ranges $\Omega_{k}$, where the index $k$ labels the
ranges and the corresponding quantum numbers, $n_{k}\in\Omega_{k}$. For
critical steps, $\delta U>\delta U_{c}$, there are five ranges of the quantum
numbers, $\Omega_{k}$, $k=1,...,5$, where the solutions  __
${}_{\zeta}\psi_{n}\left(X\right)$__ and __
${}^{\zeta}\psi_{n}\left(X\right)$__ have similar __ forms and properties for
given perpendicular momenta $p_{\perp}$ and any spin polarizations $\sigma$_._
The ranges $\Omega_{1}$ and $\Omega_{5}$ are characterized by energies bounded
from the below, $\Omega_{1}=\left\\{n:p_{0}\geq
U_{\mathrm{R}}+\pi_{\perp}\right\\},$ and by energies bounded from the above
$\Omega_{5}=\left\\{n:p_{0}\leq U_{\mathrm{L}}-\pi_{\perp}\right\\}$. The
ranges $\Omega_{2}$ and $\Omega_{4}$ are characterized by bounded energies,
namely
$\Omega_{2}=\left\\{n:U_{\mathrm{R}}-\pi_{\perp}<p_{0}<U_{\mathrm{R}}+\pi_{\perp}\right\\}$
and
$\Omega_{4}=\left\\{n:U_{\mathrm{L}}-\pi_{\perp}<p_{0}<U_{\mathrm{L}}+\pi_{\perp}\right\\}$
if $\delta U\geq 2\pi_{\perp}$ or
$\Omega_{2}=\left\\{n:U_{\mathrm{L}}+\pi_{\perp}<p_{0}<U_{\mathrm{R}}+\pi_{\perp}\right\\}$
and
$\Omega_{4}=\left\\{n:U_{\mathrm{L}}-\pi_{\perp}<p_{0}<U_{\mathrm{R}}-\pi_{\perp}\right\\}$
if $\delta U<2\pi_{\perp}$. In the ranges $\Omega_{2}$ and $\Omega_{4}$ we
deal with standing waves $\psi_{n}\left(X\right)$ completed by linear
superpositions of solutions ${}_{\zeta}\psi_{n}\left(X\right)$ and
${}^{\zeta}\psi_{n}\left(X\right)$ with corresponding longitudinal fluxes that
are equal in magnitude for a given $n$. The range $\Omega_{3}$ is nontrivial
only for critical steps and perpendicular momenta $\mathbf{p}_{\perp}$
restricted by the inequality $2\pi_{\perp}\leq\delta U$. This range is
characterized by bounded energies,
$\Omega_{3}=\left\\{n:U_{\mathrm{L}}+\pi_{\perp}\leq p_{0}\leq
U_{\mathrm{R}}-\pi_{\perp}\right\\}$. For noncritical steps $\delta U<\delta
U_{c}$, the range $\Omega_{3}$ is absent.

Stationary plane waves, ${}_{\zeta}\psi_{n}\left(X\right)$ and
${}^{\zeta}\psi_{n}\left(X\right)$, are subjected to the following
orthonormality conditions on the $x=\mathrm{const}$ hyperplane: __

$\displaystyle\left(\ {}_{\zeta}\psi_{n},\
_{\zeta^{\prime}}\psi_{n^{\prime}}\right)_{x}=\zeta\eta_{\mathrm{L}}\delta_{\zeta,\zeta^{\prime}}\delta_{n,n^{\prime}},\
\ \left(\ {}^{\zeta}\psi_{n},\
^{\zeta^{\prime}}\psi_{n^{\prime}}\right)_{x}=\zeta\eta_{\mathrm{R}}\delta_{\zeta,\zeta^{\prime}}\delta_{n,n^{\prime}};$
$\displaystyle\left(\psi,\psi^{\prime}\right)_{x}=\int\psi^{{\dagger}}\left(X\right)\gamma^{0}\gamma^{1}\psi^{\prime}\left(X\right)dtd\mathbf{r}_{\bot}\
,$ (20)

where $\eta_{\mathrm{L/R}}=\mathrm{sgn\ }\pi_{0}\left(\mathrm{L/R}\right)$ is
sign of $\pi_{0}\left(\mathrm{L/R}\right)$. We consider our theory in a large
space-time box that has a spatial volume
$V_{\bot}=\prod\limits_{j=2}^{D}K_{j}$ and the time dimension $T$, where all
$K_{j}$ and $T$ are macroscopically large. It is supposed that all the
solutions $\psi\left(X\right)$ are periodic under transitions from one box to
another. The integration over the transverse coordinates is fulfilled from
$-K_{j}/2$ to $+K_{j}/2$, and over the time $t$ from $-T/2$ to $+T/2$. Under
these suppositions, one can verify, integrating by parts, that the inner
product (20) does not depend on $x$. We assume that the macroscopic time $T$
is the system surveillance time.

Solutions (12) with the asymptotic conditions (15) have the following form:

$\ {}_{+}\varphi_{n}\left(x\right)=\
_{+}\mathcal{N}U_{n,1}\left(1+z\right)^{i\alpha_{1}}\left(1-z\right)^{i\alpha_{2}}\hat{M}_{n}w_{n,1}\left(\frac{z+1}{2}\right),\
$ $\ {}_{-}\varphi_{n}\left(x\right)=\
_{-}\mathcal{N}U_{n,2}\left(1+z\right)^{i\alpha_{1}}\left(1-z\right)^{i\alpha_{2}}\hat{M}_{n}w_{n,2}\left(\frac{z+1}{2}\right),$
$\ {}^{+}\varphi_{n}\left(x\right)=\
^{+}\mathcal{N}U_{n,3}\left(1+z\right)^{i\alpha_{1}}\left(1-z\right)^{i\alpha_{2}}\hat{M}_{n}w_{n,3}\left(\frac{z+1}{2}\right),$
$\ {}^{-}\varphi_{n}\left(x\right)=\
^{-}\mathcal{N}U_{n,4}\left(1+z\right)^{i\alpha_{1}}\left(1-z\right)^{i\alpha_{2}}\hat{M}_{n}w_{n,4}\left(\frac{z+1}{2}\right),$
(21)

where the constants $U_{n,i}$, $i=1,2,3,4$, and the normalization constants
${}_{\zeta}\mathcal{N}$ and $\ {}^{\zeta}\mathcal{N}$ are:

$\displaystyle
U_{n,1}=\frac{2^{2-i(\alpha_{1}-\alpha_{2})}\alpha_{2}}{a-(\alpha_{1}-\alpha_{2}-2\chi
eE_{0}\sigma^{2})}e^{\pi\alpha_{2}}\ ,$ $\displaystyle
U_{n,2}=\frac{2^{1-i(\alpha_{1}+3\alpha_{2})}b(ia-1)}{(2i\alpha_{2}-1)\left[b-(\alpha_{1}-\alpha_{2}+2\chi
eE_{0}\sigma^{2})\right]}e^{-\pi\alpha_{2}}\ ,$ $\displaystyle
U_{n,3}=\frac{2^{1-ia}b\left(b-a\right)}{a\left(\alpha_{1}-\alpha_{2}+2\chi
eE_{0}\sigma^{2}\right)+b\left(\alpha_{1}-\alpha_{2}-2\chi
eE_{0}\sigma^{2}\right)}e^{\pi(\alpha_{2}-a)}\ ,$ $\displaystyle
U_{n,4}=\frac{2^{-ib}(1-ia)}{1+i(b-a)}e^{\pi(\alpha_{2}-b)}\ ;$ (22) $\
{}_{\zeta}\mathcal{N}=\,{}_{\zeta}CY,\;\;\ ^{\zeta}\mathcal{N}=\
^{\zeta}CY,\;\;Y=\left(V_{\bot}T\right)^{-1/2}\ ,$ $\
{}_{\zeta}C=\left[2\left|p^{\mathrm{L}}\right|\left|\pi_{0}\left(\mathrm{L}\right)-\chi
p^{\mathrm{L}}\right|\right]^{-1/2},\;\
^{\zeta}C=\left[2\left|p^{\mathrm{R}}\right|\left|\pi_{0}\left(\mathrm{R}\right)-\chi
p^{\mathrm{R}}\right|\right]^{-1/2}\ .$ (23)

Stationary plane waves in the ranges $\Omega_{k}$, $k=1,2,4,5$ are usually
used in the potential scattering theory. In each of these ranges
sign$\eta_{\mathrm{L}}$ and sign$\eta_{\mathrm{R}}$ coincide
($\eta_{\mathrm{L/R}}=1$ for particles and $\eta_{\mathrm{L/R}}=-1$ for
antiparticles). We stress that definitions of particle and antiparticle in the
framework of one-particle quantum theory and QFT are in agreement. In the
ranges $\Omega_{1}$ and $\Omega_{2}$ there exist only states of particles
whereas in the ranges $\Omega_{4}$ and $\Omega_{5}$ there exist only states of
antiparticles. In these ranges particles and antiparticles are subjected to
the scattering and the reflection only. In fact, $\psi_{n}\left(X\right)$ for
$m\in\Omega_{2}$ are wave functions that describe an unbounded motion of
particles (electrons) in $x\rightarrow-\infty$ direction while
$\psi_{n}\left(X\right)$ for $n\in\Omega_{4}$ are wave functions that describe
an unbounded motion of antiparticles (positrons) toward $x=+\infty$. Such one-
particle interpretation does not exist in the range $\Omega_{3}$, where
sign$\eta_{\mathrm{L}}$ is opposite to sign$\eta_{\mathrm{R}}$, here one must
take a many-particle QFT consideration into account, in particular, the vacuum
instability, see Appendix A for details. Note that the range $\Omega_{3}$ is
often referred to as the Klein zone and the pair creation from the vacuum
occurs in this range, whereas the vacuum is stable in the ranges $\Omega_{k}$,
$k=1,2,4,5$.

It was demonstrated in Ref. GavGi16  (see sections V and VII and Appendices C1
and C2) by using one-particle mean currents and the energy fluxes that the
plane waves $\ {}_{\zeta}\psi_{n}\left(X\right)$ and $\
{}^{\zeta}\psi_{n}\left(X\right)$ are unambiguously identified as components
of initial and final wave packets of particles and antiparticles,

$\displaystyle\mathrm{in-solutions:\ }_{+}\psi_{n_{1}},\
^{-}\psi_{n_{1}};\mathrm{\ }_{-}\psi_{n_{5}},\ ^{+}\psi_{n_{5}};\ \
_{-}\psi_{n_{3}},\ ^{-}\psi_{n_{3}}\ ;$ $\displaystyle\mathrm{out-solutions:\
}_{-}\psi_{n_{1}},\ ^{+}\psi_{n_{1}};\ _{+}\psi_{n_{5}},\ ^{-}\psi_{n_{5}};\ \
_{+}\psi_{n_{3}},\ ^{+}\psi_{n_{3}},\;n_{k}\in\Omega_{k}.$ (24)

In the ranges $\Omega_{2}$ and $\Omega_{4}$ we deal with a total reflection.
The complete sets of in- and out-solutions must include solutions
$\psi_{n_{2}}\left(X\right)$ and $\psi_{n_{4}}\left(X\right)$.

Since each pair of solutions $\ {}_{\zeta}\psi_{n}\left(X\right)$ and $\
{}^{\zeta}\psi_{n}\left(X\right)$ with quantum numbers
$n\in\Omega_{1}\cup\Omega_{3}\cup\Omega_{5}$ are complete, there exist mutual
decompositions:

$\displaystyle\eta_{\mathrm{L}}\ ^{\zeta}\psi_{n}\left(X\right)$
$\displaystyle=$ $\
{}_{+}\psi_{n}\left(X\right)g\left({}_{+}|^{\zeta}\right)-\
_{-}\psi_{n}\left(X\right)g\left({}_{-}|^{\zeta}\right)\,,$
$\displaystyle\eta_{\mathrm{R}}\ {}_{\zeta}\psi_{n}\left(X\right)$
$\displaystyle=$ $\
{}^{+}\psi_{n}\left(X\right)g\left({}^{+}|_{\zeta}\right)-\
^{-}\psi_{n}\left(X\right)g\left({}^{-}|_{\zeta}\right)\,,$ (25)

where the decomposition coefficients $g$ are:

$g\left({}^{\zeta^{\prime}}|_{\zeta}\right)^{\ast}=g\left({}_{\zeta}|^{\zeta^{\prime}}\right)=\left(\
{}_{\zeta}\psi_{n},\ ^{\zeta^{\prime}}\psi_{n}\right)_{x}\,,\ \
n\in\Omega_{1}\cup\Omega_{3}\cup\Omega_{5}\,.$ (26)

These coefficients satisfy the following unitary relations:

$\displaystyle\left|g\left({}_{-}\left|{}^{+}\right.\right)\right|^{2}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2},\;\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}=\left|g\left({}_{-}\left|{}^{-}\right.\right)\right|^{2},$
$\displaystyle\frac{g\left({}_{+}\left|{}^{-}\right.\right)}{g\left({}_{-}\left|{}^{-}\right.\right)}=\frac{g\left({}^{+}\left|{}_{-}\right.\right)}{g\left({}^{+}\left|{}_{+}\right.\right)},\
\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}-\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}=-\eta_{\mathrm{L}}\eta_{\mathrm{R}}\
.$ (27)

One can see that all the coefficients $g$ can be expressed via only one of
them, e.g. via $g\left({}^{-}|_{+}\right)$. Using the Kummer relations (133)
for the hypergeometric equation; see Bateman , this coefficient can be found
to be:

$g\left({}^{-}|_{+}\right)=i\eta_{\mathrm{R}}\,\frac{{}_{+}\mathcal{N}}{{}^{-}\mathcal{N}}\frac{2^{i(b-\alpha_{1}+\alpha_{2})+1}\Gamma\left[i(a-b)\right]\Gamma\left(1+2i\alpha_{2}\right)}{\left(a-\alpha_{1}+\alpha_{2}+2\chi
eE_{0}\sigma^{2}\right)\Gamma\left(ia\right)\Gamma\left[-i(b-2\alpha_{2})\right]},$
(28)

where $\Gamma\left(x\right)$ is the gamma-function. Then

$\displaystyle\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}=\sinh\left(2\pi\left|p^{\mathrm{L}}\right|\sigma\right)\sinh\left(4\pi\left|p^{\mathrm{R}}\right|\sigma\right)\left|\beta_{+}\beta_{-}\right|^{-1}\
,$ $\displaystyle\beta_{\pm}=\sinh\left[\pi\sigma\left(\sqrt{2\delta
U^{2}+2\left|p^{\mathrm{R}}\right|^{2}-\left|p^{\mathrm{L}}\right|^{2}}\pm\left(2\left|p^{\mathrm{R}}\right|-\left|p^{\mathrm{L}}\right|\right)\right)\right],$
(29)

where $\delta U$ and $\left|p^{\mathrm{L/R}}\right|$ are given by Eqs. 4 and
16).

Relation (29) holds true for the quantum numbers
$n\in\Omega_{1}\cup\Omega_{3}\cup\Omega_{5}$. However, interpretations of this
relation in the range $\Omega_{3}$ and in the ranges $\Omega_{1}$ and
$\Omega_{5}$ are quite different. Note that there exists a useful relation
between absolute values of the momenta $p^{\mathrm{R}}$ and $p^{\mathrm{L}},$

$\displaystyle\left|p^{\mathrm{L}}\right|$ $\displaystyle=$
$\displaystyle\sqrt{\left|p^{\mathrm{R}}\right|^{2}+2\eta_{\mathrm{R}}\delta
U\sqrt{\left|p^{\mathrm{R}}\right|^{2}+\pi_{\bot}^{2}}+\delta U^{2}},$
$\displaystyle\left|p^{\mathrm{R}}\right|$ $\displaystyle=$
$\displaystyle\sqrt{\left|p^{\mathrm{L}}\right|^{2}-2\eta_{\mathrm{L}}\delta
U\sqrt{\left|p^{\mathrm{L}}\right|^{2}+\pi_{\bot}^{2}}+\delta U^{2}},$ (30)

As follows from Eqs. (29) and (30), if either $\left|p^{\mathrm{R}}\right|$ or
$\left|p^{\mathrm{L}}\right|$ tends to zero, one of the following limits takes
place:

$\left|g\left({}_{-}\left|{}^{+}\right.\right)\right|^{-2}\sim\left|p^{\mathrm{R}}\right|\rightarrow
0,\ \
\left|g\left({}_{-}\left|{}^{+}\right.\right)\right|^{-2}\sim\left|p^{\mathrm{L}}\right|\rightarrow
0,\ \ \forall\pi_{\bot}^{2}\neq 0.$ (31)

These properties are essential for the justification of in- and out-particle
interpretation in the general construction given in Ref. GavGi16 .

##  III Processes in stable vacuum ranges

In the ranges $\Omega_{2}$ and $\Omega_{4}$ we deal with a total reflection.
In the adjacent ranges,__ $\Omega_{1}$ and $\Omega_{5}$, a particle can be
reflected and transmitted. For example, in the range $\Omega_{1},$ the total
$\tilde{R}$ and the relative $R$ amplitudes of an electron reflection, and the
total $\tilde{T}$ and the relative $T$ amplitudes of an electron transmission
can be presented via the following matrix elements:

$\displaystyle R_{+,n}$ $\displaystyle=$
$\displaystyle\tilde{R}_{+,n}c_{v}^{-1},\ \tilde{R}_{+,n}=\langle
0,\mathrm{out}|\ _{-}a_{n}(\mathrm{out})\
_{+}a_{n}^{{\dagger}}(\mathrm{in})|0,\mathrm{in}\rangle,$ $\displaystyle
T_{+,n}$ $\displaystyle=$ $\displaystyle\tilde{T}_{+,n}c_{v}^{-1},\
\tilde{T}_{+,n}=\langle 0,\mathrm{out}|\ ^{+}a_{n}(\mathrm{out})\
_{+}a_{n}^{{\dagger}}(\mathrm{in})|0,\mathrm{in}\rangle,$ $\displaystyle
R_{-,n}$ $\displaystyle=$ $\displaystyle\tilde{R}_{-,n}c_{v}^{-1},\
\tilde{R}_{-,n}=\langle 0,\mathrm{out}|\ ^{+}a_{n}(\mathrm{out})\
^{-}a_{n}^{{\dagger}}(\mathrm{in})|0,\mathrm{in}\rangle,$ $\displaystyle
T_{-,n}$ $\displaystyle=$ $\displaystyle\tilde{T}_{-,n}c_{v}^{-1},\
\tilde{T}_{-,n}=\langle 0,\mathrm{out}|_{-}a_{n}(\mathrm{out})\
^{-}a_{n}^{{\dagger}}(\mathrm{in})|0,\mathrm{in}\rangle,$ (32)

where initial creation $a^{{\dagger}}(\mathrm{in})$ and final annihilation
$a(\mathrm{out})$ operators, initial $|0,\mathrm{in}\rangle$ and final
$|0,\mathrm{out}\rangle$ vacua, and the vacuum-to-vacuum transition amplitude
$c_{v}=\langle 0,\mathrm{out}|0,\mathrm{in}\rangle$ are defined in Appendix A.
Note that the partial vacua are stable in $\Omega_{k}$, $k=1,2,4,5$, and the
vacuum instability with $\left|c_{v}\right|\neq 1$ is due to the partial
vacuum-to-vacuum transition amplitude formed in $\Omega_{3}$. Using a linear
canonical transformation between in and out-operators in Eq. (32) (see Eq.
(4.33) in Ref. GavGi16 ) one find that the relative reflection
$\left|R_{\zeta,n}\right|^{2}$ and transmission $\left|T_{\zeta,n}\right|^{2}$
probabilities are:

$\left|T_{\zeta,n}\right|^{2}=1-\left|R_{\zeta,n}\right|^{2},\;\;\left|R_{\zeta,n}\right|^{2}=\left[1+\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\right]^{-1},\
\zeta=\pm\ ,$ (33)

where $\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$ is given by
Eq. (29). Similar expressions can be derived for positron amplitudes in the
range $\Omega_{5}$. In particular, relation (33) holds true literally for the
positrons in the range $\Omega_{5}$.

In the ranges $\Omega_{1}$ and $\Omega_{5}$ we meet a realization of rules of
the potential scattering theory in the framework of QFT and can see that
relative probabilities of the reflection and the transmission coincide with
mean currents of reflected particles $J_{R}=\left|R_{m}\right|^{2}$ and
transmitted particles $J_{T}=\left|T_{m}\right|^{2}$. The correct result
$J_{R}+J_{T}=1$ follows from the unitary relation (27).

Limits (31) imply the following properties of the coefficients
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|$:
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\rightarrow 0$ in
the range $\Omega_{1}$ if $n$ tends to the boundary with the range
$\Omega_{2}$ ($\left|p^{\mathrm{R}}\right|\rightarrow 0$);
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\rightarrow 0$ in
the range $\Omega_{5}$ if $n$ tends to the boundary with the range
$\Omega_{4}$ ($\left|p^{\mathrm{L}}\right|\rightarrow 0$). Thus, in the latter
cases the relative reflection probabilities $\left|R_{\zeta,n}\right|^{2}$
tend to the unity; i.e., they are continuous functions of the quantum numbers
$n$ on the boundaries. In addition, it follows from Eq. (28) that
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\rightarrow 0$ and,
therefore, $\left|R_{\zeta,n}\right|^{2}\rightarrow 0$ as
$p_{0}\rightarrow\pm\infty$, as it is expected.

##  IV Physical quantities specifying the vacuum instability

The vacuum instability is due to contributions formed in the range
$\Omega_{3}$. In this range the important characteristic of all the processes
are differential mean numbers $N_{n}^{\mathrm{cr}}$ of electron-positron pairs
created from the vacuum. The differential mean numbers of electrons and
positrons created from the vacuum are equal and related to the mean numbers
$N_{n}^{\mathrm{cr}}$ of created pairs,

$\displaystyle N_{n}^{a}\left(\mathrm{out}\right)=\left\langle
0,\mathrm{in}\left|\hat{N}_{n}^{a}\left(\mathrm{out}\right)\right|0,\mathrm{in}\right\rangle=\left|g\left({}_{-}\left|{}^{+}\right.\right)\right|^{-2}\
,$ $\displaystyle N_{n}^{b}\left(\mathrm{out}\right)=\left\langle
0,\mathrm{in}\left|\hat{N}_{n}^{b}\left(\mathrm{out}\right)\right|0,\mathrm{in}\right\rangle=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\
,$ $\displaystyle
N_{n}^{\mathrm{cr}}=N_{n}^{b}\left(\mathrm{out}\right)=N_{n}^{a}\left(\mathrm{out}\right),\
\ n\in\Omega_{3}\ .$ (34)

Here $\hat{N}_{n}^{a}\left(\mathrm{out}\right)$ and
$N_{n}^{b}\left(\mathrm{out}\right)$ are operators of the number of the final
electrons and positrons, given by Eq. (113) (see Appendix A for details), and
the quantity $\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$ is
given by Eq. (28). The probabilities of a particle reflection (transmission is
impossible) and a pair creation and annihilation in the Klein zone can be
expressed via differential mean numbers of created pairs
$N_{n}^{\mathrm{cr}}$; see Eq. (7.22) in Ref._GavGi16 _)_._

Unlike the case of uniform time-dependent electric fields, in the constant
inhomogeneous electric fields, there is a critical surface in space of
particle momenta, which separates the Klein zone $\Omega_{3}$ from the
adjacent ranges $\Omega_{2}$ and $\Omega_{4}$. In the ranges $\Omega_{2}$ and
$\Omega_{4}$, the work of the electric field is sufficient to ensure the total
reflection for electrons and positrons, respectively, but is not sufficient to
produce pairs from the vacuum. Accordingly, it is expected that for any
nonpathological field configuration, the pair creation vanishes close to this
critical surface. Limits (31) imply that $N_{n}^{\mathrm{cr}}\rightarrow 0$ if
$n$ tends to the boundary with either the range $\Omega_{2}$
($\left|p^{\mathrm{R}}\right|\rightarrow 0$) or the range $\Omega_{4}$
($\left|p^{\mathrm{L}}\right|\rightarrow 0$),

$N_{n}^{\mathrm{cr}}\sim\left|p^{\mathrm{R}}\right|\rightarrow 0,\ \
N_{n}^{\mathrm{cr}}\sim\left|p^{\mathrm{L}}\right|\rightarrow 0,\ \
\forall\pi_{\bot}\neq 0\ .$ (35)

Standard integral characteristics of the vacuum instability are sums over the
range $\Omega_{3}$ (see Appendix A for details), are the total number
$N^{\mathrm{cr}}$ of pairs created from the vacuum, and the vacuum-to-vacuum
transition probability $P_{v}$,

$\displaystyle
N^{\mathrm{cr}}=\sum_{n\in\Omega_{3}}N_{n}^{\mathrm{cr}},\;N_{n}^{\mathrm{cr}}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2},$
(36) $\displaystyle
P_{v}=\left|c_{v}\right|^{2}=\exp\left(\sum_{n\in\Omega_{3}}\ln\left(1-N_{n}^{\mathrm{cr}}\right)\right).$
(37)

The summations over $\Omega_{3}$ can be converted into integrals in the
standard way,

$\left(V_{\perp}T\right)^{-1}\sum{}_{p_{0},\mathbf{p}_{\perp}\in\Omega_{3}}\leftrightarrow\left(2\pi\right)^{1-d}\int
dp_{0}d\mathbf{p}_{\perp}\ ,$

in which $V_{\perp}$, $T$ are macroscopically large. It follows from Eq. (37)
that $\ln P_{v}\approx-N^{\mathrm{cr}}$ if all $N_{n}^{\mathrm{cr}}\ll 1$.

Under approximation (18) the electric field under consideration can be
neglected in the macroscopic regions $S_{\mathrm{L}}$ (at $x<x_{\mathrm{L}})$
and $S_{\mathrm{R}}$ (at $x>x_{\mathrm{R}})$, that is, particles are free in
these regions. We note that near all the work $\delta U$ is performed by the
electric field situated in a region $S_{\mathrm{int}}$ between two planes
$x=x_{\mathrm{L}}$ and $x=x_{\mathrm{R}}$. Assuming that the areas
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$ are much wider than the area
$S_{\mathrm{int}}$, this part of the field affects only coefficients $g$
entering into the mutual decompositions of the solutions given by Eq. (25).
Created electrons and positrons leaving the area $S_{\mathrm{int}}$ enter the
areas $S_{\mathrm{L}}$ and $S_{\mathrm{R}}$, respectively, and continue to
move with constant velocities. The positron of a pair created with quantum
number $n$ moves in the $x$ direction with a velocity
$v^{\mathrm{R}}=\left|p^{\mathrm{R}}/\pi_{0}\left(\mathrm{R}\right)\right|$
while the electron belonging to the same pair moves in the opposite direction
with a velocity $-v^{\mathrm{L}}$,
$v^{\mathrm{L}}=\left|p^{\mathrm{L}}/\pi_{0}\left(\mathrm{L}\right)\right|$.
It is shown that the microscopical parameter $T$ can be interpreted as the
time of the observation of the created particles leaving the area
$S_{\mathrm{int}}$; see GavGi20 .

Following the way used in Ref. GavGi20 , we can calculate the current
densities and the energy flux densities of electrons and positrons, after the
instant when these fluxes become completely separated and already have left
the region $S_{\mathrm{int}}$. The motion of the positrons forms the flux
density

$\left\langle j_{x}\right\rangle_{n}=N_{n}^{\mathrm{cr}}(TV_{\perp})^{-1}$
(38)

in the area $S_{\mathrm{R}}$, while the electron motion forms the flux density
$-\left(j_{x}\right)_{n}$ in the area $S_{\mathrm{L}}$. Here it is taken into
account that differential mean numbers of created electrons and positrons with
a given $n$ are equal. The total flux densities of the positrons and electrons
are

$\left\langle j_{x}\right\rangle=\sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}=N^{\mathrm{cr}}(TV_{\perp})^{-1}$ (39)

and $-\left\langle j_{x}\right\rangle$, respectively. The current density of
both created electrons and positrons is $J_{x}^{\mathrm{cr}}=e\left\langle
j_{x}\right\rangle$. It is conserved in the $x$-direction.

During the time $T,$ the created positrons carry the charge $e\left\langle
j_{x}\right\rangle_{n}T$ over the unit area $V_{\bot}$ of the surface
$x=x_{\mathrm{R}}$. This charge is evenly distributed over the cylindrical
volume of the length $v^{\mathrm{R}}T$. Thus, the charge density of the
positrons created with a given $n$ is $ej_{n}^{0}\left(\mathrm{R}\right)$,
where $j_{n}^{0}\left(\mathrm{R}\right)=\left\langle
j_{x}\right\rangle_{n}/v^{\mathrm{R}}$ is the number density of the positrons.
During the time $T,$ the created electrons carry the charge $e\left\langle
j_{x}\right\rangle_{n}T$ over the unit area $V_{\bot}$ of the surface
$x=x_{\mathrm{L}}$. Taking into account that this charge is evenly distributed
over the cylindrical volume of the length $v^{\mathrm{L}}T$, we can see that
the charge density of the electrons created with a given $n$ is
$-ej_{n}^{0}\left(\mathrm{L}\right)$, where
$j_{n}^{0}\left(\mathrm{L}\right)=\left\langle
j_{x}\right\rangle_{n}/v^{\mathrm{L}}$ is the number density of the electrons.
The total charge density of the created particles reads:

$J_{0}^{\mathrm{cr}}\left(x\right)=e\left\\{\begin{array}[]{c}-\sum_{n\in\Omega_{3}}j_{n}^{0}\left(\mathrm{L}\right),\
\ x\in S_{\mathrm{L}}\\\
\sum_{n\in\Omega_{3}}j_{n}^{0}\left(\mathrm{R}\right),\ x\in
S_{\mathrm{R}}\end{array}\right..\ $ (40)

Due to a relation between the velocities $v^{\mathrm{L}}$ and
$v^{\mathrm{R}}$, the total number densities of the created electrons and
positrons are the same,

$\sum_{n\in\Omega_{3}}j_{n}^{0}\left(\mathrm{L}\right)=\sum_{n\in\Omega_{3}}j_{n}^{0}\left(\mathrm{R}\right)\
.$

The created electrons and positrons are spatially separated and carry a charge
that tends to weaken the external electric field.

In the same manner, one can derive some representation for the nonzero
components of energy-momentum tensor of the created particles:

$\displaystyle
T_{\mathrm{cr}}^{00}(x)=\left\\{\begin{array}[]{c}\sum_{n\in\Omega_{3}}j_{n}^{0}\left(\mathrm{L}\right)\pi_{0}\left(\mathrm{L}\right),\
x\in S_{\mathrm{L}}\\\
\sum_{n\in\Omega_{3}}j_{n}^{0}\left(\mathrm{R}\right)\left|\pi_{0}\left(\mathrm{R}\right)\right|,\mathrm{\
}x\in S_{\mathrm{R}}\end{array}\right.,$ (43) $\displaystyle
T_{\mathrm{cr}}^{11}(x)=\left\\{\begin{array}[]{c}\sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}\left|p^{\mathrm{L}}\right|,\mathrm{\ }x\in
S_{\mathrm{L}}\\\ \sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}\left|p^{\mathrm{R}}\right|,\mathrm{\ }x\in
S_{\mathrm{R}}\end{array}\right.,$ (46) $\displaystyle
T_{\mathrm{cr}}^{kk}(x)=\left\\{\begin{array}[]{c}\sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}\left(p_{k}\right)^{2}/\left|p^{\mathrm{L}}\right|,\mathrm{\
}x\in S_{\mathrm{L}}\\\ \sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}\left(p_{k}\right)^{2}/\left|p^{\mathrm{R}}\right|,\mathrm{\
}x\in S_{\mathrm{R}}\end{array}\right.,\ \ k\neq 1,$ (49) $\displaystyle
T_{\mathrm{cr}}^{10}(x)=\left\\{\begin{array}[]{c}-\sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}\pi_{0}\left(\mathrm{L}\right),\mathrm{\ }x\in
S_{\mathrm{L}}\\\ \sum_{n\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{n}\left|\pi_{0}\left(\mathrm{R}\right)\right|,\mathrm{\
}x\in S_{\mathrm{R}}\end{array}\right..$ (52)

Here $T_{\mathrm{cr}}^{00}(x)$ and $T_{\mathrm{cr}}^{kk}(x)$, $k=1,\ldots,D$,
(there is no summation over $k$) are energy density and components of the
pressure of the particles created in the areas $S_{\mathrm{L}}$ and
$S_{\mathrm{R}}$ respectively, whereas $T_{\mathrm{cr}}^{10}(x)v_{s}$, for
$x\in S_{\mathrm{L}}$ or $x\in S_{\mathrm{R}}$, is the energy flux density of
the created particles through the surfaces $x=x_{\mathrm{L}}$ or
$x=x_{\mathrm{R}}$ respectively. In a strong field, or in a field with the
sufficiently large potential step $\delta U$ , the energy density and the
pressure along the direction of the axis $x$ are near equal,
$T_{\mathrm{cr}}^{00}(x)\approx T_{\mathrm{cr}}^{11}(x)$, in the areas
$S_{\mathrm{L}}$__ and __ $S_{\mathrm{R}}$ respectively,__

##  V Particle creation due to a weakly inhomogeneous electric field

###  V.1 Intensity of the particle creation over the Klein zone

The above study of the vacuum instability caused by the asymmetric $x$-step
can be useful for a consideration of the particle creation by a weakly
inhomogeneous electric field between two capacitor plates separated by a
sufficiently large length. Indeed, if the parameter $\sigma$ is taken to be
sufficiently large,

$\sigma\gg\left(eE_{0}\right)^{-1/2}\max\left\\{1,m^{2}/eE_{0}\right\\},$ (53)

the step can be considered as a regularization (like the Sauter potential with
appropriate parameters) of a weakly inhomogeneous constant electric field
between the plates. For example, for such big $\sigma$ we can consider the
behavior of mean numbers of electron-positron pairs created over the Klein
zone __ $\Omega_{3}$. For this purpose, consider arguments of the functions
$\beta_{\pm}$ in the denominator of expression (29). Absolute values of
$\left|p^{\mathrm{R}}\right|$ and $\left|p^{\mathrm{L}}\right|$ are related by
Eq. (30). One can see from Eq. (30) that
$d\left|p^{\mathrm{L}}\right|/d\left|p^{\mathrm{R}}\right|<0,$ and at any
given $\mathbf{p}_{\bot}$ these quantities are restricted inside the range
$\Omega_{3}$ as

$0\leq\left|p^{\mathrm{R/L}}\right|\leq
p^{\mathrm{\max}},\;\;p^{\mathrm{\max}}=\sqrt{\delta U\left(\delta
U-2\pi_{\bot}\right)}.$ (54)

These relations for big $\sigma$ are

$\pi\sigma\left(\sqrt{\delta U\left(\delta
U\mathbb{-\pi}_{\perp}\right)}-p^{\mathrm{\max}}\right)\gg 1,\ \
\pi\sigma\left(\sqrt{\delta U\left(\delta
U\mathbb{+}2\mathbb{\pi}_{\perp}\right)}-p^{\mathrm{\max}}\right)\gg 1,$ (55)

and imply that

$\pi\sigma\left[\sqrt{2\delta
U^{2}+2\left|p^{\mathrm{R}}\right|^{2}-\left|p^{\mathrm{L}}\right|^{2}}\pm\left(2\left|p^{\mathrm{R}}\right|-\left|p^{\mathrm{L}}\right|\right)\right]\gg
1.$ (56)

We get from (29) that

$N_{n}^{\mathrm{cr}}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\approx\frac{4\sinh\left(2\pi\left|p^{\mathrm{L}}\right|\sigma\right)\sinh\left(4\pi\left|p^{\mathrm{R}}\right|\sigma\right)}{\exp\left[2\pi\sigma\sqrt{2\delta
U^{2}+2\left|p^{\mathrm{R}}\right|^{2}-\left|p^{\mathrm{L}}\right|^{2}}\right]}\
.$ (57)

It follows from Eq. (57) that the quantities $N_{n}^{\mathrm{cr}}$ are
exponentially small,

$N_{n}^{\mathrm{cr}}\approx
2\left(4\pi\sigma\right)^{2}\left|p^{\mathrm{L}}p^{\mathrm{R}}\right|\exp\left(-2\sqrt{2}\pi\delta
U\sigma\right)\ ,$ (58)

if the range $\Omega_{3}$ is small enough

$eE\sigma-2\pi_{\bot}\rightarrow 0\Longrightarrow\pi\sigma
p^{\mathrm{\max}}\ll 1\Longrightarrow\left|p^{\mathrm{R/L}}\right|\ll 1\ .$
(59)

Then we can consider the opposite case of big ranges $\Omega_{3}$ when

$\pi\sigma p^{\mathrm{\max}}\gg 1$ (60)

and the quantities $N_{n}^{\mathrm{cr}}$ are not small. Such ranges do exist
if

$eE\sigma\gg m$ (61)

and

$\sigma\pi_{\bot}<K_{\bot}\ ,$ (62)

where $K_{\bot}$ is a given arbitrary number, restricted as $m\sigma\ll
K_{\bot}\ll eE\sigma^{2}$.

Let us study the behavior of $N_{n}^{\mathrm{cr}}$ on the boundaries of the
Klein domain $\Omega_{3}$, when $\left|p^{\mathrm{R}}\right|\rightarrow 0$ or
$\left|p^{\mathrm{L}}\right|\rightarrow 0$. Let
$\pi\sigma\left|p^{\mathrm{R/L}}\right|<K_{1}$, where $K_{1}\geq 1$ is some
arbitrary number satisfying the inequality

$K_{1}\ll eE_{0}\sigma^{2}\ .$ (63)

In close proximity to these boundaries,
$\pi\sigma\left|p^{\mathrm{L/R}}\right|<K_{0}$, $K_{0}<1$, we obtain that the
value $N_{n}^{\mathrm{cr}}$ is exponentially small,

$N_{n}^{\mathrm{cr}}\approx 2e^{-4\pi m\sigma},\ \
\pi\sigma\left|p^{\mathrm{R}}\right|<K_{0}\ ;\ \ N_{n}^{\mathrm{cr}}\approx
2e^{-2m\sigma},\ \ \pi\sigma\left|p^{\mathrm{L}}\right|<K_{0}\ .$ (64)

For boundary regions located closer to the center of the Klein zone
$\Omega_{3}$, $1\lesssim\pi\sigma\left|p^{\mathrm{L/R}}\right|<K_{1}$,
$K_{1}>1$ the following approximation is valid

$\displaystyle N_{n}^{\mathrm{cr}}$ $\displaystyle\approx$
$\displaystyle\exp\left\\{-4\pi\sigma\left[\sqrt{\left(p^{\mathrm{R}}\right)^{2}+\pi_{\bot}^{2}}-\left|p^{\mathrm{R}}\right|\right]\right\\},\
\ 1\lesssim\pi\sigma\left|p^{\mathrm{R}}\right|\lesssim K_{1}\ ,$
$\displaystyle N_{n}^{\mathrm{cr}}$ $\displaystyle\approx$
$\displaystyle\exp\left\\{-2\pi\sigma\left[\sqrt{\left(p^{\mathrm{L}}\right)^{2}+\pi_{\bot}^{2}}-\left|p^{\mathrm{L}}\right|\right]\right\\},\
\ 1\lesssim\pi\sigma\left|p^{\mathrm{L}}\right|\lesssim K_{1}\ .$ (65)

The numbers $N_{n}^{\mathrm{cr}}$ increase as $n$ moves away from the
boundaries of the Klein region $\Omega_{3}$ and for any fixed value of
$\pi_{\bot}$, they reach their maximum value when
$\pi\sigma\left|p^{\mathrm{L/R}}\right|\rightarrow K_{1}$. In turn, this
maximum value increases as $\pi_{\bot}\rightarrow m$. In this range we can
estimate $N_{n}^{\mathrm{cr}}$ as

$N_{n}^{\mathrm{cr}}<\exp\left\\{-4\pi\sigma m\left[\frac{\pi\sigma
m}{2K_{1}}-\left(\frac{\pi\sigma m}{2K_{1}}\right)^{3}\right]\right\\},\ \
\pi\sigma\left|p^{\mathrm{R}}\right|\rightarrow K_{1}\ ,$ (66)

Similarly, in the domain $\pi\sigma\left|p^{\mathrm{L}}\right|\rightarrow k$
we have the estimate

$N_{n}^{\mathrm{cr}}<\exp\left\\{-2\pi\sigma m\left[\frac{\pi\sigma
m}{2K_{1}}-\left(\frac{\pi\sigma m}{2K_{1}}\right)^{3}\right]\right\\},\ \
\pi\sigma\left|p^{\mathrm{L}}\right|\rightarrow K_{1}\ .$ (67)

The right sides of inequalities (66) and (67) are exponentially small if

$K_{1}\ll\pi m\sigma/2$ (68)

for any $K_{1}$.

Consequently, the main contribution to the pairs creation is formed in the
subrange of the Klein zone $D\subset\Omega_{3}$, where the energy $\pi_{\bot}$
is restricted by inequality (62) and the energy $p_{0}$ is restricted as:

$-eE_{0}\sigma^{2}+K<\sigma p_{0}<-K,\ \
\,K=\sqrt{K_{1}^{2}+(\sigma\pi_{\bot})^{2}}\ .$ (69)

Inequalities (62) and (63) imply that $K\ll eE\sigma^{2}$. In such a case, we
can approximate numbers (57) as follows

$N_{n}^{\mathrm{cr}}\approx
N_{p_{0},\mathbf{p}_{\bot}}^{\mathrm{as}}=e^{-\pi\tau},\
\,\tau=2\sigma\left(\sqrt{2\delta
U^{2}+2\left|p^{\mathrm{R}}\right|^{2}-\left|p^{\mathrm{L}}\right|^{2}}-\left|p^{\mathrm{L}}\right|-2\left|p^{\mathrm{R}}\right|\right)\
.$ (70)

In the point $x_{\mathrm{M}}=\sigma\ln 2$ the electric field (2) has the
maximum value $E_{\max}=E_{0}/(3\sqrt{3})$ and the corresponding kinetic
energy reads:

$\pi_{0}^{\prime}=p_{0}-U\left(x_{\mathrm{M}}\right)=p_{0}+\frac{eE_{0}\sigma}{\sqrt{3}}\
.$

The function $\tau$ takes its minimum value at the point $\pi_{0}^{\prime}=0$,

$\min\tau=\left.\tau\right|_{\pi_{0}^{\prime}=0}=\lambda=\frac{\pi_{\bot}^{2}}{eE_{\max}}\
.$ (71)

Further, $\tau$ monotonically increases with $\left|\pi_{0}^{\prime}\right|$
approaching the boundary of the subrange $D$. On the one side of the center of
the Klein zone $\Omega_{3}$, the function $\tau$ takes the maximum value

$\tau_{\max}^{-}=\left.\tau\right|_{\sigma\left|p_{0}\right|\rightarrow
eE_{0}\sigma^{2}-K}\simeq
2\left(K-K_{1}/\sigma+\frac{\lambda}{4\sqrt{3}}\right),\ $

while on the other side, the maximum value is:

$\tau_{\max}^{+}=\left.\tau\right|_{\sigma\left|p_{0}\right|\rightarrow
K}\simeq 4\left(K-K_{1}/\sigma\right)\ .$

In the wide range $D$, where

$K/\sigma-
eE_{0}\sigma\left(1-1/\sqrt{3}\right)<\pi_{0}^{\prime}<eE_{0}\sigma/\sqrt{3}-K/\sigma\
,\ $ (72)

the numbers $N_{n}^{\mathrm{cr}}$ practically do not depend on the parameter
$\sigma$ and have the form of the differential number of created particles in
an uniform electric field Nikis70b ; Nikis79 ,

$N_{n}^{\mathrm{cr}}\approx e^{-\pi\lambda}\ .$ (73)

###  V.2 Integral quantities

The total number of created pairs is given by integral (36). The main
contribution to the integral is due to the subrange $D\subset\Omega_{3}$,
defined by Eqs. (62) and (69). In this subrange the functions
$N_{n}^{\mathrm{cr}}$ can be approximated by Eq. (70), and integral (36) can
be represented as:

$\displaystyle N^{\mathrm{cr}}$ $\displaystyle\approx$
$\displaystyle\frac{V_{\bot}TJ_{(d)}}{(2\pi)^{d-1}}\int_{\alpha\pi_{\bot}<K_{\bot}}\left(I_{p_{\bot}}^{+}+I_{p_{\bot}}^{-}\right)d\mathbf{p}_{\bot}\
,$ $\displaystyle I_{p_{\bot}}^{+}$ $\displaystyle=$
$\displaystyle\int_{0}^{eE_{0}\sigma/\sqrt{3}-K/\sigma}e^{-\pi\tau}d\pi_{0}^{\prime}\
,$ $\displaystyle I_{p_{\bot}}^{-}$ $\displaystyle=$
$\displaystyle\int_{-\left[eE_{0}\sigma\left(1-1/\sqrt{3}\right)-K/\sigma\right]}^{0}e^{-\pi\tau}d\pi_{0}^{\prime}\
,$ (74)

where $\tau$ is given by Eq. (70). To calculate $I_{p_{\bot}}^{+}$ and
$I_{p_{\bot}}^{-}$, it is convenient to use the representation
$\tau=\lambda\left(q+1\right)$ and to pass from the integration over
$\pi_{0}^{\prime}$ to the integration over the parameter $q$ (the transition
to such a variable provides exponential decrease of the integrand with
increasing $q$, and the expansion of the pre-exponential factor in powers of
$q$ has a form of an asymptotic series).

Finding $\pi_{0}^{\prime}$ as a function of $q$, one has to take into account
that in the region $D$ the following expansions are valid up to linear terms
in reciprocal powers of large parameters:

$\displaystyle\left|p^{\mathrm{L}}\right|\approx\frac{1}{\epsilon_{1}}-\frac{\epsilon_{1}\pi_{\perp}^{2}}{2}+O(\epsilon_{1}),\
\
\epsilon_{1}=\left[\pi_{0}^{{}^{\prime}}+eE_{0}\sigma\left(1-1/\sqrt{3}\right)\right]^{-1}\
,$
$\displaystyle\left|p^{\mathrm{R}}\right|\approx\frac{1}{\epsilon_{2}}-\frac{\epsilon_{2}\pi_{\perp}^{2}}{2}+O(\epsilon_{2}),\
\ \epsilon_{2}=\left[eE_{0}\sigma/\sqrt{3}-\pi_{0}^{{}^{\prime}}\right]^{-1}\
,$ $\displaystyle\sqrt{2\delta
U^{2}+2\left|p^{\mathrm{R}}\right|^{2}-\left|p^{\mathrm{L}}\right|^{2}}\approx\frac{1}{\epsilon_{3}}-\frac{\epsilon_{3}\pi_{\perp}^{2}}{2}+O(\epsilon_{3}),\
\
\epsilon_{0}=\left[eE_{0}\sigma\left(1+1/\sqrt{3}\right)-\pi_{0}^{{}^{\prime}}\right]^{-1}\
.$

Then we obtain the following approximation for $\tau:$

$\tau\approx\frac{2\sqrt{3}\lambda}{2\sqrt{3}+9\left(r-\sqrt{3}\right)r^{2}},\
\ r=\frac{\pi_{0}^{{}^{\prime}}}{eE_{0}\sigma}\ .$ (75)

Besides, we have to find $\pi_{0}^{\prime}$ as a function of $q$ using Eq.
(75). Such a function can be found from the cubic equation

$r^{3}-\sqrt{3}r^{2}+\frac{2q}{3\sqrt{3}\left(q+1\right)}=0\ .$ (76)

Note that when $\pi_{0}^{{}^{\prime}}\rightarrow
eE_{0}\sigma/\sqrt{3}-K/\sigma$ and
$\pi_{0}^{{}^{\prime}}\rightarrow-\left[eE_{0}\sigma\left(1-1/\sqrt{3}\right)-K/\sigma\right]$,
the parameter $\tau$ reaches the limiting values
$\tau_{\max}^{\pm}=\lambda\left(q_{\max}^{\pm}+1\right)$, respectively.
However, since contributions of the factor $\exp\left(-\pi\tau\right)$ to
integrals (74) outside of range $D$ are exponentially small, one can extend
limits of the integration over $q$ to $\pm\infty$. Equation (76) has three
real roots:

$\displaystyle
r_{1}=\frac{2}{\sqrt{3}}\cos\frac{\alpha\left(q\right)}{3}+1/\sqrt{3},\
\alpha\left(q\right)=\arccos\left[\left(q+1\right)^{-1}\right]\ ,$
$\displaystyle
r_{2}=-\frac{2}{\sqrt{3}}\cos\left[\frac{\alpha\left(q\right)}{3}+\frac{\pi}{3}\right]+1/\sqrt{3}\
,$ $\displaystyle
r_{3}=-\frac{2}{\sqrt{3}}\cos\left[\frac{\alpha\left(q\right)}{3}-\frac{\pi}{3}\right]+1/\sqrt{3}\
;$ (77)

see, e.g., Korn .

Since $0<q<+\infty$, the inequality $0\leq\alpha\left(q\right)\leq\pi/2$ holds
true, which implies:

$0\leq\frac{\alpha\left(q\right)}{3}\leq\frac{\pi}{6},\ \
\frac{\pi}{3}\leq\left[\frac{\alpha\left(q\right)}{3}+\frac{\pi}{3}\right]\leq\frac{\pi}{2},\
\
-\frac{\pi}{3}\leq\left[\frac{\alpha\left(q\right)}{3}-\frac{\pi}{3}\right]\leq-\frac{\pi}{6}\
.$ (78)

Then

$1\leq r_{1}\leq\sqrt{3},\ \ 0\leq r_{2}\leq\frac{1}{\sqrt{3}},\ \
\left(\frac{1}{\sqrt{3}}-1\right)\leq r_{3}\leq 0$ (79)

such that roots $r_{2}$ and $r_{3}$ represent the quantity
$\pi_{0}^{{}^{\prime}}$ in the subranges
$\pi_{0}^{{}^{\prime}}\in\left(-eE_{0}\sigma\left(1-1/\sqrt{3}\right),0\right)\
$and $\pi_{0}^{{}^{\prime}}\in\left(0,eE_{0}\sigma/\sqrt{3}\right)$,
respectively.

Thus, the integrals $I_{p_{\bot}}^{+}$ and $I_{p_{\bot}}^{-}$ take the forms:

$I_{p_{\perp}}^{\pm}=\pm\frac{2\delta
U}{3\sqrt{3}}\int_{0}^{+\infty}dq\frac{\left(q+1\right)^{-2}}{\sqrt{1-\left(q+1\right)^{-2}}}\sin\left[\frac{\alpha\left(q\right)}{3}\pm\frac{\pi}{3}\right]\exp\left[-\pi\lambda\left(q+1\right)\right]\
,$ (80)

and their sum can be represented as:

$I_{p_{\perp}}^{-}+I_{p_{\perp}}^{+}=\frac{2\delta
U}{3}\int_{0}^{+\infty}dq\frac{\left(1+q\right)^{-2}}{\sqrt{1-\left(1+q\right)^{-2}}}\cos\frac{\alpha\left(q\right)}{3}\exp\left[-\pi\lambda\left(q+1\right)\right]\
.$ (81)

Substituting Eq. (81) into Eq. (74) and integrating over
$dp_{\perp}^{\left(d-2\right)}$, we obtain:

$\displaystyle N^{\mathrm{cr}}\approx
V_{\bot}T\rho^{\mathrm{D}},\;\rho^{\mathrm{D}}=\beta\frac{\delta
U}{eE_{\max}}w,\ \
\beta=\frac{J_{\left(d\right)}\left[eE_{\max}\right]^{d/2}}{\left(2\pi\right)^{d-1}}\exp\left[-\frac{\pi
m^{2}}{eE_{\max}}\right]\ ,$ $\displaystyle
w\left(d\right)=\frac{2}{3}\int_{0}^{+\infty}dq\left(q^{2}+2q\right)^{-1/2}\left(q+1\right)^{-d/2}\cos\frac{\alpha\left(q\right)}{3}\exp\left[-\frac{\pi
m^{2}}{eE_{\max}}q\right]\ .$ (82)

The corresponding probability $P_{\mathrm{v}}$ of the vacuum to remain a
vacuum reads:

$P_{\mathrm{v}}=\exp\left[-\mu N^{\mathrm{cr}}\right],\ \
\mu=\sum_{l=0}^{\infty}\left(l+1\right)^{-d/2}\exp\left(-l\frac{\pi
m^{2}}{eE_{\max}}\right)\ .$ (83)

In general, the current density $J_{x}^{\mathrm{cr}}$ of created particles is
given by Eq. (39), while the charge polarization due to the separation of
created positrons and electrons is presented by the charge density
$\rho^{\mathrm{cr}}\left(x\right)$ given by Eq. (40). The nonzero components
of energy-momentum tensor of the created particles are given by Eq. (52). In
the subrange $D\subset\Omega_{3}$ the velocities $v^{\mathrm{L}}$ and
$v^{\mathrm{R}}$ tend to the speed of the light $c=1$ such that
$\left|p^{\mathrm{L/R}}\right|\approx\left|\pi_{0}^{\prime}\left(\mathrm{L/R}\right)\right|\approx\left|U_{\mathrm{L/R}}^{\prime}\right|$,
with $U_{\mathrm{L}}^{\prime}=-eE_{0}\sigma\left(1-1/\sqrt{3}\right)$ and
$U_{\mathrm{R}}^{\prime}=eE_{0}\sigma/\sqrt{3}$ according to inequality (72).
Using representation (82), we find:

$\displaystyle J_{x}^{\mathrm{cr}}\approx
e\rho^{\mathrm{D}},\;J_{0}^{\mathrm{cr}}\left(x\right)\approx\left\\{\begin{array}[]{c}-J_{x}^{\mathrm{cr}},\
\ x\in S_{\mathrm{L}}\\\ J_{x}^{\mathrm{cr}},\ x\in
S_{\mathrm{R}}\end{array}\right.\ ;$ (86) $\displaystyle
T_{\mathrm{cr}}^{00}(x)\approx
T_{\mathrm{cr}}^{11}(x)\approx\left\\{\begin{array}[]{c}\rho^{\mathrm{D}}\left|U_{\mathrm{L}}^{\prime}\right|,\
x\in S_{\mathrm{L}}\\\
\rho^{\mathrm{D}}\left|U_{\mathrm{R}}^{\prime}\right|,\mathrm{\ }x\in
S_{\mathrm{R}}\end{array}\right.\ ,$ (89) $\displaystyle
T_{\mathrm{cr}}^{kk}(x)\approx\left\\{\begin{array}[]{c}\tilde{\rho}^{\mathrm{D}}\left|U_{\mathrm{L}}^{\prime}\right|^{-1},\mathrm{\
}x\in S_{\mathrm{L}}\\\
\tilde{\rho}^{\mathrm{D}}\left|U_{\mathrm{R}}^{\prime}\right|^{-1},\mathrm{\
}x\in S_{\mathrm{R}}\end{array}\right.,\ \ k\neq 1\ ,$ (92) $\displaystyle
T_{\mathrm{cr}}^{10}(x)\approx\left\\{\begin{array}[]{c}-\rho^{\mathrm{D}}\left|U_{\mathrm{L}}^{\prime}\right|,\
x\in S_{\mathrm{L}}\\\
\rho^{\mathrm{D}}\left|U_{\mathrm{R}}^{\prime}\right|,\mathrm{\ }x\in
S_{\mathrm{R}}\end{array}\right.\ ,\
\tilde{\rho}^{\mathrm{D}}=\beta\frac{\delta
U\mathbb{\sigma}}{2\pi}w\left(d+2\right).$ (95)

It can be seen that the transversal components of the pressure of the created
particles are much less than the longitudinal pressure in the areas
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$, respectively,
$T_{\mathrm{cr}}^{kk}(x)\ll T_{\mathrm{cr}}^{11}(x)$. It is in accordance with
the relation $T_{\mathrm{cr}}^{00}(x)\approx T_{\mathrm{cr}}^{11}(x)$.

Unlike the case with the Sauter field, the energy density and the longitudinal
component of the pressure in the areas $S_{\mathrm{L}}$ and $S_{\mathrm{R}}$,
given by Eq. (95), are not equal,
$T_{\mathrm{cr}}^{00}(x_{\mathrm{L}})=2T_{\mathrm{cr}}^{00}(x_{\mathrm{R}})$
and
$T_{\mathrm{cr}}^{11}(x_{\mathrm{L}})=2T_{\mathrm{cr}}^{11}(x_{\mathrm{R}})$,
that is, the energy density and longitudinal pressure of created electrons is
two times more than the energy density and longitudinal pressure of created
positrons due to the asymmetric form of the field (2). The direction of this
field is chosen along the $x$-axis. Choosing the opposite direction of the
field, one finds that the energy density and longitudinal pressure of created
positrons is two times more than the energy density and the longitudinal
pressure of created electrons.

One can show that representations (82) - (95) can be obtained in the framework
of a new kind of a locally constant field approximation (LCFA) constructed in
Ref. GavGitShi19b ; see Appendix B. This can be considered as an additional
argument in favor of the validity of such an approximation.

Besides, it turns out that LCFA allows one to see relation to the Heisenberg-
Euler effective action method. Note that in the framework of the LCFA the
probability $P_{v}$, given by Eq. (83), can be represented via the imaginary
part of a one-loop effective action $S$ by the seminal Schwinger formula
Schw51 ,

$P_{\mathrm{v}}=\exp\left(-2\mathrm{Im}S\right).$ (96)

Thus, all the vacuum mean values, obtained in the framework of the LCFA, can
be associated with the effective action approach. If the total number of
created particles is small, $N^{\mathrm{cr}}\ll 1$, then $1-P_{v}\approx
N^{\mathrm{cr}}$_._ Therefore, knowledge of the probability $P_{v}$ allows one
to estimate the total number of created particles $N^{\mathrm{cr}}$. It is, in
this case, the effective action approach to calculating $P_{v}$ turns out to
be useful. We note that this approach is a base of a number of approximation
methods; see, e.g., Ref. Adv-QED22  for a review. In this relation, it should
be noted that the probability $P_{v}$ by itself is not very useful in the case
of strong fields when $P_{v}\ll 1$. In the latter case it is necessary to
directly calculate the vacuum mean values of physical quantities, using either
exact solutions, as it is done above, or the new kind of the LCFA GavGitShi19b
.

##  VI The Klein step

The Klein paradox is known from the work by Klein Klein27  who considered, in
the framework of the one-particle relativistic theory, reflection and
transmission probabilities of charged relativistic particles incident on a
sufficiently high rectangular potential step (the Klein step) of the form

$qA_{0}\left(x\right)=\left\\{\begin{array}[]{l}U_{\mathrm{L}},\ x<0\\\
U_{\mathrm{R}},\ x>0\end{array}\right.\ ,$ (97)

where $U_{\mathrm{R}}$ and $U_{\mathrm{L}}$ are some constants. The field (97)
represents a kind of $x$-step. According to calculations of Klein and other
authors, for certain energies and sufficient high magnitude
$U=U_{\mathrm{R}}-U_{\mathrm{L}}$ of the Klein step, it seems that there are
more reflected fermions than incident. This fact many articles and books where
treated as a paradox (the Klein paradox); see Refs. DomCal99 ; HansRavn81 )
for historical review.__ This paradox and other misunderstandings in
considering quantum effects in fields of strong $x$-steps can be consistently
solved as many particle effects in the $QED$ with an unstable vacuum; see
GavGi16 . Obviously that the Klein step is a limiting case of a very sharp
peak field. It is important to have in hands examples of potentials
representing very sharp peak field that can be considered as their
regularizations. Such an example given by Sauter potential (1) was presented
in Ref. GavGi16 ; see Refs. GavGitSh17 ; AdoGavGit20  for regularizations by
piecewise forms of analytic functions. The $x$-step (2) under consideration
represents a new example of such regularizations given by an analytic
function.

Let us study characteristics of the vacuum instability caused by the field (2)
with $\sigma$ sufficiently small, $\sigma\rightarrow 0$_._ If
$U_{\mathrm{R}}=0$ and $U_{\mathrm{L}}=-\delta U=\mathbb{-}eE_{0}\sigma$ are
given constant and

$\delta U\sigma\ll 1\ $ (98)

the field imitates sufficiently well the asymmetric Klein step (97) and
coincides with the latter as $\sigma\rightarrow 0$.

In the ranges $\Omega_{1}$ and $\Omega_{5}$ the energy $\left|p_{0}\right|$ is
not restricted from the above, that is why, in what follows, we consider only
the subranges,
where$\max\left\\{\sigma\left|p^{\mathrm{L}}\right|,\sigma\left|p^{\mathrm{R}}\right|\right\\}\ll
1$. In the leading-term approximation in __ $\sigma$ it follows from Eqs. (28)
that

$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\approx\frac{4k}{\left(1-k\right)^{2}},\
\
k=\frac{\left|p^{\mathrm{R}}\right|}{\left|p^{\mathrm{L}}\right|}\frac{\pi_{0}\left(\mathrm{L}\right)+\pi_{\bot}}{\pi_{0}\left(\mathrm{R}\right)+\pi_{\bot}}\
,$ (99)

where $k$ is called the kinematic factor.

Note that $k$ are positive and do not achieve the unit values, $k\neq 1$ in
the ranges $\Omega_{1}$ and $\Omega_{5}$. In these ranges, the coefficients
$g$ satisfy the same relations,

$\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}+1\
.$ (100)

Therefore, reflection and transmission probabilities derived from Eqs. (99)
have the same forms

$\displaystyle\left|T_{\zeta,n}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{-2}=\frac{4k}{\left(1+k\right)^{2}},$
$\displaystyle\left|R_{\zeta,n}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{-2}=\frac{\left(1-k\right)^{2}}{\left(1+k\right)^{2}}\
.$ (101)

To compare our exact results with results of the nonrelativistic consideration
obtained in any textbook for one dimensional quantum motion, we set
$p_{\bot}=0,$ then $\pi_{\bot}=m,$ $\pi_{0}\left(\mathrm{L}\right)=m+E$, and
$\pi_{0}\left(\mathrm{R}\right)=m+E-\delta U$. In the nonrelativistic limit,
when $\delta U,E\ll m$, we obtain

$k=k^{\mathrm{NR}}=\sqrt{\frac{E-\delta U}{E}}\ ,$

which can be identified with the nonrelativistic results, e.g., see LanLiQM .

Let us consider the range $\Omega_{3}$. Here the quantum numbers
$\mathbf{p}_{\bot}$ are restricted by the inequality$\ 2\pi_{\bot}\leq\delta
U$ and for any of such $\pi_{\bot}$ the quantum numbers $p_{0}$ obey the
strong inequality (LABEL:3.0). In this range the quantity
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$ represents the
differential mean numbers of electron-positron pairs created from the vacuum,
$N_{n}^{\mathrm{cr}}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$.
In this range for any given $\pi_{\bot}$ the absolute values of
$\left|p^{\mathrm{R}}\right|$ and $\left|p^{\mathrm{L}}\right|$ are restricted
from the above, see (54). Therefore, condition (98) implies Eq. (LABEL:exs43).
It follows from Eq. (28) that in the leading approximation the following
equation holds true

$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\approx\frac{4\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}{\delta
U^{2}-\left(\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right)^{2}}=\frac{4\left|k\right|}{\left(1+\left|k\right|\right)^{2}}.$
(102)

Note that expression (102) differs from expression (99) only by the sign of
the kinematic factor $k$. This factor is positive in the ranges $\Omega_{1}$
and $\Omega_{5}$, and it is negative in the range $\Omega_{3}$. In the range
$\Omega_{3}$, the difference
$\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|$ may be zero at
$p_{0}=U_{\mathrm{L}}/2$, which corresponds to $k=-\left(\delta
U+2\pi_{\bot}\right)/\left(\delta U-2\pi_{\bot}\right)$. Namely in this case
the quantity $\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$ has a
maximum at a given $\pi_{\bot}$,

$\max\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}=1-\left(2\pi_{\bot}/\delta
U\right)^{2}.$ (103)

Note that expressions (99), (101) and (102) coincide up to the redesigning of
the constants $U_{\mathrm{L}/\mathrm{R}}$ with expressions, corresponding to
other regularizations of the Klein step; see Ref. GavGi16 ; GavGitSh17 ;
AdoGavGit20 . We see that the Klein step is (in a sense) a limiting case of
various sharp peak fields under condition that the magnitude $\delta U$ is a
given finite constant and the peak of the fields are sufficiently sharp.

##  VII Concluding remarks

A new exactly solvable case in strong-field $QED$ with $x$-step is presented.
This step can be seen as a certain analytic "deformation" of the Sauter field.
In contrast to the Sauter field the potential field under consideration is
asymmetric with respect of the axis $x$ reflection. Bearing in mind numerous
examples of using the results of the exactly solvable problem with the Sauter
field in physical applications related to the problem of vacuum instability,
we believe that the new exact solvable case will also be useful in such
applications. It can be treated as a new regularization of the Klein step.
Exact solutions of the Dirac equation used in the above nonperturbative
calculations, are presented in the form of stationary plane waves with special
left and right asymptotics and identified as components of initial and final
wave packets of particles and antiparticles in the framework of the strong-
field $QED$ GavGi16 . We show that in spite of the fact that the symmetry with
respect to positive and negative bands of energies is broken, distribution of
created pairs and other physical quantities can be presented by elementary
functions. We consider the processes of transmission and reflection in the
ranges of the stable vacuum and study physical quantities specifying the
vacuum instability as well. We find the differential mean numbers of electron-
positron pairs created from the vacuum, the components of current density and
energy-momentum tensor of the created electrons and positrons leaving the area
of the strong field under consideration. We study the particular case of the
particle creation due to a weakly inhomogeneous electric field and obtain
explicitly the total number, the current density and energy-momentum tensor of
created particles. Unlike the symmetric case of the Sauter field the
asymmetric form of the field under consideration causes the energy density and
longitudinal pressure of created electrons to be not equal to the energy
density and longitudinal pressure of created positrons.

###### Acknowledgements.

Gitman is grateful to CNPq for continued support.

##  Appendix A Basic elements of a nonperturbative approach to $QED$ with
$x$-steps

In this appendix, we briefly present some basic constructions of quantization
in terms of particles for QED with $x$-steps; see sections IV-VII in Ref.
GavGi16  for details.

The time-independent inner product for any pair of solutions of the Dirac
equation, $\psi_{n}\left(X\right)$ and
$\psi_{n^{\prime}}^{\prime}\left(X\right)$, is defined on the $t=$const
hyperplane as follows:

$\left(\psi_{n},\psi_{n^{\prime}}^{\prime}\right)=\int_{V_{\bot}}d\mathbf{r}_{\bot}\int\limits_{-K^{\left(\mathrm{L}\right)}}^{K^{\left(\mathrm{R}\right)}}\psi_{n}^{{\dagger}}\left(X\right)\psi_{n^{\prime}}^{\prime}\left(X\right)dx,\
\ $ (104)

where the integral over the spatial volume $V_{\bot}$ is completed by an
integral over the interval
$\left[K^{\left(\mathrm{L}\right)},K^{\left(\mathrm{R}\right)}\right]$ in the
$x$ direction. The parameters $K^{\left(\mathrm{L}/\mathrm{R}\right)}$ are
assumed sufficiently large in final expressions. Assuming that the principal
value of integral (104) is determined by integrals over the areas where the
field $E(x)$ is negligible small it is possible to evaluate this value using
only the asymptotic behavior (15) of functions in the regions where particles
are free. The field $E(x)$ in the area where it is strong enough affects only
coefficients $g$ entering into the mutual decompositions of the solutions
given by Eq. (25). One can see that the norms of the plane waves $\
{}_{\zeta}\psi_{n}\left(X\right)$ and $\ {}^{\zeta}\psi_{n}\left(X\right)$
with respect to the inner product (104) are proportional to the
macroscopically large parameters
$\tau^{\left(\mathrm{L}\right)}=K^{\left(\mathrm{L}\right)}/v^{\mathrm{L}}$
and
$\tau^{\left(\mathrm{R}\right)}=K^{\left(\mathrm{R}\right)}/v^{\mathrm{R}}$,
where
$v^{\mathrm{L}}=\left|p^{\mathrm{L}}/\pi_{0}\left(\mathrm{L}\right)\right|>0$
and
$v^{\mathrm{R}}=\left|p^{\mathrm{R}}/\pi_{0}\left(\mathrm{R}\right)\right|>0$
are absolute values of the longitudinal velocities of particles in the regions
where particles are free; see Sect. IIIC.2 and Appendix B in Ref. GavGi16  for
details.

It was shown (see Appendix B in Ref. GavGi16 ) that the following couples of
plane waves are orthogonal with respect to the inner product (104)

$\left({}_{\zeta}\psi_{n},^{-\zeta}\psi_{n}\right)=0,\ \
n\in\Omega_{1}\cup\Omega_{5}\ ;\ \
\left({}_{\zeta}\psi_{n},^{\zeta}\psi_{n}\right)=0,\ \ n\in\Omega_{3}\ ,$
(105)

if the parameters of the volume regularization
$\tau^{\left(\mathrm{L/R}\right)}$ satisfy the condition

$\tau^{\left(\mathrm{L}\right)}-\tau^{\left(\mathrm{R}\right)}=O\left(1\right),$
(106)

where $O\left(1\right)$ denotes terms that are negligibly small in comparison
with the macroscopic quantities $\tau^{\left(\mathrm{L/R}\right)}$. One can
see that $\tau^{\left(\mathrm{R}\right)}$ and $\tau^{\left(\mathrm{L}\right)}$
are macroscopic times of motion of particles and antiparticles in the areas
$S_{\mathrm{R}}$ and $S_{\mathrm{L}}$, respectively and they are equal,

$\tau^{\left(\mathrm{L}\right)}=\tau^{\left(\mathrm{R}\right)}=\tau.$ (107)

It allows one to introduce an unique time of motion $\tau$ for all the
particles in the system under consideration. This time can be interpreted as a
system monitoring time during its evolution and as such is fixed as $\tau=T$
in the framework of the renormalization procedure; see Ref. GavGi20  for
details.

The renormalization and volume regularization procedures are associated with
the introduction of a modified inner product and a parameter $\tau$ of the
regularization. Based on physical considerations, we fix this parameter. It
turns out that in the Klein range this parameter can be interpreted as the
time of the observation of the pair production process.

Under condition (106) the following orthonormality relations on the $t=$const
hyperplane are

$\displaystyle\left(\ {}_{\zeta}\psi_{n},\
_{\zeta}\psi_{n^{\prime}}\right)=\left(\ {}^{\zeta}\psi_{n},\
^{\zeta}\psi_{n^{\prime}}\right)=\delta_{n,n^{\prime}}\mathcal{M}_{n}\ ,\ \
n\in\Omega_{1}\cup\Omega_{3}\cup\Omega_{5}\ ;$
$\displaystyle\left(\psi_{n},\psi_{n^{\prime}}\right)=\delta_{n,n^{\prime}}\mathcal{M}_{n},\;n\in\Omega_{2}\cup\Omega_{4}\
,$
$\displaystyle\mathcal{M}_{n}=2\frac{\tau}{T}\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2},\
\ n\in\Omega_{1}\cup\Omega_{5},$
$\displaystyle\mathcal{M}_{n}=2\frac{\tau}{T}\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2},\
\ n\in\Omega_{3},$ $\displaystyle\mathcal{M}_{n}=2\frac{\tau}{T},\ \
n\in\Omega_{2},\mathrm{\;}\mathcal{M}_{n}=2\frac{\tau}{T},\ \ n\in\Omega_{4}.$
(108)

All the wave functions having different quantum numbers $n$ are orthogonal,
and

$\displaystyle\left(\ {}_{\zeta}\psi_{n},\ ^{-\zeta}\psi_{n}\right)=0,\ \
n\in\Omega_{1}\cup\Omega_{5}\ ,\ _{\ \zeta}\psi_{n}\ \mathrm{and}\ \
^{-\zeta}\psi_{n}\ \mathrm{independent,}$ $\displaystyle\left(\
{}_{\zeta}\psi_{n},\ ^{\zeta}\psi_{n}\right)=0,\ \ n\in\Omega_{3}\ ,\
_{\zeta}\psi_{n}\ \mathrm{and}\ \ ^{\zeta}\psi_{n}\ \mathrm{independent.}$
(109)

We denote __ the corresponding quantum numbers by $n_{k}$, so that
$n_{k}\in\Omega_{k}$. Then we identify components of the initial and final
wave packets of particles and antiparticles in Eq. (24).

We decompose the Heisenberg operator $\hat{\Psi}\left(X\right)$ in two sets of
solutions $\left\\{\ {}_{\zeta}\psi_{n}\left(X\right)\right\\}$ and $\left\\{\
{}^{\zeta}\psi_{n}\left(X\right)\right\\}$ of the Dirac equation (7) complete
on the $t=\mathrm{const}$ hyperplane. Operator-valued coefficients in such
decompositions do not depend on coordinates. Our division of the quantum
numbers $n$ in five ranges $\Omega_{k}$, implies the representation for
$\hat{\Psi}\left(X\right)$ as a sum of five operators
$\hat{\Psi}_{k}\left(X\right)$, $k=1,2,3,4,5$,

$\hat{\Psi}\left(X\right)=\sum_{k=1}^{5}\hat{\Psi}_{k}\left(X\right).$ (110)

For each of three operators $\hat{\Psi}_{k}\left(X\right)$,$\ k=1,3,5,$ there
exist two possible decompositions (24) according to the existence of two
different complete sets of solutions with the same quantum numbers $n$ in the
ranges $\Omega_{1}$, $\Omega_{3}$, and $\Omega_{5}$,

$\displaystyle\hat{\Psi}_{1}\left(X\right)$ $\displaystyle=$
$\displaystyle\sum_{n_{1}}\mathcal{M}_{n_{1}}^{-1/2}\left[\
{}_{+}a_{n_{1}}(\mathrm{in})\ _{+}\psi_{n_{1}}\left(X\right)+\
^{-}a_{n_{1}}(\mathrm{in})\ ^{-}\psi_{n_{1}}\left(X\right)\right]$
$\displaystyle=$ $\displaystyle\sum_{n_{1}}\mathcal{M}_{n_{1}}^{-1/2}\left[\
{}^{+}a_{n_{1}}(\mathrm{out})\ ^{+}\psi_{n_{1}}\left(X\right)+\
_{-}a_{n_{1}}(\mathrm{out})\ _{-}\psi_{n_{1}}\left(X\right)\right],$
$\displaystyle\hat{\Psi}_{3}\left(X\right)$ $\displaystyle=$
$\displaystyle\sum_{n_{3}}\mathcal{M}_{n_{3}}^{-1/2}\left[\
{}^{-}a_{n_{3}}(\mathrm{in})\ ^{-}\psi_{n_{3}}\left(X\right)+\
_{-}b_{n_{3}}^{\dagger}(\mathrm{in})\ _{-}\psi_{n_{3}}\left(X\right)\right]$
$\displaystyle=$ $\displaystyle\sum_{n_{3}}\mathcal{M}_{n_{3}}^{-1/2}\left[\
{}^{+}a_{n_{3}}(\mathrm{out})\ ^{+}\psi_{n_{3}}\left(X\right)+\
_{+}b_{n_{3}}^{\dagger}(\mathrm{out})\ _{+}\psi_{n_{3}}\left(X\right)\right],$
$\displaystyle\hat{\Psi}_{5}\left(X\right)$ $\displaystyle=$
$\displaystyle\sum_{n_{5}}\mathcal{M}_{n_{5}}^{-1/2}\left[\
{}^{+}b_{n_{5}}^{{\dagger}}(\mathrm{in})\ ^{+}\psi_{n_{5}}\left(X\right)+\
_{-}b_{n_{5}}^{{\dagger}}(\mathrm{in})\ _{-}\psi_{n_{5}}\left(X\right)\right]$
(111) $\displaystyle=$
$\displaystyle\sum_{n_{5}}\mathcal{M}_{n_{5}}^{-1/2}\left[\
{}_{+}b_{n_{5}}^{{\dagger}}(\mathrm{out})\ _{+}\psi_{n_{5}}\left(X\right)+\
^{-}b_{n_{5}}^{{\dagger}}(\mathrm{out})\
^{-}\psi_{n_{5}}\left(X\right)\right].$

There may exist only one complete set of solutions with the same quantum
numbers $n_{2}$ and $n_{4}$. Therefore, we have only one possible
decomposition for each of the two operators $\hat{\Psi}_{i}\left(X\right),$
$i=2,4$,

$\hat{\Psi}_{2}\left(X\right)=\sum_{n_{2}}\mathcal{M}_{n_{2}}^{-1/2}a_{n_{2}}\psi_{n_{2}}\left(X\right),\
\
\hat{\Psi}_{4}\left(X\right)=\sum_{n_{4}}\mathcal{M}_{n_{4}}^{-1/2}b_{n_{4}}^{\dagger}\psi_{n_{4}}\left(X\right).$
(112)

We interpret all $a$ and $b\ $as annihilation and all $a^{{\dagger}}$ and
$b^{{\dagger}}$ as creation operators. All $a$ and $a^{{\dagger}}$ are
interpreted as describing electrons and all $b$ and $b^{{\dagger}}$ as
describing positrons. All the operators labeled by the argument in are
interpreted as in-operators, whereas all the operators labeled by the argument
out as out-operators. This identification is confirmed by a detailed
mathematical and physical analysis of solutions of the Dirac equation with
subsequent QFT analysis of correctness of such an identification in Ref.
GavGi16 .

Taking into account the orthogonality and orthonormalization relations, we
find that the standard anticommutation relations for the Heisenberg operator
(110) yield the standard anticommutation rules for the introduced creation and
annihilation in- or out-operators.

We define two vacuum vectors $\left|0,\mathrm{in}\right\rangle$ and
$\left|0,\mathrm{out}\right\rangle$, one of which is the zero-vector for all
in-annihilation operators and the other is zero-vector for all
$\mathrm{out}$-annihilation operators. Besides, both vacua are zero-vectors
for the annihilation operators $a_{n_{2}}$ and $b_{n_{4}}$. One can verify
that the introduced vacua have minimum (zero by definition) kinetic energy and
zero electric charge and all the excitations above the vacuum have positive
energies. Then we postulate that the state space of the system under
consideration is the Fock space constructed, say, with the help of the vacuum
$\left|0,\mathrm{in}\right\rangle$ and the corresponding creation operators.
This Fock space is unitarily equivalent to the other Fock space constructed
with the help of the vacuum $\left|0,\mathrm{out}\right\rangle$ and the
corresponding creation operators if the total number of particles created by
the external field is finite.

Because any annihilation operators with quantum numbers $n_{k}$ corresponding
to different $k$ anticommute between themselves, we can represent the
introduced vacua $\left|0,\mathrm{in}\right\rangle$ and
$\left|0,\mathrm{out}\right\rangle$ as tensor products of the corresponding
partial vacua in the five ranges $\Omega_{k}$, $k=1,...,5$. The partial vacua
are stable in $\Omega_{k}$, $k=1,2,4,5$, and the vacuum instability with
$\left|c_{v}\right|\neq 1$ is due to the partial vacuum-to-vacuum transition
amplitude formed in $\Omega_{3}$. In the range $\Omega_{3}$ operators of the
number of final electrons and positrons are

$\hat{N}_{n}^{a}\left(\mathrm{out}\right)=\ ^{+}a_{n}^{\dagger}(\mathrm{out})\
^{+}a_{n}(\mathrm{out}),\;N_{n}^{b}\left(\mathrm{out}\right)=\
_{+}b_{n}^{\dagger}(\mathrm{out})\ _{+}b_{n}(\mathrm{out}).$ (113)

Using the linear canonical transformation between in and out-operators of
creation and annihilation (see Eq. (7.4) in Ref. GavGi16 ) one sees that the
differential mean numbers of electrons and positrons created from vacuum are
presented by Eq. (34). The vacuum-to-vacuum transition amplitude, given by Eq.
(7.21) in Ref. GavGi16 ), is

$c_{v}=\langle
0,\mathrm{out}|0,\mathrm{in}\rangle=\prod\limits_{n\in\Omega_{3}}g\left({}^{-}\left|{}_{-}\right.\right)g\left({}^{-}\left|{}_{+}\right.\right)^{-1}\,.$
(114)

Then the probability for a vacuum to remain a vacuum can be presented by Eq.
(37).

##  Appendix B Locally constant field approximation

A new kind of a locally constant field approximation (LCFA) was formulated in
Ref. _GavGitShi19b. _Here we pretend to show that the density of created pairs
(82) and the probability of the vacuum to remain a vacuum (83), obtained from
exact equations for the slowly varying field in the leading-term
approximation, are in agreement with results following in the framework of
LCFA.

We call the electric field $E(x)$ a weakly inhomogeneous electric field on a
spatial interval $\Delta l$ if the following condition holds true:

$\left|\frac{\overline{\partial_{x}E(x)}\Delta l}{\overline{E(x)}}\right|\ll
1,\ \ \Delta l/\Delta l_{\mathrm{st}}^{\mathrm{m}}\gg 1\ ,$ (115)

where $\overline{E(x)}$ and $\overline{\partial_{x}E(x)}$ are the mean values
of $E(x)$ and $\partial_{x}E(x)$ on the spatial interval $\Delta l$,
respectively, and $\Delta l$ is significantly larger than the length scale
$\Delta l_{\mathrm{st}}^{\mathrm{m}}$, which is

$\Delta l_{\mathrm{st}}^{\mathrm{m}}=\Delta
l_{\mathrm{st}}\max\left\\{1,m^{2}/e\overline{E(x)}\right\\},\ \ \Delta
l_{\text{{st}}}=\left[e\overline{E(x)}\right]^{-1/2}\ .$ (116)

Note that the length scale $\Delta l_{\mathrm{st}}^{\mathrm{m}}$ appears in
Eq. (115) as the length scale when the perturbation theory with respect to the
electric field breaks down and the Schwinger (nonperturbative) mechanism is
primarily responsible for the pair creation. In what follows, we show that
this condition is sufficient. We are primarily interested in strong electric
fields, $m^{2}/e\overline{E(x)}\lesssim 1$. In this case, the second
inequality in Eq. (115) is simplified to the form $\Delta l/\Delta
l_{\mathrm{st}}\gg 1$, in which the mass $m$ is absent. In such cases, the
potential of the corresponding electric step hardly differs from the potential
of a uniform electric field,

$U(x)=-eA_{0}(x)\approx U_{\mathrm{const}}(x)=e\overline{E(x)}x+U_{0},$ (117)

on the interval $\Delta l$, where $U_{0}$ is a given constant.

For an arbitrary weakly inhomogeneous strong electric field, in the leading-
term approximation, were derived universal formulas for the total density of
created pairs

$\rho^{\mathrm{\Omega}}\approx\frac{J_{(d)}}{(2\pi)^{d-1}}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}dx\
eE(x)\int d\mathbf{p}_{\bot}N_{n}^{\mathrm{uni}},\ \
N_{n}^{\mathrm{uni}}=\exp\left[-\pi\frac{\pi_{\bot}^{2}}{eE(x)}\right]\ ,$
(118)

and an expression for the probability $P_{\mathrm{v}}$ given by Eq. (37) of
the vacuum to remain a vacuum,

$P_{\mathrm{v}}\approx\exp\left\\{-\frac{V_{\bot}TJ_{(d)}}{(2\pi)^{d-1}}\sum_{l=1}^{\infty}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}dx\frac{\left[eE(x)\right]^{d/2}}{l^{d/2}}\exp\left[-\pi\frac{lm^{2}}{eE(x)}\right]\right\\}\
;$ (119)

see Ref. GavGitShi19b . In Eqs. (118) and (119) integration limits are
specified over the region $S_{\mathrm{int}}=(x_{\mathrm{L}},x_{\mathrm{R}})$,
in which the electrical the field is not zero. In our case
$x_{\mathrm{L}}=-\infty$, $x_{\mathrm{R}}=+\infty$.

Let’s compare Eqs. (118) and (119) with the results obtained above in (82). To
do this, let’s represent (118) in the form

$\displaystyle\rho^{\mathrm{\Omega}}\approx\frac{J_{\left(d\right)}}{\left(2\pi\right)^{d-1}}\int
d\mathbf{p}_{\bot}\left(J_{p_{\bot}}^{+}+J_{p_{\bot}}^{-}\right)\,,$
$\displaystyle
J_{p_{\bot}}^{+}=\int_{x_{\mathrm{M}}}^{\infty}dx\left[eE\left(x\right)\right]N_{n}^{\mathrm{univ}}=-e\int_{x_{\mathrm{M}}}^{\infty}dA_{0}(x)N_{n}^{\mathrm{univ}}\,$
$\displaystyle
J_{p_{\bot}}^{-}=\int_{-\infty}^{x_{\mathrm{M}}}dx\left[eE\left(x\right)\right]N_{n}^{\mathrm{univ}}=-e\int_{x_{\mathrm{M}}}^{\infty}dA_{0}(x)N_{n}^{\mathrm{univ}}\
.$ (120)

Note that the functions $A_{0}(x)$ and $E(x)$ can be related to each other
using the cubic equation

$y^{3}-y-\frac{2}{3\sqrt{3}(q+1)}=0,\ \ y=-A_{0}(x)/(E_{0}\sigma),\ \
q=\left(3\sqrt{3}\frac{E(x)}{E_{0}}\right)^{-1}-1\ .$ (121)

We can express $A_{0}\left(x\right)$ as a function of the field
$E\left(x\right)$ or as a function of the variable $q$ using solutions of
equation (121). This equation has three real solutions,

$\displaystyle y_{1}=\frac{2}{\sqrt{3}}\cos\frac{\alpha\left(q\right)}{3},\
y_{2}=-\frac{2}{\sqrt{3}}\cos\left[\frac{\alpha\left(q\right)}{3}+\frac{\pi}{3}\right]\
,$ $\displaystyle
y_{3}=-\frac{2}{\sqrt{3}}\cos\left[\frac{\alpha\left(q\right)}{3}-\frac{\pi}{3}\right],\
\alpha\left(q\right)=\arccos\left[\left(q+1\right)^{-1}\right]\ ,$ (122)

see, e.g., Korn . Since $A_{0}\left(x\right)$ is negative, only the solutions
$y_{2,3}$ are relevant. One can see that for solutions $y_{2,3}$ the
differential $dA_{0}\left(x\right)$ takes the form:

$\displaystyle
dA\left(x\right)=-2(E_{\max}\sigma)\sin\left[\frac{\alpha\left(q\right)}{3}+\frac{\pi}{3}\right]\frac{\left(q+1\right)^{-2}}{\sqrt{1-\left(q+1\right)^{-2}}}dq,\
\ x>0\ ,$ $\displaystyle
dA\left(x\right)=-2(E_{\max}\sigma)\sin\left[\frac{\alpha\left(q\right)}{3}-\frac{\pi}{3}\right]\frac{\left(q+1\right)^{-2}}{\sqrt{1-\left(q+1\right)^{-2}}}dq\
\ x\leq 0\ ,$ $\displaystyle E(x)=-dA(x)=\frac{E_{\max}}{q+1}\ .$ (123)

Passing from the integration over $x$ to the integration over the parameter
$q$ in Eq. (120), we find:

$J_{p_{\bot}}^{\pm}=I_{p_{\bot}}^{\pm}\ ,$ (124)

where the quantities $I_{p_{\bot}}^{\pm}$ are given by Eq. (80).

It follows from Eq. (124) that the density of the created pairs (118) and the
probability of the vacuum to remain a vacuum (119) obtained with the help of
the approximation for weakly inhomogeneous strong electric field coincide with
expressions (82) and (83), respectively.

##  Appendix C Some properties of hypergeometric functions

The hypergeometric function
$F\left(a,b,c;z\right)=\,_{2}F_{1}\left(a,b,c;z\right)$ (here and in what
follows it is supposed that parameters $a$ and $b$ are not equal to $0,-1,-2$,
$\ldots$) is defined by series

$F\left(a,b,c;z\right)=\sum_{n=0}^{+\infty}\frac{\left(a\right)_{n}\left(b\right)_{n}}{\left(c\right)_{n}}\frac{z^{n}}{n!}=\frac{\Gamma\left(c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}\sum_{n=0}^{+\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{\Gamma\left(c+n\right)}\frac{z^{n}}{n!},\
\ \left|z\right|<1.$ (125)

Note that in the solutions (13) and (14) the arguments $1-\xi^{-1}$ and
$\xi^{-1}$ in the corresponding hypergeometric functions are less than unity
and the series (125) converges.

At $\left|z\right|=1$ the series (125) converges absolutely when
$\mathrm{Re}\left(c-a-b\right)>0$. The integral representation

$F\left(a,b,c;z\right)=\frac{\Gamma\left(c\right)}{\Gamma\left(b\right)\Gamma\left(c-b\right)}\int_{0}^{1}t^{b-1}\left(1-t\right)^{c-b-1}\left(1-zt\right)^{-a}dt,\
\ \left(\mathrm{Re}c>\mathrm{Re}b>0\right)$ (126)

gives an analytical continuation for the function $F\left(a,b,c;z\right)$ to
the complex $z$-plane with a cut along the real axis from $1$ to $\infty$
(since the right-hand side is an unambiguous analytic function in the domain
$\left|\arg\left(1-z\right)\right|\leq\pi$). From the integral representation
(126) it is easy to see that $\lim_{z\rightarrow 0}F\left(a,b,c;z\right)=1.$
The formula for differentiating the hypergeometric function has the form:

$\frac{d}{dz}F\left(a,b,c;z\right)=\frac{ab}{c}F\left(a+1,b+1,c+1;z\right).$
(127)

It is follows from (126) that

$F\left(a,b,c;z\right)=\left(1-z\right)^{c-a-b}F\left(c-a,c-b,c;z\right),\ \
\left|z\right|<1.$ (128)

Hypergeometric function can be transformed as

$\displaystyle F\left(a,b,c;z\right)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}F\left(a,b,a+b-c+1;1-z\right)$
(129) $\displaystyle+$
$\displaystyle\left(1-z\right)^{c-a-b}\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}F\left(c-a,c-b,c-a-b+1;1-z\right)\,,$
$\displaystyle\left(\left|\arg\left(1-z\right)\right|<\pi\right),$

The hypergeometric equation in its general form,

$z\left(1-z\right)w^{\prime\prime}\left(z\right)+\left[c-\left(a+b+1\right)z\right]w^{\prime}\left(z\right)-abw\left(z\right)=0,$
(130)

has three regular singular points $z=0,1,\infty$. When none of the numbers
$c$, $c-a-b$, $a-b$ is integer, the general solution $w\left(z\right)$ of the
hypergeometric equation (130) can be obtained as

$\displaystyle
w\left(z\right)=c_{1}w_{1}\left(z\right)+c_{2}w_{2}\left(z\right),\ \text{
}z\rightarrow 0,$ $\displaystyle
w\left(z\right)=c_{1}w_{3}\left(z\right)+c_{2}w_{4}\left(z\right),\ \
z\rightarrow 1,$ $\displaystyle
w\left(z\right)=c_{1}w_{5}\left(z\right)+c_{2}w_{6}\left(z\right),\ \
z\rightarrow\infty\text{.}$ (131)

where $c_{1}$ and $c_{2}$ are some constants, and the functions
$w_{j}\left(z\right)$, $j=1,\ldots,6$, have the form:

$\displaystyle w_{1}\left(z\right)=F\left(a,b,c;z\right),\ \
w_{2}\left(z\right)=z^{1-c}F\left(a-c+1,b-c+1,2-c;z\right),$ $\displaystyle
w_{3}\left(z\right)=F\left(a,b,a+b+1-c;1-z\right),\ $ $\displaystyle
w_{4}\left(z\right)=\left(1-z\right)^{c-a-b}F\left(c-b,c-a,c-a-b+1,1-z\right),$
$\displaystyle w_{5}\left(z\right)=z^{-a}F\left(a,a-c+1,a-b+1,z^{-1}\right),\
$ $\displaystyle
w_{6}\left(z\right)=z^{-b}F\left(b,b-c+1,b-a+1,z^{-1}\right).$ (132)

The Kummer relations and for the hypergeometric equation Bateman  allow us to
represent the functions $w_{1}\left(z\right)$ and $w_{2}\left(z\right)$ via
the functions $w_{3}\left(z\right)$ and $w_{4}\left(z\right)$,

$\displaystyle w_{1}\left(z\right)$ $\displaystyle=$ $\displaystyle
e^{i\pi\left(2\alpha_{1}-b\right)}\frac{\Gamma\left(2\left(\alpha_{1}+1\right)-a-b\right)\Gamma\left(b-a+1\right)}{\Gamma\left(2-a\right)\Gamma\left(2\alpha_{1}-a+1\right)}w_{4}\left(z\right)$
$\displaystyle-$ $\displaystyle
e^{i\pi\left(2\alpha_{1}-a\right)}\frac{\Gamma\left(2\left(\alpha_{1}+1\right)-a-b\right)\Gamma\left(a-b-1\right)}{\Gamma\left(1-b\right)\Gamma\left(2\alpha_{1}-b\right)}w_{3}\left(z\right),$
$\displaystyle w_{2}\left(z\right)$ $\displaystyle=$ $\displaystyle
e^{i\pi\left(a-1\right)}\frac{\Gamma\left(a+b-2\alpha_{1}\right)\Gamma\left(b-a+1\right)}{\Gamma\left(b-2\alpha_{1}+1\right)\Gamma\left(b\right)}w_{4}\left(z\right)$
(133) $\displaystyle+$ $\displaystyle e^{i\pi
b}\frac{\Gamma\left(a+b-2\alpha_{1}\right)\Gamma\left(a-b-1\right)}{\Gamma\left(a-2\alpha_{1}\right)\Gamma\left(a-1\right)}w_{3}\left(z\right).$

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