11institutetext: Institute for Theoretical Physics, University of Wroclaw,
50-204 Wroclaw, Poland 22institutetext: Bogoliubov Laboratory for Theoretical
Physics, Joint Institute for Nuclear Research, RU - 141980 Dubna, Russia
33institutetext: National Research Nuclear University (MEPhI), RU - 115409
Moscow, Russia 44institutetext: Saratov State University, RU - 410026 Saratov,
Russia 55institutetext: A.I. Alikhanyan National Science Laboratory (YerPhI),
0036 Yerevan, Armenia 66institutetext: Byurakan Astronomical Observatory,
National Academy of Sciences, 0213 Byurakan, Armenia 77institutetext: Tomsk
State University, RU - 634050 Tomsk, Russia
77email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Approximate Solutions of a Kinetic Theory for Graphene

D. B. Blaschke 112233 V. V. Dmitriev 44 N. T. Gevorgyan 5566 B. Mahato 11 A.
D. Panferov 44 S. A. Smolyansky 4477 V. A. Tseryupa 44

###### Abstract

The effective mass approximation is analysed in a nonperturbative kinetic
theory approach to strong field excitations in graphene [1, 2]. This problem
is highly actual for the investigation of quantum radiation from graphene [3],
where the collision integrals in the photon kinetic equation are rather
complicated functionals of the distribution functions of the charge carriers.
These functions are needed in the explicit analytical definition as solutions
of the kinetic equations for the electron-hole excitations. In the present
work it is shown that the suggested approach is rather effective in a certain
range of parameters for the pulse of an external electromagnetic field. For
example, the applicability condition of the approximation in the case of a
harmonic field is $\hbar\omega^{2}/(\sqrt{2}eE_{0}v_{F})<1,$ were $v_{F}$ is
the Fermi velocity. In the standard massive quantum electrodynamics the
usability of the analogical approximation is very narrow.

###### Keywords:

Graphene Kinetic theory Strong field.

##  1 Introduction

It is well known that the interaction of carriers in graphene with with an
external electromagnetic field is nonanalytic in the coupling constant [4, 5].
This renders applications of perturbation theory unjustified and stimulates
the search for nonperturbative approaches. Paradigms are here exactly solvable
quantum field theory models [6, 7]. However, these solutions are limited to
narrow classes of external field models (constant electric field, Eckart’s
potential). Alternative cases are founded either on direct application of the
basic equations of motion [8, 9] or on the nonperturbative kinetic theory [1,
2, 3, 10] constructed in analogy to the standard strong field QED [11, 12, 13,
14].

A higher level of description of the graphene excitations is connected with
the investigation of different mechanisms of radiation. In the simplest case
the question is about the emission of a quasiclassical electromagnetic field
by the plasma currents in graphene [15, 16]. There is also the radiation of a
quantized field that is the result of the direct interaction of the charge
carriers with the photon field. A consistent realization of this approach is
based on a truncation procedure of the Bogoliubov-Born-Green-Kirkwood-Yvon
(BBGKY) chain of equations for the correlation functions of the $eh\gamma$ –
system [3]. This leads to a closed system of kinetic equations (KEs) with
collision integrals (CIs) of non-Markovian type in the electron-hole and
photon sectors and with the Maxwell equation for the acting inner (plasma)
field. As far as the evolution of the $eh$-plasma is accompanied by high-
frequency quantum (vacuum, as in the case of the electron - positron plasma)
oscillations, any solution of this KE system turns out very susceptible to the
selection not only of the model and the parameters of the external field but
also to the necessary roughening in the process of calculating the physical
quantities.

Below we will consider this problem on the examples of two approximative
methods for the solution of the basic KEs describing the production of
$eh$-pairs under the action of an external field: the low density
approximation [17] and the method of the asymptotic decompositions [18] (Sect.
2). The ideas for these approaches are borrowed from the standard QED. In the
considered case of the massless theory, an essential role plays the effective
mass approximation [19]. The results of analytical calculations of the basic
functions of the kinetic theory are compared among themselves and with the
exactly solvable model (Sect. 3). The considered examples show that the
introduction of the effective electromagnetic mass allows to obtain rather
simple expansions for the distribution functions of the $eh$-plasma in
graphene for time dependent electric fields with different pulse shape.

##  2 The basic KE and its approximate solutions

The basic KE for description of excitation and evolution of the $eh$-plasma in
graphene under the action of an external quasiclassical spatially uniform
time-dependent electric field with the vector potential
$A^{\mu}=(0,A^{(1)}(t),A^{(2)}(t),0)$ in the Hamiltonian gauge $A^{(0)}=0$ can
be written in the integro-differential form [1, 2, 3, 10]

$\dot{f}(\mathbf{p},t)=\frac{1}{2}\lambda(\mathbf{p},t)\int_{t_{0}}^{t}\lambda(\mathbf{p},t^{\prime})[1-2f(\mathbf{p},t^{\prime})]\cos\theta(\mathbf{p};t,t^{\prime})dt^{\prime}$
(1)

or as the equivalent system of ordinary differential equations

$\displaystyle\dot{f}(\mathbf{p},t)=\frac{1}{2}\lambda(\mathbf{p},t)u(\mathbf{p},t),$
(2)
$\displaystyle\dot{u}(\mathbf{p},t)=\lambda(\mathbf{p},t)\left[1-2f(\mathbf{p},t)\right]-\frac{2\varepsilon(\mathbf{p},t)}{\hbar}\nu(\mathbf{p},t),\
\dot{\nu}(\mathbf{p},t)=\frac{2\varepsilon(\mathbf{p},t)}{\hbar}u(\mathbf{p},t).$

Here $f(\mathbf{p},t)$ is the distribution function of charge carriers
introduced by taking into account the electroneutrality
$f(\mathbf{p},t)=f_{e}(\mathbf{p},t)=f_{h}(-\mathbf{p},t)$. The excitation
function $\lambda(\mathbf{p},t)$ in the low energy model is determined as

$\lambda(\mathbf{p},t)=\frac{ev_{F}^{2}[E^{(1)}(t)P^{2}-E^{(2)}(t)P^{1}]}{\varepsilon^{2}(\mathbf{p},t)},$
(3)

where $v_{F}=10^{6}\ $ m/s is the Fermi velocity,
$E^{(k)}=-\frac{1}{c}\dot{A}^{(k)}(t)$ is the field strength $(k=1,2)$,
$P^{k}=p^{k}+\frac{e}{c}A^{k}(t)$ is the quasi-momentum and
$\varepsilon(\mathbf{p},t)=v_{F}\sqrt{P^{2}}$ is the quasi-energy. The
electron charge is $-e$. Finally, the quantity
$\theta(\mathbf{p};t,t^{\prime})$ in the KE (1) is the phase,

$\theta(\mathbf{p};t,t^{\prime})=\frac{2}{\hbar}\int_{t^{\prime}}^{t}dt^{\prime\prime}\varepsilon(\mathbf{p},t^{\prime\prime}).$
(4)

The KE (1) is an integro-differential equation of the non-Markovian type with
a fastly oscillating kernel. There is an integral of motion

$(1-2f)^{2}+u^{2}+v^{2}={\rm const},$ (5)

where the constant is fixed with the corresponding initial condition.

The KEs (1), (2) for the case of graphene were obtained in the works [1, 2, 3,
10] by the Bogoliubov method of canonical transformations that can by realized
in the massless $D=2+1$ theory in an explicit form. On the other hand the
system (2) can be reduced from the general system of twelve KEs in the
standard QED [20].

The massless low energy model of graphene with the lightlike dispersion law
$\varepsilon(\mathbf{p},t)$ leads to the absence of the critical field that is
characteristic for massive QED and results in a specific feature of the
momentum dependence of the excitation function $\lambda(\mathbf{p},t)$ (3):
this function decreases in the ultraviolet area, $\lambda(\mathbf{p},t)\sim
1/P\rightarrow 0$ at $P\rightarrow\infty$ and is singular in the infra-red
area $\lambda(\mathbf{p},t)\rightarrow\infty$ at $P\rightarrow 0$. The last
distinction also leads to a nonanalytic structure of the theory in its
dependence on the coupling constant and to the absence of the standard
perturbation theory.

Let us write also the differential equation of the third order that is
equivalent to the system of equations (2),

$\stackrel{{\scriptstyle...}}{{g}}+\ddot{g}\left[\varepsilon\lambda\left(\frac{1}{\varepsilon\lambda}\right)^{\prime}-\frac{\dot{\lambda}}{\lambda}\right]+\dot{g}\left[4\varepsilon^{2}+\lambda^{2}-\varepsilon\lambda\left(\frac{\dot{\lambda}}{\varepsilon\lambda^{2}}\right)^{\prime}\right]+g\varepsilon\lambda\left(\frac{\lambda}{\varepsilon}\right)^{\prime}=0,$
(6)

where $g=1-2f$. In Eq. (6) the ′ denotes also the time derivative.

At the present time, an exact solution of the KEs (1), (2) is not known.
However, below we will assume that the well-known exact solutions of the Dirac
equation for a constant electric field and the Eckart potential are at the
same time solutions of the KEs (1), (2). This assumption gives a basis for
comparing these exact solutions with the known approximate solutions of the
KEs (1), (2) and to construct then some new classes of approximate solutions.

In order to estimate the effectivity of the approximate solutions, we will
compare them to the exact solutions (analytical and numerical). Such a
comparison will be made on the level of an integral macroscopic quantity. The
number density of pairs $n(t)$ will be considered as the simplest quantity of
such type,

$n(t)=\frac{N_{f}}{(2\pi\hbar)^{2}}\int f(\mathbf{p},t)d\mathbf{p},$ (7)

where $N_{f}=4$ is the number of flavours. The integration allows here to
smoothen out some insignificant details in the momentum dependence of the
distribution functions in different approximations.

As the next step, we will consider two approximate methods of solving the KEs
(1), (2). To this end, we will consider the case of a linearly polarized
electric field $A^{(1)}=0,\ A^{(2)}(t)=A(t)$.

###  2.1 Low density approximation

This approximation corresponds to the limit $f\ll 1$ in the r.h.s. KE (1). It
was introduced in the strong field vacuum production of charged particles [17]
and was used many times in strong field QED and in the kinetic theory of
excitations in graphene. It leads to the quadrature formula [1, 2]

$f(t)=\frac{1}{4}\left[\int_{-\infty}^{t}dt^{\prime}\lambda_{s}(t^{\prime})\right]^{2}+\frac{1}{4}\left[\int_{-\infty}^{t}dt^{\prime}\lambda_{c}(t^{\prime})\right]^{2},$
(8)

where

$\lambda_{c}(t)=\lambda(t)\cos\theta(t,-\infty),\ \
\lambda_{s}(t)=\lambda(t)\sin\theta(t,-\infty).$ (9)

In particular, it follows from Eq. (8), that $f(\mathbf{p},t)\geq 0$.

In the low-density approximation, the two last KEs of the system (2) separate
from it,

$\dot{u}=\lambda-\frac{2\varepsilon}{\hbar}v,\
\dot{v}=\frac{2\varepsilon}{\hbar}u.$ (10)

Then the distribution function $f$ can be found from the first equation of the
system (2) alone.

The system of equations (10) corresponds to the ordinary differential
equation,

$\frac{\hbar}{2\varepsilon}\frac{d}{dt}\left(\frac{\hbar}{2\varepsilon}\frac{du}{dt}\right)+u-\frac{\hbar}{2\varepsilon}\frac{d}{dt}\left(\frac{\hbar\lambda}{2\varepsilon}\right)=0$
(11)

or

${\cal D}^{2}u+u-{\cal D}\left(\frac{\hbar\lambda}{2\varepsilon}\right)=0,\
{\cal D}=\frac{\hbar}{2\varepsilon}\frac{d}{dt}.$ (12)

###  2.2 Effective electromagnetic mass approximation

This approximation was used already in the case of the harmonic model of an
external field in the analysis of the radiation effects in the $e^{-}e^{+}$
plasma [21] and in the $eh$ plasma in graphene [3]. Below we will consider a
generalization of this approach to other models of the external field.

The idea of the method is that the time dependent value of the square of the
kinetic momentum $P^{2}(t)$ in the definition of the quasienergy
$\varepsilon(\mathbf{p},t)$ gets substituted by corresponding time average
$\prec P^{2}(t)\succ$, where the symbol $\prec...\succ$ means the averaging
procedure over a characteristic time of the external field. In the case of the
linearly polarized electric field we obtain

$\prec P^{2}_{2}(t)\succ=p^{2}_{2}+(e/c)^{2}\prec A^{2}(t)\succ,$ (13)

if $\prec A(t)\succ=0$. Implying the substitution

$P^{2}_{2}(t)\rightarrow\prec P^{2}_{2}(t)\succ,$ (14)

in the square of quasienergy
$\varepsilon^{2}(\mathbf{p},t)=v^{2}_{F}[p^{2}_{1}+P^{2}_{2}(t)]$, one can
introduce the longitudinal (with respect to external field) effective
electromagnetic mass

$(e/c)^{2}\prec A^{2}(t)\succ=m^{2}_{\ast}v^{2}_{F},$ (15)

or

$m_{\ast}^{2}=\frac{e^{2}}{c^{2}v_{F}^{2}}\frac{1}{2T}\int_{-T}^{T}dtA^{2}(t).$
(16)

Thus, this approximation corresponds to change

$\varepsilon(\mathbf{p},t)\rightarrow\varepsilon_{\ast}(\mathbf{p})=v_{F}\sqrt{p_{1}^{2}+(p_{2}^{2}+m^{2}_{\ast}v^{2}_{F})}.$
(17)

The appearance of the longitudinal mass is a reflection of the anisotropy of
the system stipulated by the presence of the external field and leads to a
reduction of the mobility of charge carriers along the direction of the action
of the external field. In the limiting case $p_{2}\ll m_{\ast}v_{F}$, we
obtain a strong anisotropic momentum dependence of the quasienergy,

$\varepsilon_{\ast}(\mathbf{p})=v_{F}\sqrt{m^{2}_{\ast}v^{2}_{F}+p_{1}^{2}}.$
(18)

The approximation of the effective mass (14), (15) is valid in field models
with square integrable functions $A(t)$ only.

The transition amplitude (3) in the effective mass approximation in the
linearly polarized external field will be

$\lambda_{\ast}(\mathbf{p},t)=-\frac{ev^{2}_{F}p_{1}}{\varepsilon^{2}_{\ast}(\mathbf{p})}E(t)\equiv\Lambda(\mathbf{p})E(t),$
(19)

where $p_{1}=\cos\varphi$ and $\varphi$ is the polar angle.

The problem of evaluating the distribution function is now brought to the
calculation of the integral

$J(\mathbf{p},t)=\int_{-\infty}^{t}dt^{\prime}E(t^{\prime})\cos\theta_{\ast}(\mathbf{p};t^{\prime},-\infty),$
(20)

where $\theta_{\ast}(\mathbf{p};t^{\prime},-\infty)$ is defined by the
relation (4) with the replacement
$\varepsilon(\mathbf{p},t)\rightarrow\varepsilon_{\ast}(\mathbf{p})$. The
analogous integral with the replacement
$\cos\theta_{\ast}\rightarrow\sin\theta_{\ast}$ is equal to zero, if
$E(t)=E(-t)$. The distribution function (8) will then be

$f(\mathbf{p},t)=\frac{1}{4}\Lambda^{2}(\mathbf{p})J^{2}(\mathbf{p},t).$ (21)

According to the relations (19) and (21), the anisotropy of the distribution
function $\sim\cos^{2}\varphi$ is universal and does not depend on the
selection of the external field model. Another distinctive feature of the
amplitude (19) is the singular slit of the surfaces $\lambda(\mathbf{p},t)$ on
the plane $(p_{1},p_{2})$ along the axis $p_{1}=0$ (or $\varphi=\pi/2$) for
$p_{2}\neq 0$, i.e. $\lambda(\mathbf{p},t)|_{p_{1}=0}=0,$ and
$\lambda(\mathbf{p},t)|_{p_{1}\neq 0}\neq 0$. As it follows from the KEs (1)
and (2), this peculiarity is reproduced also in the distribution function,
$f(\mathbf{p},t)|_{p_{1}=0}=0$. This means that quasiparticles are not created
in the directions of the external field action in the strict sense. Fig. 1
demonstrates this on the example of the Sauter impulse. This slit is evident
in the figures shown below in the case of the massless version of the theory.
The introduction of a mass results in a widening of this slit and in the
appearance of the energy gap. Let us remark, that the exact solution of the
problem [7] in the case of the Sauter pulse field has the singular line
$p_{1}=0$ in the case of the linearly polarized external field. The presence
of this infinitely thin slit is not reflected in calculations of the integral
”observable” macroscopical averages of the type of the pair number density
(7).

Figure 1: The residual distribution function $f_{out}(x_{1},x_{2})$ for the
Sauter field model with $E=1000\ {\rm V/cm},\kappa=10^{12}\ {\rm Hz}$, where
$x_{1,2}=p_{1,2}/m_{\ast}v_{F}$ are dimensionless momenta.

###  2.3 Method of asymptotic decompositions

This method is adopted from the standard strong field QED [18]. We consider
now the dimensionless excitation amplitude in the exact case (3)

$\Lambda(\mathbf{p},t)=e\hbar
v^{2}_{F}E(t)p^{1}/2\varepsilon^{3}(\mathbf{p},t)$ (22)

and in the effective mass approximation

$\Lambda_{\ast}(\mathbf{p},t)=e\hbar
v^{2}_{F}E(t)p^{1}/2\varepsilon^{3}_{\ast}(\mathbf{p},t).$ (23)

In contrast to $\Lambda(\mathbf{p},t)$, the amplitude
$\Lambda_{\ast}(\mathbf{p},t)$ (23) is limited everywhere,

$\Lambda_{\ast}(\mathbf{p},t)\leq\Lambda_{\ast}^{max}=\frac{e\hbar
E_{0}}{3\sqrt{3}m^{2}_{\ast}v^{3}_{F}},$ (24)

where $\Lambda_{\ast}^{max}$ is the maximal value of the amplitude (23) in the
point of time where $E(t)=E_{0}$ (see Fig.1).

In order to clarify the physical meaning of the parameter
$\Lambda_{\ast}^{max}$ given in (24), let us consider the case of a harmonic
field, where the momentum of the electromagnetic field is equal to
$p_{A}=(e/c)A(t)\sim eE_{0}/\omega$. It corresponds to the contribution of the
electromagnetic field in the quasienergy
$\varepsilon_{A}=v_{F}p_{A}=eE_{0}v_{F}/\omega$. Then the relation (24) can be
rewritten as

$\Lambda_{\ast}^{max}=\frac{2\hbar\omega^{2}}{3\sqrt{3}eE_{0}v_{F}}\sim\frac{\hbar\omega}{\varepsilon_{A}},$
(25)

which corresponds to the ratio of the energy of the absorbed quant of external
field to the part of energy of quasiparticle acquired as a result of
acceleration in this field.

In the case of a sufficiently large low-frequency external field,
$\hbar\omega\ll eE_{0}v_{F}$, one can search a solution of the KE system (2)
by means of an asymptotic decomposition of the functions $f,\ u,\ v$ for the
small parameter $\Lambda_{\ast}(\mathbf{p},t)\ll\Lambda_{\ast}^{max}\ll 1$,

$f=\sum_{n=0}f_{n},\quad u=\sum_{n=0}u_{n},\quad v=\sum_{n=0}v_{n}.$ (26)

Substituting these decompositions in the KE system (2) and equating the
contributions of the same orders, one can obtain the leading terms of the
asymptotic decompositions (26) as

$\displaystyle f_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Lambda^{2}_{\ast}(\mathbf{p},t)=\frac{e^{2}v^{4}_{F}E^{2}(t)p_{1}^{2}}{16\varepsilon^{6}_{\ast}},$
(27) $\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2\varepsilon_{\ast}}\dot{\Lambda}_{\ast}(\mathbf{p},t)=\frac{ev^{2}_{F}\dot{E}(t)p_{1}}{4\varepsilon^{4}_{\ast}},\quad
v_{2}=\Lambda_{\ast}(\mathbf{p},t).$

Formally, these expressions have the same form as the analogous results in
standard QED which were obtained in the framework of the asymptotic
decompositions of the functional series in $E_{0}/E_{c}\ll 1$, where
$E_{c}=m^{2}c^{3}/e\hbar$ is the critical field. In a similar way one can
obtain the post-leading terms.

The obtained asymptotic solutions of the KE (2) can be used for estimating the
convergences of the integral macroscopical physical values (e.g., the
densities of the conduction and polarization currents an so on) and also in
analytical calculations in theory of radiation and other transport phenomena.

Let us consider now the realization of the effective mass approximation for
different external field models.

##  3 Approximate solutions of KEs for different external field models in
graphene

###  3.1 The Sauter pulse

The field of this pulse is given by

$A(t)=-(cE_{0}/\kappa)\tanh\kappa t,\ E(t)=E_{0}/\cosh^{2}\kappa t~{}.$ (28)

It is a classical example of the external field model leading to an exact
solution of the basic equations of motion of QED [11]. The analogous solution
for the massless graphene model was obtained in the work [7].

The effective electromagnetic mass (16) in this model is

$m_{\ast}=eE_{0}/v_{F}\kappa.$ (29)

The corresponding integral (20) is

$J(\mathbf{p},t)=\frac{E_{0}}{\kappa}\int_{-\infty}^{\kappa
t}dx\frac{\cos[\Omega(\mathbf{p})/\kappa]x}{\cosh^{2}x},$ (30)

where $\Omega(\mathbf{p})=2\varepsilon_{\ast}(\mathbf{p})/\hbar$. The
corresponding distribution function $f(\mathbf{p},t)$ is defined then
according to the Eq. (21). The vacuum polarization functions $u(\mathbf{p},t)$
and $v(\mathbf{p},t)$ can be restored then with help of the Eqs. (2). In the
asymptotic limit $t\rightarrow\infty$ it follows that

$J_{out}(\mathbf{p})=\frac{2E_{0}}{\kappa}\frac{\pi\Omega(\mathbf{p})/2\kappa}{\sinh[\pi\Omega(\mathbf{p})/2\kappa]}.$
(31)

Then the distribution function (21) in the out-state will be

$f_{out}(\mathbf{p})=\left[\frac{eE_{0}v^{2}_{F}p_{1}}{\kappa\varepsilon^{2}_{\ast}(\mathbf{p})}\frac{\pi\Omega(\mathbf{p})/2\kappa}{\sinh[\pi\Omega(\mathbf{p})/2\kappa]}\right]^{2}.$
(32)

The corresponding expression for the pair number density (7) written in terms
of the dimensionless momentum $x_{k}=p_{k}/m_{\ast}v_{F}$ is

$n_{out}=\frac{N_{f}}{\pi}\left(\frac{\eta^{2}\kappa}{8\pi
v_{F}}\right)^{2}\int_{0}^{\infty}\frac{dx\
x^{3}}{1+x^{2}}\frac{1}{\left[\sinh\left(\frac{\eta}{2}\sqrt{1+x^{2}}\right)\right]^{2}},$
(33)

where

$\eta=\frac{2\pi m_{\ast}v^{2}_{F}}{\hbar\kappa}=\frac{2\pi
eE_{0}v_{F}}{\hbar\kappa^{2}}.$ (34)

The point $\eta=1$ separates two domains: the domain $\eta<1$, where the
energy of quasiparticles acquired in the external field per unit of time
$eE_{0}v_{F}/\kappa$ is less than the energy of an absorbed quant of the field
$\hbar\kappa$, and the domain $\eta>1$, where the field acceleration mechanism
dominates.

Figure 2: Sauter field model for two values $\kappa=10^{12}\ {\rm s}^{-1}$ and
$\kappa=10^{13}\ {\rm s}^{-1}$. Blue lines are numerical solutions of KE (1),
orange lines are approximate solutions (33), green lines are approximate
solutions (35).

Some results of the numerical comparison of the exact and approximate (34)
dependencies $n_{out}(\eta)$ are given in Fig. 2. From here it follows that
the effective electromagnetic mass approximation is valid in the domain
$\eta\leq 1$ and can be dubbed ”slow switching” with $\kappa\gg(2\pi
eE_{0}v_{F}/\hbar)^{1/2}$. Assuming that $\eta\ll 1$ on the r.h.s. of Eq.
(33), one can obtain

$n_{out}=-\frac{N_{f}}{\pi}\left(\frac{eE_{0}}{2\hbar\kappa}\right)^{2}\left[1+\frac{1}{2}\ln\frac{1}{24}+\ln\eta\right],\quad\eta\ll
1.$ (35)

This corresponds to the result obtained in [7].

###  3.2 The Gaussian pulse

This field model

$A(t)=-\sqrt{\frac{\pi}{2}}cE_{0}\tau{\rm
erf}\left(\frac{t}{\sqrt{2}\tau}\right),\ E(t)=E_{0}\exp(-t^{2}/2\tau^{2})$
(36)

results in the effective mass

$\displaystyle m_{\ast}$ $\displaystyle=$
$\displaystyle\frac{eE_{0}\tau\pi^{1/4}}{v_{F}}\left(\sqrt{2}\ {\rm
erf}(1/\sqrt{2})e^{-1/2}+\sqrt{\pi}\ {\rm erf}^{2}(1/\sqrt{2})/2-{\rm
erf}(1)\right)^{1/2}$ (37) $\displaystyle\sim$ $\displaystyle 0.5257\
\frac{eE_{0}\tau}{v_{F}}.$

The distribution function will be

$f(\mathbf{p},t)=\frac{1}{2}\left[\frac{eE_{0}v^{2}_{F}\tau
p_{1}}{\varepsilon^{2}_{\ast}(\mathbf{p})}\
I\left(\frac{t}{\sqrt{2}\tau},\sigma\right)\right]^{2},$ (38)

where $\sigma=\Omega(\mathbf{p})\tau$ and

$I\left(\frac{t}{\sqrt{2}\tau},\sigma\right)=\int_{0}^{t/\sqrt{2}\tau}dx\cos(\sqrt{2}\sigma
x)e^{-x^{2}}.$ (39)

A simple result follows from Eq. (38) and (39) in the asymptotic case
$t\rightarrow\infty$,

$f_{out}(\mathbf{p})=\frac{\pi}{2}\left\\{\frac{eE_{0}\tau
v_{F}^{2}p_{1}}{\varepsilon^{2}_{\ast}(\mathbf{p})}\exp\left[-\frac{2\varepsilon^{2}_{\ast}\tau^{2}}{\hbar^{2}}\right]\right\\}^{2}.$
(40)

From here one can find after simple calculations the pair number density in
the out-state,

$n_{out}=-N_{f}\left(\frac{e\tau
E_{0}}{4\hbar}\right)^{2}\left[e^{-\xi^{2}}+(1+\xi^{2}){\rm
Ei}(-\xi^{2})\right],$ (41)

where $\xi=2m_{\ast}v^{2}_{F}\tau/\hbar$ and ${\rm Ei}(-\xi^{2})$ is the
exponential integral function. This result in the region $\xi\ll 1$
corresponds to Eq. (35) for the case of the Sauter pulse.

Figure 3: The pair number densities $n_{out}$ for the numerical (blue lines)
and approximate (orange lines) solutions for the Sauter ($\kappa=10^{12}\
\rm{s}^{-1}$) and Gauss ($\tau=10^{-12}\ \rm{s}$) pulses.

In Fig. 3 we compare the behaviour of the pair number densities $n_{out}$ for
the exact and approximate solutions for the Sauter ($\kappa=10^{12}\
\rm{s}^{-1}$) and Gauss ($\tau=10^{-12}\ \rm{s}$) pulses.

###  3.3 The harmonic field model

The harmonic field model

$A(t)=-(cE_{0}/\omega)\sin\omega t,\ E(t)=E_{0}\cos\omega t;$ (42)

corresponds to the effective mass

$m_{\ast}=eE_{0}/\sqrt{2}v_{F}\omega.$ (43)

The distribution function in this field model was obtained in the work [3],

$f(\mathbf{p},t)=f^{(0)}(\mathbf{p})+f^{(2)}(\mathbf{p},t),$ (44)

where the function

$f^{(0)}(\mathbf{p})=\frac{(e\hbar
E_{0}v_{F}^{2}p_{1})^{2}(4\varepsilon_{\ast}^{2}+\hbar^{2}\omega^{2})}{8\varepsilon_{\ast}^{4}(4\varepsilon_{\ast}^{2}-\hbar^{2}\omega^{2})^{2}}$
(45)

corresponds to a stationary background distribution while the function

$f^{(2)}(\mathbf{p},t)=\frac{(e\hbar
E_{0}v_{F}^{2}p_{1})^{2}}{8\varepsilon_{\ast}^{4}(4\varepsilon_{\ast}^{2}-\hbar^{2}\omega^{2})}\cos
2\omega t$ (46)

corresponds to the breathing mode on the doubled frequency of the external
field. The residual functions $u,\ v$ can be reconstructed using Eqs.
(44)-(46) and the KE system (2)

$u(\mathbf{p},t)=\frac{e\hbar^{2}v_{F}^{2}p_{1}}{\varepsilon_{\ast}^{2}(4\varepsilon_{\ast}^{2}-\hbar^{2}\omega^{2})}\dot{E}(t),$
(47)
$v(\mathbf{p},t)=\frac{2e\hbar^{2}v_{F}^{4}p_{1}}{\varepsilon_{\ast}(4\varepsilon_{\ast}^{2}-\hbar^{2}\omega^{2})}E(t).$
(48)

The distribution function (44)-(45) corresponds to the first and third
harmonics of the current density and radiation spectrum of the plasma
oscillations [3]. These results are valid in the case

$\hbar\omega^{2}/(\sqrt{2}eE_{0}v_{F})<1.$ (49)

This limitation holds also for other field models, if the quantity $\omega$ is
interpreted as the corresponding characteristic frequency of the field
alteration.

A general feature of the two outlined approximate approaches is the $E^{2}$ \-
proportionality of all the resulting distribution functions,
$f(\mathbf{p},t)\sim(eE_{0})^{2}$. This feature was obtained in the work [22]
on the basis of an analysis of the numerical solutions of the corresponding
KEs in standard QED, see also [23].

The effectiveness of the low density approximation in the standard strong
field QED for rather weak fields $E_{0}\ll E_{c}$ has been investigated in the
work [24]. The additional introduction of the effective mass approximation
results herein a strong restriction of the domain of applicability of the
method.

##  4 Conclusion

In the present work we have obtained some simple and rather universal results
of approximate solutions for the distribution functions of charge carriers in
graphene based on nonperturbative KEs. This was achieved by a combination of
the low density approximation and the concept of an effective electromagnetic
mass. Such an approach is effective for a rather wide classes of external
field models with the parameters limited by the relation for the top (49) of
the characteristic frequency $\omega$ of the external field.

The considered approximation is of particular interest for the investigation
of such complicated nonlinear single-photon effects in graphene as the
emission (absorption) and annihilation (photoproduction) [3] and the more
complex two-photon processes. Such kind of nonlinear phenomena in graphene
became accessible for experimental verification recently, see [15, 16].

### Acknowledgements

The work of D.B. was supported by the Russian Federal Program ”Priority-2030”.
N.G. received support from Volkswagen Foundation (Hannover, Germany) under
collaborative research grant No. 97029. B.M. acknowledges a stipend from the
International Max Planck Research School for ”Many-Particle Systems in
Structured Environments” at the Max-Planck Institute for Physics of Complex
Systems (Dresden, Germany).

## References

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[a]Konstantin Kouzakov

# Elastic neutrino-atom scattering as a probe of neutrino millicharge and
magnetic moment

Georgy Donchenko  Alexander Studenikin

###### Abstract

Neutrino scattering on atomic systems at low-energy transfer is a powerful
tool for searching the neutrino electromagnetic interactions. The regime of
coherent elastic neutrino-atom scattering, i.e., when the atom recoils as a
pointlike particle, can be effectively fulfilled in the case of tritium
antineutrinos. We present theoretical calculations for coherent elastic
neutrino-atom scattering processes on such targets as the H, 2H, 3He, and 4He
atoms. We show how the atomic effects and neutrino electromagnetic properties,
namely the neutrino millicharge and magnetic moment, may manifest themselves
in the atomic-recoil spectra. Our results can be used in planning the
experiments on coherent elastic neutrino-atom scattering (in particular, with
superfluid He-4).

##  1 Introduction

The search for light particles of dark matter requires the detectors that are
sensitive to low recoil energies ($\lesssim$100 meV). This can be achieved,
for example, by using a superfluid He-4 target [1]. Another possible
application of the superfluid He-4 detector could be the study of the low-
energy neutrino scattering, in particular of the coherent elastic neutrino-
atom scattering (CE$\nu$AS) [2, 3, 4, 5] that has not been observed so far.
Below we inspect the sensitivity of the CE$\nu$AS processes on light atomic
systems to such neutrino electromagnetic properties as millicharge $e_{\nu}$
and magnetic moment $\mu_{\nu}$ [6]. For this purpose we account for the
indicated neutrino properties in the CE$\nu$AS cross section and present the
corresponding numerical results.

##  2 Effects of neutrino millicharge and magnetic moment in CEvAS

We consider an elastic neutrino-atom collision in the following kinematical
regime:

$E_{\nu}\ll m,\qquad T\leq\frac{2E_{\nu}^{2}}{m}\ll E_{\nu},\qquad
E_{\nu}\ll\frac{1}{R_{\rm nuc}},$

where $E_{\nu}$ is the neutrino energy, $T$ is the energy transfer, $m$ is the
atomic mass, and $R_{\rm nuc}$ is the nuclear radius.

According to [2, 3, 4, 7, 8, 9], the CE$\nu$AS differential cross section is
given by

$\frac{d\sigma}{dT}=\frac{d\sigma^{(w,e_{\nu})}}{dT}+\frac{d\sigma^{(\mu_{\nu})}}{dT}.$
(1)

Here the weak interaction and neutrino millicharge contribution is

$\frac{d\sigma^{(w,e_{\nu})}}{dT}=\frac{G_{F}^{2}m}{\pi}\left[C_{V}^{2}\left(1-\frac{mT}{2E_{\nu}^{2}}\right)+C_{A}^{2}\left(1+\frac{mT}{2E_{\nu}^{2}}\right)\right],$
(2)

with

$\displaystyle C_{V}$
$\displaystyle=Z\left(\frac{1}{2}-2\sin^{2}\theta_{W}\right)-\frac{1}{2}N+Z\left(\mp\frac{1}{2}+2\sin^{2}\theta_{W}\right)F_{\rm
el}(q^{2})+\frac{\sqrt{2}\pi\alpha Ze_{\nu}}{G_{F}mT}[1-F_{\rm el}(q^{2})],$
$\displaystyle C_{A}^{2}$ $\displaystyle=(C_{A}^{\rm
nuc})^{2}+\frac{1}{4}\sum\limits_{n,l}\left[\left(L^{nl}_{+}-L^{nl}_{-}\right)F_{\rm
el}^{nl}(q^{2})\right]^{2},$ $\displaystyle(C_{A}^{\rm nuc})^{2}$
$\displaystyle=\frac{g_{A}^{2}}{4}\left[(Z_{+}-Z_{-})-(N_{+}-N_{-})\right]^{2},$

where $q$ is the momentum transfer, with $q^{2}=2mT$, the plus (minus) stands
for $\nu=\nu_{e}$ ($\nu=\nu_{\mu,\tau}$), and $Z$ ($N$) is the number of
protons (neutrons) in the atomic nucleus. $F_{\rm el}(q^{2})$ is the Fourier
transform of the electron density, $g_{A}=1.25$, $Z_{\pm}$ and $N_{\pm}$ are
the numbers of protons and neutrons (electrons) with spin parallel ($+$) or
antiparallel ($-$) to the nucleus spin (the total electron spin).
$L^{nl}_{\pm}$ is the number of electrons in the $nl$ atomic orbital with spin
parallel ($+$) or antiparallel ($-$) to the electron spin, and $F_{\rm
el}^{nl}(q^{2})$ is the Fourier transform of the $nl$ electron density. The
neutrino millicharge $e_{\nu}$ is in units of $e$.

Figure 1: Differential cross sections for the CEvAS processes within Standard
Model ($e_{\nu}=0$ and $\mu_{\nu}=0$ ) and with account for the neutrino
millicharge ($e_{\nu}=\pm 10^{-15}e$) and magnetic moment
($\mu_{\nu}=10^{-12}\mu_{B}$).

The neutrino magnetic moment contribution is

$\frac{d\sigma^{(\mu_{\nu})}}{dT}=\frac{\pi\alpha^{2}Z^{2}}{m_{e}^{2}}|\mu_{\nu}|^{2}\left(\frac{1}{T}-\frac{1}{E_{\nu}}\right)\left[1-F_{\rm
el}(q^{2})\right]^{2},$ (3)

where the neutrino magnetic moment $\mu_{\nu}$ is in units of $\mu_{B}$. In
contrast to the case of neutrino millicharge, the neutrino magnetic moment
interaction flips the neutrino helicity, and therefore it does not interfere
with the weak interaction channel.

In Fig. 1 we present the numerical results for the differential cross section
(1) in the case of an electron antineutrino with $E_{\nu}=10$ keV that is
typical for the tritium neutrino source. It can be seen that the atomic recoil
spectra in CE$\nu$AS processes on the H, 2H, 3He, and 4He atomic systems are
very sensitive to the neutrino millicharge and magnetic moment. Measuring
these spectra may allow us to test the $e_{\nu}$ and $\mu_{\nu}$ values at a
level of $10^{-15}e$ and $10^{-12}\mu_{B}$, respectively, or even below that
level.

The obtained results will be used in the search for the electromagnetic
properties of neutrinos in the experiment involving an intense tritium
neutrino source and a superfluid 4He target. This experiment is currently
being prepared in the framework of the research program of the National Center
for Physics and Mathematics in Sarov, Russia.

## Acknowledgments

The work is supported by the Russian Science Foundation under grant No.
22-22-00384. G.D. acknowledges the support from the National Center for
Physics and Mathematics (Project “Study of coherent elastic neutrino-atom and
-nucleus scattering and neutrino electromagnetic properties using a high-
intensity tritium neutrino source”).

## References

  * [1] H. J. Maris et al., Phys. Rev. Lett. 119 (2017) 181303.
  * [2] Yu. V. Gaponov and V. N. Tikhonov, Sov. J. Nucl. Phys. (Engl. Transl.) 26 (1977) 314 [Yadernaya fizika 26 (1977) 594 - in Russian].
  * [3] L. M. Sehgal and M. Wanninger, Phys. Lett. B 171 (1986) 107.
  * [4] M. Cadeddu, F. Dordei, C. Giunti, K. Kouzakov, E. Picciau, and A. Studenikin, Phys. Rev. D 100 (2019) 073014.
  * [5] E. Picciau, M. Cadeddu, F. Dordei, C. Giunti, K. Kouzakov, and A. Studenikin, PoS(ICHEP2020) 390 (2021) 211.
  * [6] C. Giunti and A. Studenikin, Rev. Mod. Phys. 87 (2015) 531.
  * [7] K. Kouzakov and A. Studenikin, Phys. Rev. D, 95 (2017) 055013.
  * [8] G. Donchenko, K. Kouzakov, and A. Studenikin, J. Phys.: Conf. Ser. 2156 (2021) 012231.
  * [9] G. Donchenko, K. Kouzakov, and A. Studenikin, PoS(EPS-HEP2021) 398 (2022) 204.


# On discrete boundary value problems with nonlocal conditions in a quarter-
plane

Vladimir Vasilyev  Chair of Applied Mathematics and Computer Modeling
Belgorod State National Research University
Pobedy street 85, Belgorod 308015, Russia<EMAIL_ADDRESS>and  Anastasia
Mashinets  Chair of Applied Mathematics and Computer Modeling
Belgorod State National Research University
Pobedy street 85, Belgorod 308015, Russia<EMAIL_ADDRESS>
###### Abstract.

We consider discrete analogue of model pseudo-differential equations in
discrete plane sector using discrete variant of Sobolev–Slobodetskii spaces.
Starting from the concept of wave factorization for elliptic periodic symbol
we describe solvability conditions for the equations and corresponding
discrete boundary value problems. We give also a comparison between discrete
and continuous solutions in appropriate discrete normed spaces.

###### Key words and phrases:

elliptic symbol, invertibility, two-dimensional periodic Riemann problem,
digital pseudo-differential operator, discrete equation, periodic wave
factorization

###### 2010 Mathematics Subject Classification:

Primary: 35S15; Secondary: 65T50

##  1\. Introduction

A theory of pseudo-differential operators and equations [9, 16, 17, 6] not so
long history than other mathematical subjects of analysis. Nevertheless, these
operators and related boundary value problems widely arise in a lot of applied
problems in physics and technique (see, for example [14] and references
therein). Discrete aspects of the theory are reflected in mathematical papers
more weak [11, 2] although these studies are closely related to the theory of
Fourier series [5]. In our opinion the discrete theory is very important since
it permits to use computer calculations to solve concrete applied problems.

We interested in studying discrete pseudo-differential equations and their
solvability in appropriate discrete functional spaces. There are certain
approaches to studying discrete boundary value problems for partial
differential equations including finite difference method [13, 12]. But these
approaches are not applicable to studying discrete boundary value problems for
elliptic pseudo-differential equations. According to this statement the first
author with colleagues has started to develop discrete theory for elliptic
pseudo-differential equations [21, 15]. This is main motivation, and we have
started from certain canonical domains. First considerations were related to
discrete $m$-dimensional space and half-space, and here we consider discrete
quadrant.

We consider a special type of boundary conditions, namely integral conditions
on a boundary. These conditions are nonlocal, and it seems, such conditions
are artificial. But there are a lot of applied problems for partial
differential equations with such boundary conditions [1, 4, 3], therefore it
is natural way. Moreover, these conditions appear in a natural way to
determine arbitrary functions in a general solution of an elliptic pseudo-
differential equation.

##  2\. Cones, periodic symbols, digital operators and equations

###  2.1. Discrete spaces and transforms

Let $\mathbb{Z}^{2}$ be an integer lattice in a plane. Let
$K=\\{x\in\mathbb{R}^{2}:x=(x_{1},x_{2}),x_{1}>0,x_{2}>0\\}$ be a quadrant,
$K_{d}=h\mathbb{Z}^{2}\cap K,h>0$. We consider functions of discrete variable
$u_{d}(\tilde{x}),\tilde{x}=(\tilde{x}_{1},\tilde{x}_{2})\in h\mathbb{Z}^{2}$.

Let us denote $\mathbb{T}^{2}=[-\pi,\pi]^{2},\hbar=h^{-1}$. We consider
functions defined in $\hbar\mathbb{T}^{2}$ as periodic functions defined in
$\mathbb{R}^{2}$ with basic square of periods $\hbar\mathbb{T}^{2}$.

One can define the discrete Fourier transform for the function $u_{d}$

$(F_{d}u_{d})(\xi)\equiv\tilde{u}_{d}(\xi)=\sum\limits_{\tilde{x}\in
h\mathbb{Z}^{2}}e^{-i\tilde{x}\cdot\xi}u_{d}(\tilde{x})h^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \xi\in\hbar\mathbb{T}^{2},$

if the latter series converges, and the function $\tilde{u}_{d}(\xi)$ is a
periodic function in $\mathbb{R}^{2}$ with basic square of periods
$\hbar\mathbb{T}^{2}$. Such discrete Fourier transform preserves all
properties of integral Fourier transform, and the inverse discrete Fourier
transform looks as follows

$(F_{d}^{-1}\tilde{u}_{d})(\tilde{x})=\frac{1}{(2\pi)^{2}}\int\limits_{\hbar\mathbb{T}^{2}}e^{i\tilde{x}\cdot\xi}\tilde{u}_{d}(\xi)d\xi,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{x}\in h\mathbb{Z}^{2}.$

The discrete Fourier transform gives one-to-one correspondence between spaces
$L_{2}(h\mathbb{Z}^{2})$ and $L_{2}(\hbar\mathbb{T}^{2})$ with norms

$||u_{d}||_{2}=\left(\sum\limits_{\tilde{x}\in
h\mathbb{Z}^{2}}|u_{d}(\tilde{x})|^{2}h^{2}\right)^{1/2},\quad||\tilde{u}_{d}||_{2}=\left(\int\limits_{\xi\in\hbar\mathbb{T}^{2}}|\tilde{u}_{d}(\xi)|^{2}d\xi\right)^{1/2}.$

We need more general discrete functional spaces and we introduce such spaces
using divided differences [13].

The divided differences of first order look as follows

$(\Delta_{1}^{(1)}u_{d})(\tilde{x})=h^{-1}(u_{d}(\tilde{x}_{1}+h,\tilde{x}_{2})-u_{d}(\tilde{x}_{1},\tilde{x}_{2})),$
$(\Delta_{2}^{(1)}u_{d})(\tilde{x})=h^{-1}(u_{d}(\tilde{x}_{1},\tilde{x}_{2}+h)-u_{d}(\tilde{x}_{1},\tilde{x}_{2})),$

and their discrete Fourier transforms are given by formulas

$\widetilde{(\Delta_{k}^{(1)}u_{d})}(\xi)=h^{-1}(e^{-ih\cdot\xi_{k}}-1)\tilde{u}_{d}(\xi),k=1,2.$

The divided difference of second order is a divided difference of first order
from divided difference of first order

$(\Delta_{1}^{(2)}u_{d})(\tilde{x})=h^{-2}(u_{d}(\tilde{x}_{1}+2h,\tilde{x}_{2})-2u_{d}(\tilde{x}_{1}+h,\tilde{x}_{2})+u_{d}(\tilde{x}_{1},\tilde{x}_{2})),$
$(\Delta_{2}^{(2)}u_{d})(\tilde{x})=h^{-2}(u_{d}(\tilde{x}_{1},\tilde{x}_{2}+2h)-2u_{d}(\tilde{x}_{1},\tilde{x}_{2}+h)+u_{d}(\tilde{x}_{1},\tilde{x}_{2})),$

with the Fourier transform

$\widetilde{(\Delta_{k}^{(2)}u_{d})}(\xi)=h^{-2}(e^{-ih\cdot\xi_{k}}-1)^{2}\tilde{u}_{d}(\xi),k=1,2.$

Discrete analogue of the Laplacian is the following

$(\Delta_{d}u_{d})(\tilde{x})=(\Delta_{1}^{(2)}u_{d})(\tilde{x})+(\Delta_{2}^{(2)}u_{d})(\tilde{x}),$

so that its Fourier transform is

$\widetilde{(\Delta_{d}u_{d})}(\xi)=h^{-2}((e^{-ih\cdot\xi_{1}}-1)^{2}+(e^{-ih\cdot\xi_{2}}-1)^{2})\tilde{u}_{d}(\xi).$

We use such discrete objects for constructing discrete Sobolev–Slobodetskii
spaces to study wide class of discrete equations.

First, we introduce discrete analogue of the Schwartz space
$S(h\mathbb{Z}^{2})$ as a set of discrete functions with finite semi-norms

$|u_{d}|=\sup\limits_{\tilde{x}\in
h\mathbb{Z}^{2}}(1+|\tilde{x}|)^{l}|\Delta^{({\bf k})}u_{d}(\tilde{x})|$

for arbitrary $l\in\mathbb{N},{\bf k}=(k_{1},k_{2}),k_{r}\in\mathbb{N},r=1,2$,

$\Delta^{({\bf
k})}u_{d}(\tilde{x})=\Delta^{k_{1}}_{1}\Delta^{k_{2}}_{2}u_{d}(\tilde{x}).$

Definition 1. A discrete distribution is called a linear continuous functional
defined on the space $S(h\mathbb{Z}^{2})$.

A set of such distributions will be denoted by $S^{\prime}(h\mathbb{Z}^{2})$,
and a value of the discrete distribution $f_{d}$ on the test discrete function
$u_{d}\in S(h\mathbb{Z}^{2})$ will be denoted by $(f_{d},u_{d})$.

One can introduce a concept of a support for a discrete distribution. Namely,
a support of the discrete function $u_{d}\in S(h\mathbb{Z}^{2})$ is a subset
of the set $h\mathbb{Z}^{2}$ such that $u_{d}(\tilde{x})\neq 0$ for all points
$\tilde{x}$ from this subset. For an arbitrary set $M\subset\mathbb{R}^{2}$ we
denote $M_{d}=M\cap h\mathbb{Z}^{2}$, and then one says that $f_{d}=0$ in the
discrete domain $M_{d}$ if $(f_{d},u_{d})=0,\forall u_{d}\in S(M_{d}),$ where
$S(M_{d})\subset S(h\mathbb{Z}^{2})$ consists of discrete functions with
supports in $M_{d}$. If $\widetilde{M}_{d}$ is a union of such $M_{d}$ where
$f_{d}=0$ then support of the discrete distribution $f_{d}$ is the set
$h\mathbb{Z}^{2}\setminus\widetilde{M}_{d}$.

Similarly [26] we can define standard operations in the space
$S^{\prime}(h\mathbb{Z}^{2})$, but differentiation will be changed by divided
difference of first order. These operations are described in [21] in details,
a convergence is meant as a weak convergence in the space
$S^{\prime}(h\mathbb{Z}^{2})$.

Example 1. If the function $f_{d}(\tilde{x})$ is locally summable then it
generates the discrete distribution

(1) $(f_{d},u_{d})=\sum\limits_{\tilde{x}\in
h\mathbb{Z}^{2}}f_{d}(\tilde{x})u_{d}(\tilde{x})h^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall u_{d}\in
S(h\mathbb{Z}^{2}).$

But there are different possibilities, for example, analogue of the Dirac
mass-function

$(\delta_{d},u_{d})=u_{d}(0),$

which can not be represented by the formula (1).

Let
$\zeta^{2}=h^{-2}((e^{-ih\cdot\xi_{1}}-1)^{2}+(e^{-ih\cdot\xi_{2}}-1)^{2})$.
We introduce the following definition.

Definition 2. The space $H^{s}(h\mathbb{Z}^{2})$ consists of discrete
distributions and it is a closure of the space $S(h\mathbb{Z}^{2})$ with
respect to the norm

(2)
$||u_{d}||_{s}=\left(\int\limits_{\hbar\mathbb{T}^{2}}(1+|\zeta^{2}|)^{s}|\tilde{u}_{d}(\xi)|^{2}d\xi\right)^{1/2}.$

Let us remind that a lot of properties of such discrete spaces were studied in
[7], Varying the parameter $h$ in (2) we obtain different norms which are
equivalent to the $L_{2}$-norm. But constants in this equivalence depend on
$h$. In our constructions all constants do not depend on $h$.

Definition 3. T͡he space $H^{s}(K_{d})$ consists of discrete distributions
from $H^{s}(h\mathbb{Z}^{2})$ such that their supports belong to the set
$\overline{K_{d}}$. A norm in the space $H^{s}(K_{d})$ is induced by the norm
of the space $H^{s}(h\mathbb{Z}^{2})$. The space $H^{s}_{0}(K_{d})$ consists
of discrete distributions $f_{d}\in S^{\prime}(h\mathbb{R}^{2})$ with supports
inside of $K_{d}$, and these discrete distributions must admit a continuation
into the space $H^{s}(h\mathbb{Z}^{2})$. A norm in the space
$H^{s}_{0}(K_{d})$ is given by the formula

$||f_{d}||^{+}_{s}=\inf||\ell f_{d}||_{s},$

where infimum is taken for all continuations $\ell$.

The Fourier image of the space $H^{s}(K_{d})$ will be denoted by
$\widetilde{H}^{s}(K_{d})$.

###  2.2. Symbols, operators and projectors

Let $\widetilde{A}_{d}(\xi)$ be a measurable periodic function in
$\mathbb{R}^{2}$ with basic square of periods $\hbar\mathbb{T}^{2}$. Such
functions we call symbols.

Definition 4. A digital pseudo-differential operator $A_{d}$ with the symbol
$A_{d}(\xi)$ in the discrete quadrant $K_{d}$ is called an operator of the
following type

(3) $(A_{d}u_{d})(\tilde{x})=\sum\limits_{\tilde{y}\in
h\mathbb{Z}^{2}}h^{2}\int\limits_{\hbar\mathbb{T}^{2}}\widetilde{A}_{d}(\xi)e^{i(\tilde{x}-\tilde{y})\cdot\xi}\tilde{u}_{d}(\xi)d\xi,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{x}\in K_{d},$

We say that the operator $A_{d}$ is elliptic one if

$ess\leavevmode\nobreak\ \inf_{\xi\in\hbar\mathbb{T}^{2}}|A_{d}(\xi)|>0.$

A more general digital pseudo-differential operator with the symbol
$\widetilde{A}_{d}(\tilde{x},\xi)$ depending on a spatial variable $\tilde{x}$

$(A_{d}u_{d})(\tilde{x})=\sum\limits_{\tilde{y}\in
h\mathbb{Z}^{2}}h^{2}\int\limits_{\hbar\mathbb{T}^{2}}A_{d}(\tilde{x},\xi)e^{i(\tilde{x}-\tilde{y})\cdot\xi}\tilde{u}_{d}(\xi)d\xi,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{x}\in K_{d},$

can be defined in the same way, but here we consider only operators of type
(3).

We consider symbols satisfying the condition

(4) $c_{1}(1+|\zeta^{2}|)^{\alpha/2}\leq|A_{d}(\xi)|\leq
c_{2}(1+|\zeta^{2}|)^{\alpha/2}$

with constants $c_{1},c_{2}$ non-depending on $h$. The number
$\alpha\in\mathbb{R}$ is called an order of digital pseudo-differential
operator $A_{d}$.

The following simple result can be proved easily.

Lemma 1. A digital pseudo-differential operator $A_{d}$ with the symbol
$\widetilde{A}_{d}(\xi)$ is a linear bounded operator
$H^{s}(h\mathbb{Z}^{2})\to H^{s-\alpha}(h\mathbb{Z}^{2})$ with a norm non-
depending on $h$.

We study a solvability of the discrete equation

(5) $(A_{d}u_{d})(\tilde{x})=v_{d}(\tilde{x}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{x}\in K_{d},$

in the space $H^{s}(K_{d})$ assuming that $v_{d}\in H^{s-\alpha}_{0}(K_{d})$.

We will use certain special domain in two-dimensional complex space
$\mathbb{C}^{2}.$ A domain of the type
${\mathcal{T}}_{h}(K)=\hbar\mathbb{T}^{2}+iK$ is called a tube domain over the
quadrant $K$, and we will consider analytical functions $f(x+i\tau)$ in the
domain ${\mathcal{T}}_{h}(K)=\hbar\mathbb{T}^{2}+iK$.

Let us introduce the periodic Bochner kernel similar [26]

$B_{h}(z)=\sum\limits_{\tilde{x}\in
K_{d}}e^{i\tilde{x}\cdot(\xi+i\tau)}h^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\xi\in\hbar\mathbb{T}^{2},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tau\in K,$

and corresponding integral operator

$(B_{h}\tilde{u}_{d})(\xi)=\lim\limits_{\tau\to 0,\tau\in
K}\frac{1}{4\pi^{2}}\int\limits_{\hbar\mathbb{T}^{2}}B_{h}(\xi+i\tau-\eta)\tilde{u}_{d}(\eta)d\eta.$

Lemma 2. For the quadrant $K$ the operator $B_{h}$ has the following form

$(B_{h}\tilde{u}_{d})(\xi)=\frac{h^{2}}{8\pi^{2}}\int\limits_{\mathbb{T}^{2}}\tilde{u}_{d}(\eta)d\eta+\lim\limits_{\tau\to
0+}\frac{ih}{8\pi^{2}}\int\limits_{\mathbb{T}^{2}}\cot\frac{h(\xi_{1}-\eta_{1}+i\tau_{1})}{2}\tilde{u}_{d}(\eta)d\eta+$
$+\lim\limits_{\tau\to
0+}\frac{ih}{8\pi^{2}}\int\limits_{\mathbb{T}^{2}}\cot\frac{h(\xi_{2}-\eta_{2}+i\tau_{2})}{2}\tilde{u}_{d}(\eta)d\eta-$
$-\lim\limits_{\tau\to
0+}\frac{h^{2}}{8\pi^{2}}\int\limits_{\mathbb{T}^{2}}\cot\frac{h(\xi_{1}-\eta_{1}+i\tau_{1})}{2}\cot\frac{h(\xi_{2}-\eta_{2}+i\tau_{2})}{2}\tilde{u}_{d}(\eta)d\eta,$

and $B_{h}$ is a linear bounded operator
$H^{s}(\hbar\mathbb{T}^{2})\rightarrow H^{s}(\hbar\mathbb{T}^{2})$ for
$|s|<1/2$. Moreover, the operator $B_{h}$ is a projector
$\widetilde{H}^{s}(h\mathbb{Z}^{2})\rightarrow\widetilde{H}^{s}(K_{d})$.

Proof. Corresponding calculations for one-dimensional discrete cone were done
in [20]. We use these evaluations adapting to our two-dimensional case. Since

$\sum\limits_{\tilde{x}_{k}\in
h\mathbb{Z}_{+}}e^{-i\tilde{x}_{k}z_{k}}h=\frac{h}{2}-\frac{ih}{2}\cot\frac{hz_{k}}{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
z_{k}=\xi_{k}+i\tau_{k},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ k=1,2.$

then multiplying two factors and applying the Fourier property on
correspondence between a product and convolution we obtain the assertion.

Boundedness of the one-dimensional operator with the kernel
$h\cot\frac{hz}{2}$for $|s|<1/2$ was proved in [7], Theorem 6; two-dimensional
case can be considered by the same method. $\blacksquare$

Remark 1. The operator $B_{h}$ is so called periodic bi-singular operator.
Using classical results for Cauchy type integral [8, 10] one can evaluate the
boundary value, but it is not important this time. Since these formulas are
very huge we can do some simplifications without lost of generality. For
example, we can consider the space $S_{1}(h\mathbb{Z}^{2})\subset
S_{1}(h\mathbb{Z}^{2})$ with zeroes in coordinate axes and consider the space
$H^{s}(h\mathbb{Z}^{2})$ as closure of the set $S_{1}(h\mathbb{Z}^{2})$
assuming that all functions of discrete variable vanish on coordinate axes.
For this case the first three summands in $B_{h}$ will be zero.

Lemma 3. If $|s|<1/2$ then the space $\widetilde{H}^{s}(h\mathbb{Z}^{2})$ is
uniquely represented as the direct sum

$\widetilde{H}^{s}(h\mathbb{Z}^{2})=\widetilde{H}^{s}(K_{d})\oplus\widetilde{H}^{s}(h\mathbb{Z}^{2}\setminus
K_{d})$

Proof. It is simple consequence of Lemma 2. Indeed, the unique representation
of the function $\tilde{f}\in\widetilde{H}(h\mathbb{Z}^{2})$ is the following

$\tilde{f}=B_{h}\tilde{f}+(I-B_{h})\tilde{f}.$

A uniqueness of the such representation is possible only for $|s|<1/2$.
$\blacksquare$

To describe a solvability picture for the discrete equation (5) we need some
additional elements of multidimensional complex analysis. We give it in the
next section.

##  3\. Periodic wave factorization

This concept is a periodic analogue of the wave factorization [18]. Some first
preliminary considerations and results were described in [25, 24, 23, 22].

Definition 5. A periodic wave factorization for the elliptic symbol
$A_{d}(\xi)\in E_{\alpha}$ is called its representation in the form

$A_{d}(\xi)=A_{d,\neq}(\xi)A_{d,=}(\xi),$

where the factors $A_{d,\neq}(\xi),A_{d,=}(\xi)$ admit analytical continuation
into tube domains ${\mathcal{T}}_{h}(K),{\mathcal{T}}_{h}(-K)$ respectively
with estimates

$c_{1}(1+|\hat{\zeta}^{2}|)^{\frac{\ae}{2}}\leq|A_{d,\neq}(\xi+i\tau)|\leq
c^{\prime}_{1}(1+|\hat{\zeta}^{2}|)^{\frac{\ae}{2}},$
$c_{2}(1+|\hat{\zeta}^{2}|)^{\frac{{\alpha-\ae}}{2}}\leq|A_{d,=}(\xi-i\tau)|\leq
c^{\prime}_{2}(1+|\hat{\zeta}^{2}|)^{\frac{{\alpha-\ae}}{2}},$

and constants $c_{1},c^{\prime}_{1},c_{2},c^{\prime}_{2}$ non-depending on
$h$, where

$\hat{\zeta}^{2}\equiv\hbar^{2}\left((e^{-ih(\xi_{1}+i\tau_{1})}-1)^{2}+(e^{-ih(\xi_{2}+i\tau_{2})}-1)^{2}\right),$
$\xi=(\xi_{1},\xi_{2})\in\hbar\mathbb{T}^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tau-(\tau_{1},\tau_{2})\in K.$

The number $\ae\in{\mathbb{R}}$ is called an index of periodic wave
factorization.

Unfortunately, we have no an algorithm to construct the factorization. But the
are certain examples of periodic symbols which admit such factorization. We
give one of them.

If $f$ is an arbitrary function of a discrete variable, $f\in
S(h\mathbb{Z}^{2})$, $supp\leavevmode\nobreak\ f\subset K_{d}\cup(-K_{d})$
then we have

$f=\chi_{+}f+\chi_{-}f,$

where $\chi_{\pm}$ are indicators of $\pm K_{d}$. Applying the discrete
Fourier transform we obtain the representation
$\tilde{f}=\tilde{f}_{+}+\tilde{f}_{-}$, and $\tilde{f}_{\pm}$ admit an
analytical continuation into ${\mathcal{T}}_{h}(\pm K)$ according to Lemma 2.
Thus, we can write $\exp\tilde{f}=\exp\tilde{f}_{+}\cdot\exp\tilde{f}_{-}$,
therefore we obtain periodic wave factorization with index zero for the
function $\exp\tilde{f}$.

Everywhere below we assume existence of such periodic wave factorization for
the symbol $A_{d}(\xi)$ with index $\ae$.

###  3.1. A unique solvability

This section is devoted to most simple case when a solution of the equation
(5) exists and it is unique.

Theorem 1. Let $|\ae-s|<1/2$. Then the equation (5) has a unique solution for
arbitrary right hand side $v_{d}\in H^{s-\alpha}_{0}(K_{d})$, and it is given
by the formula

$\tilde{u}_{d}(\xi)-A^{-1}_{d,\neq}(\xi)B_{h}(A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)),$

where $\ell v_{d}$ is an arbitrary continuation of $v_{d}$ into
$H^{s-\alpha}(h\mathbb{Z}^{2})$.

Proof Let $\ell v_{d}$ be an arbitrary continuation of $v_{d}\in
H^{s-\alpha}_{0}(K_{d})$ into $H^{s-\alpha}(h\mathbb{Z}^{2})$. Let us
introduce the function

$w_{d}(\tilde{x})=(\ell v_{d})(\tilde{x})-(A_{d}u_{d})(\tilde{x}),$

so that $w(\tilde{x})=0$ for $\tilde{x}\notin K_{d}$.

Now we write the equation (5) in the form

$(A_{d}u_{d})(\tilde{x})+w_{d}(\tilde{x})=(\ell
v_{d})(\tilde{x}),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{x}\in h\mathbb{Z}^{2},$

and after applying the discrete Fourier transform and periodic wave
factorization we obtain

(6)
$A_{d,\neq}(\xi)\tilde{u}_{d}(\xi)+A^{-1}_{d,=}(\xi)\tilde{w}_{d}(\xi)=A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\xi\in\hbar\mathbb{T}^{2},$

We have the following inclusions according to Lemma 1 and Lemma 2

$A_{d,\neq}(\xi)\tilde{u}_{d}(\xi)\in\widetilde{H}^{s-\ae}(K_{d}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
A^{-1}_{d,=}(\xi)\tilde{w}_{d}(\xi)\in\widetilde{H}^{s-\ae}(h\mathbb{Z}^{2}\setminus
K_{d}),$ $A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)\in\widetilde{H}^{s-\ae}(h\mathbb{Z}^{2}),$

and then according to Lemma 3 the right hand side of the equality (6) is
uniquely represented by the sum

$A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)=f^{+}_{d}(\xi)+f^{-}_{d}(\xi),$

where

$f^{+}_{d}(\xi)=B_{h}(A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
f^{-}_{d}(\xi)=(I-B_{h})(A^{-1}_{d,=}(\xi)\widetilde{(\ell v_{d})}(\xi))$

.

Further, we rewrite the equality (6)

$A_{d,\neq}(\xi)\tilde{u}_{d}(\xi)-f^{+}_{d}(\xi)=f^{-}_{d}(\xi)-A^{-1}_{d,=}(\xi)\tilde{w}_{d}(\xi)$

and using the uniqueness of the representation as the direct sum
$\widetilde{H}^{s-\ae}(K_{d})\oplus\widetilde{H}^{s-\ae}(h\mathbb{Z}^{2}\setminus
K_{d})$ we conclude that both left hand side and right hand side should be
zero. Thus,

$\tilde{u}_{d}(\xi)=A^{-1}_{d,\neq}(\xi)B_{h}(A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)),$

and Theorem 1 is proved. $\blacksquare$

##  4\. Discrete boundary value problem

In this section we consider more interesting case when the equation (5) has a
lot of solutions.

###  4.1. Form of a discrete solution

This section uses some results from [21] concerning a form of a discrete
distribution supported at the origin.

Theorem 2. Let $\ae-s=n+\delta,n\in\mathbb{N},|\delta|<1/2$. Then a general
solution of the equation (5) has the following form

$\tilde{u}_{d}(\xi)=A^{-1}_{d,\neq}(\xi)Q_{n}(\xi)B_{h}(Q_{n}^{-1}(\xi)A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi))+$
$+A^{-1}_{d,\neq}(\xi)\left(\sum\limits_{k=0}^{n-1}\tilde{c}_{k}(\xi_{1})\hat{\zeta}_{2}^{k}+\tilde{d}_{k}(\xi_{2})\hat{\zeta}_{1}^{k}\right),$

where $Q_{n}(\xi)$ is an arbitrary polynomial of order $n$ of variables
$\zeta_{k}=\hbar(e^{-ih\xi_{k}}-1),k=1,2,$ satisfying the condition (4) for
$\alpha=n$, $\tilde{c}_{k}(\xi_{1}),\tilde{d}_{k}(\xi_{2}),k=0,1,\cdots,n-1,$
– are arbitrary functions from $H^{s_{k}}(h\mathbb{T}),s_{k}=s-\ae+k-1/2$.

The a priori estimate

$||u_{d}||_{s}\leq
const\left(||f||^{+}_{s-\alpha}+\sum\limits_{k=0}^{n-1}([c_{k}]_{s_{k}}+[d_{k}]_{s_{k}})\right),$

holds, where $[\cdot]_{s_{k}}$ denotes a norm in $H^{s_{k}}(h\mathbb{T})$, and
$const$ does not depend on $h$.

Proof. We start from the equality (6). Let $Q_{n}(\xi)$ be an arbitrary
polynomial of order $n$ of variables
$\zeta_{k}=\hbar(e^{-ih\xi_{k}}-1),k=1,2,$ satisfying the condition (4) for
$\alpha=n$. We multiply the equality (6) by $Q^{-1}_{n}(\xi)$

(7)
$Q^{-1}_{n}(\xi)A_{d,\neq}(\xi)\tilde{u}_{d}(\xi)+Q^{-1}_{n}(\xi)A^{-1}_{d,=}(\xi)\tilde{w}_{d}(\xi)=Q^{-1}_{n}(\xi)A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\xi\in\hbar\mathbb{T}^{2},$

We have in view of Lemma 1

$Q^{-1}_{n}(\xi)A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)\in\widetilde{H}^{s-\ae+n}(h\mathbb{Z}^{2}),$

and since $s-\ae+n=-\delta$ then according to Lemma 3 we write the unique
decomposition

$Q^{-1}_{n}(\xi)A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)=F^{+}_{d}(\xi)+F^{-}_{d}(\xi),$

where

$F^{+}_{d}(\xi)=B_{h}(Q^{-1}_{n}(\xi)A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
F^{-}_{d}(\xi)=(I-B_{h})(Q^{-1}_{n}(\xi)A^{-1}_{d,=}(\xi)\widetilde{(\ell
v_{d})}(\xi)).$

Taking into account this fact we rewrite the equality (7) in the form

$A_{d,\neq}(\xi)\tilde{u}_{d}(\xi)+A^{-1}_{d,=}(\xi)\tilde{w}_{d}(\xi)=Q_{n}(\xi)F^{+}_{d}(\xi)+Q_{n}(\xi)F^{-}_{d}(\xi),$

and further,

$A_{d,\neq}(\xi)\tilde{u}_{d}(\xi)-Q_{n}(\xi)F^{+}_{d}(\xi)=Q_{n}(\xi)F^{-}_{d}(\xi)-A^{-1}_{d,=}(\xi)\tilde{w}_{d}(\xi),$

Since
$F^{+}_{d}(\xi)\in\widetilde{H}^{s-\ae+n}(K_{d}),F^{-}_{d}(\xi)\in\widetilde{H}^{s-\ae+n}(h\mathbb{Z}^{2}\setminus
K_{d})$ then according to Lemma 1 we conclude
$Q_{n}(\xi)F^{+}_{d}(\xi)\in\widetilde{H}^{s-\ae}(K_{d}),Q_{n}(\xi)F^{-}_{d}(\xi)\in\widetilde{H}^{s-\ae}(h\mathbb{Z}^{2}\setminus
K_{d})$. Applying the inverse discrete Fourier transform we obtain an equality
for two discrete distributions. The left hand side vanishes at least under one
condition $\tilde{x}_{1}<0$ or $\tilde{x}_{2}<0$, and the right hand side
vanishes under the condition .$\tilde{x}_{1}>0,\tilde{x}_{2}>0$, Thus, it
should be a discrete distribution supported on sides of the discrete quadrant
$\\{(\tilde{x}_{1},\tilde{x}_{2})\in
h\mathbb{Z}^{2}:\\{\tilde{x}_{1}>0,\tilde{x}_{2}=0\\}\cup\\{\tilde{x}_{1}=0,\tilde{x}_{2}>0\\}\\}$.
Using corresponding result from [21] we obtain the following form for this
discrete distribution

$\sum\limits_{k=0}^{n-1}\left(c_{k}(\tilde{x}_{1})(\Delta_{2}^{(k)}\delta_{d})(\tilde{x}_{2})+d_{k}(\tilde{x}_{2})(\Delta_{1}^{(k)}\delta_{d})(\tilde{x}_{1})\right),$

where all summands should be elements of the space
$H^{s-\ae}(h\mathbb{Z}^{2})$.

The left question is how much summands we need in the right-hand side.
Counting principle is a very simple because every summand should belong to the
space $\widetilde{H}^{s}(\hbar\mathbb{T}^{2})$.

Let us consider the summand $c_{k}(\xi_{1})\zeta_{2}^{k}$. Taking into account
that order of $A^{-1}_{d,+}(\xi)$ is $-\ae$ we need to verify the finiteness
of the $H^{s-\ae}$-norm for $c_{k}(\xi_{1})\zeta_{2}^{k}$. We have

$||c_{k}(\Delta^{(k)}_{2}\delta_{d})||^{2}_{s-\ae}=\int\limits_{\hbar\mathbb{T}^{2}}(1+|\zeta^{2}|)^{s-\ae}||c_{k}(\xi_{1})\zeta_{2}^{k}|^{2}d\xi$
$=\int\limits_{\hbar\mathbb{T}^{2}}(1+|\zeta^{2}|)^{s-\ae}||c_{k}(\xi_{1})|^{2}|\zeta_{2}^{k}|^{2}d\xi\leq
a_{1}\hbar^{2(s-\ae+k+1/2)}\int\limits_{\hbar\mathbb{T}}|c_{k}(\xi_{1})|^{2}d\xi_{1}$
$\leq
a_{2}\int\limits_{\hbar\mathbb{T}}(1+|{\zeta_{1}}^{2}|)^{s-\ae+k+1/2}|c_{k}(\xi_{1})|^{2}d\xi_{1},$

and the constants $a_{1},a_{2}$ do not depend on $h$.

The last summand should be $(n-1)$th because for $n$th summand we obtain a
positive growth: for $k=n$ we have
$s_{n}=s-\ae-n+1/2=-n-\delta+n+1/2=-\delta+1/2>0$.

A priori estimates can be obtained in the same way described in [21].
$\blacksquare$

###  4.2. The Dirichlet discrete boundary condition

We consider here first simple case with discrete Dirichlet boundary
conditions. We suppose in this section that
$\ae-s=1+\delta,|\delta|<1/2,v_{d}\equiv 0.$ It follows from Theorem 2 that we
have the following general solution of the equation (5)

(8)
$\tilde{u}_{d}(\xi)=A^{-1}_{d,\neq}(\xi)(\tilde{c}_{0}(\xi_{1})+\tilde{d}_{0}(\xi_{2})),$

where $c_{0},d_{0}\in H^{s-\ae-1/2}(\hbar\mathbb{Z})$ are arbitrary functions.
To determine uniquely these functions we add the discrete Dirichlet conditions
on angle sides

(9) ${u_{d}}_{|_{\tilde{x}_{1}=0}}=f_{d}(\tilde{x}_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{u_{d}}_{|_{\tilde{x}_{2}=0}}=g_{d}(\tilde{x}_{1}).$

Thus, we have the discrete Dirichlet problem (5),(9).

First, we apply the discrete Fourier transform to discrete conditions (9) and
obtain the following form

(10)
$\int\limits_{-\hbar\pi}^{\hbar\pi}\tilde{u}_{d}(\xi_{1},\xi_{2})d\xi_{1}=\tilde{f}_{d}(\xi_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\int\limits_{-\hbar\pi}^{\hbar\pi}\tilde{u}_{d}(\xi_{1},\xi_{2})d\xi_{2}=\tilde{g}_{d}(\xi_{1}).$

Substituting (10) into (8) we obtain the following relations

$\int\limits_{-\hbar\pi}^{\hbar\pi}\tilde{u}_{d}(\xi)d\xi_{1}=\int\limits_{-\hbar\pi}^{\hbar\pi}A^{-1}_{d,\neq}(\xi)\tilde{c}_{0}(\xi_{1})d\xi_{1}+\tilde{d}_{0}(\xi_{2})\int\limits_{-\hbar\pi}^{\hbar\pi}A^{-1}_{d,\neq}(\xi)d\xi_{1}$
$\int\limits_{-\hbar\pi}^{\hbar\pi}\tilde{u}_{d}(\xi)d\xi_{2}=\tilde{c}_{0}(\xi_{1})\int\limits_{-\hbar\pi}^{\hbar\pi}A^{-1}_{d,\neq}(\xi)d\xi_{2}+\int\limits_{-\hbar\pi}^{\hbar\pi}A^{-1}_{d,\neq}(\xi)\tilde{d}_{0}(\xi_{2})d\xi_{2}$

Let us denote

$\int\limits_{-\hbar\pi}^{\hbar\pi}A^{-1}_{d,\neq}(\xi)d\xi_{1}\equiv\tilde{a}_{0}(\xi_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\int\limits_{-\hbar\pi}^{\hbar\pi}A^{-1}_{d,\neq}(\xi)d\xi_{2}\equiv\tilde{b}_{0}(\xi_{1})$

and suppose that $\tilde{a}_{0}(\xi_{2}),\tilde{b}_{0}(\xi_{1})\neq
0,\forall\xi_{1}\neq 0,\xi_{2}\neq 0$.

Therefore, we have the following system of two linear integral equations with
respect to two unknown functions
$\tilde{c}_{0}(\xi_{1}),\tilde{d}_{0}(\xi_{2})$

(11)
$\left\\{\begin{array}[]{rcl}\int\limits_{-\hbar\pi}^{\hbar\pi}M_{1}(\xi)\tilde{c}_{0}(\xi_{1})d\xi_{1}+\tilde{d}_{0}(\xi_{2})=\tilde{F}_{d}(\xi_{2})\\\
\tilde{c}_{0}(\xi_{1})+\int\limits_{-\hbar\pi}^{\hbar\pi}M_{2}(\xi)\tilde{d}_{0}(\xi_{2})d\xi_{2}=\tilde{G}_{d}(\xi_{1}),\end{array}\right.$

where we have used the following notations

$\tilde{F}_{d}(\xi_{2})=\tilde{f}_{d}(\xi_{2})\tilde{a}_{0}^{-1}(\xi_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{G}_{d}(\xi_{1})=\tilde{g}_{d}(\xi_{1})\tilde{b}_{0}^{-1}(\xi_{1}),$
$M_{1}(\xi)=A^{-1}_{d,\neq}(\xi)\tilde{a}_{0}^{-1}(\xi_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
M_{2}(\xi)=A^{-1}_{d,\neq}(\xi)\tilde{b}_{0}^{-1}(\xi_{1}).$

Unique solvability conditions for the system (11) will be equivalent to unique
solvability for the discrete Dirichlet problem (5),(9).

Thus, we obtain the following result.

Proposition 1. Let $f_{d},g_{d}\in H^{s-1/2}(\mathbb{R}_{+}),s>1/2,v_{d}\equiv
0$. Then the discrete Dirichlet problem (5),(9) is reduced to the equivalent
system of linear integral equations (11).

###  4.3. Non-local discrete boundary condition

We consider here the $\ae-s=1+\delta,|\delta|<1/2$ for the equation (5) with
different boundary conditions, namely

(12) $\begin{array}[]{rcl}\sum\limits_{\tilde{x}_{1}\in
h\mathbb{Z}_{+}}u_{d}(\tilde{x}_{1},\tilde{x}_{2})h=f_{d}(\tilde{x}_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \sum\limits_{\tilde{x}_{2}\in
h\mathbb{Z}_{+}}u_{d}(\tilde{x}_{1},\tilde{x}_{2})h=g_{d}(\tilde{x}_{1}),\\\
\sum\limits_{\tilde{x}\in
h\mathbb{Z}_{++}}u_{d}(\tilde{x}_{1},\tilde{x}_{2})h^{2}=0.\end{array}$

These additional conditions will help us to determine uniquely the unknown
functions $c_{0},d_{0}$ in the solution (8).

Indeed, using the discrete Fourier transform we rewrite the conditions (12) as
follows

(13) $\tilde{u}_{d}(0,\xi_{2})=\tilde{f}_{d}(\xi_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{u}_{d}(\xi_{1},0)=\tilde{g}_{d}(\xi_{1}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{u}_{d}(0,0)=0.$

Now we substitute the formulas (13) into (8). The first two equality are

$\tilde{u}_{d}(0,\xi_{2})=A^{-1}_{d,\neq}(0,\xi_{2})(\tilde{c}_{0}(0)+\tilde{d}_{0}(\xi_{2}))=\tilde{f}_{d}(\xi_{2}),$
$\tilde{u}_{d}(\xi_{1},0)=A^{-1}_{d,\neq}(\xi_{1},0)(\tilde{c}_{0}(\xi_{1})+\tilde{d}_{0}(0))=\tilde{g}_{d}(\xi_{1}).$

It implies the following relations according to the third condition

$\tilde{f}_{d}(0)=\tilde{g}_{d}(0),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and from which}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{c}_{0}(0)+\tilde{d}_{0}(0)=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{c}_{0}(0)=\tilde{d}_{0}(0)=0.$

Then we have at least formally

(14)
$\tilde{u}_{d}(\xi)=A^{-1}_{d,\neq}(\xi)\left(A_{d,\neq}(\xi_{1},0)\tilde{g}_{d}(\xi_{1})+A_{d,\neq}(0,\xi_{2})\tilde{f}_{d}(\xi_{2})\right)$

It is left to formulate and to prove exactly the obtained result.

Theorem 3. Let $f_{d},g_{d}\in H^{s+1/2}(h\mathbb{Z}),v_{d}\equiv 0$. Then the
discrete problem (5),(12) has unique solution which is given by the formula
(14).

The a priori estimate

$||u_{d}||_{s}\leq const(||f_{d}||_{s+1/2}+||g_{d}||_{s+1/2})$

holds with a const non-depending on $h$

Proof. We need to prove the a priori estimate only. Let us consider the first
summand

$||A^{-1}_{d,\neq}(\xi)A_{d,\neq}(\xi_{1},0)\tilde{g}_{d}(\xi_{1})||^{2}_{s}=$

$=\int\limits_{\hbar\mathbb{T}^{2}}|A^{-1}_{d,\neq}(\xi_{1},\xi_{2})A_{d,\neq}(\xi_{1},0)\tilde{g}_{d}(\xi_{1})|^{2}(1+|\zeta^{2}|)^{s}d\xi_{1}d\xi_{2}\leq$
$\leq C\hbar^{2s}\int\limits_{\hbar\mathbb{T}^{2}}|g_{d}(\xi_{1})|^{2}d\xi\leq
C_{1}\hbar^{2s+1}\int\limits_{-\hbar\pi}^{\hbar\pi}|g_{d}(\xi_{1})|^{2}d\xi_{1}\leq$
$\leq
C_{2}\int\limits_{-\hbar\pi}^{\hbar\pi}|g_{d}(\xi_{1})|^{2}(1+|\zeta_{1}^{2}|)^{s+1/2}d\xi_{1}=||g_{d}|^{2}_{s+1/2}.$

The second summand has the same estimate. $\blacksquare$

##  5\. A comparison between discrete and continuous solutions

The continuous analogue of the discrete boundary value problem is the
following [19].

Let $A$ be a pseudo-differential operator with the symbol
$A(\xi),\xi=(\xi_{1},\xi_{2})$ satisfying the condition

$c_{1}(1+|\xi)^{\alpha}\leq|A(\xi)|\leq c_{2}(1+|\xi)^{\alpha}.$

and admitting the wave factorization with respect to the quadrant $K$ with
index $\ae$.

We consider the equation

(15) $(Au)(x)=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
x\in K,$

with the following additional conditions

(16)
$\int\limits_{0}^{+\infty}u(x_{1},x_{2})dx_{1}=f(x_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\int\limits_{0}^{+\infty}u(x_{1},x_{2})dx_{2}=g(x_{1}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \int\limits_{-K}u(x)dx=0.$

A solution of the problem (15),(16) is sought in the space $H^{s}(K)$ [18] and
boundary functions are taken from the space $H^{s+1/2}(\mathbb{R}_{+})$. Such
problem was considered in [19] and it has the solution

(17)
$\tilde{u}(\xi)=A^{-1}_{\neq}(\xi)\left(A_{\neq}(\xi_{1},0)\tilde{g}(\xi_{1})+A_{\neq}(0,\xi_{2})\tilde{f}(\xi_{2})\right)$

under condition that the symbol $A(\xi)$ admits the wave factorization with
respect to the quadrant $K$

$A(\xi)=A_{\neq}(\xi)A_{=}(\xi)$

with index $\ae$ such that $\ae-s=1+\delta,|\delta|<1/2$.

To construct a discrete boundary value problem which is good approximation to
(15),(16) we need to choose $A_{d}(\xi)$ and $f_{d},g_{d}$ in a special way.
First, we introduce the operator $l_{h}$ which acts as follows. For a function
$u$ defined in $\mathbb{R}$ we take its Fourier transform $\tilde{f}$ then we
take its restriction on $\hbar T$ and periodically extend it to $\mathbb{R}$.
Finally, we take its inverse discrete Fourier transform and obtain the
function of discrete variable $(l_{h}u)(\tilde{x}),\tilde{x}\in h\mathbb{R}$.
Thus, we put

$f_{d}=l_{h}f,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
g_{d}=l_{h}g.$

Second, the symbol of digital operator $A_{d}$ we construct in the same way.
If we have the wave factorization for the symbol $A(\xi)$ then we take
restrictions of factors on $\hbar\mathbb{T}^{2}$ and the periodic symbol
$A_{d}(\xi)$ is a product of these restrictions. For such $f_{d},g_{d}$ and
the symbol $A_{d}(\xi)$ we obtain the following result.

Theorem 4. Let $f,g\in S(\mathbb{R}),\ae>1$. Then we have the following
estimate for solutions $u$ and$u_{d}$ of the continuous problem (15),(16) and
the discrete one (5),(12)

$|u(\tilde{x})-u_{d}(\tilde{x})|\leq C(f,g)h^{\beta},$

where the const $C(f,g)$ depends on functions $f,g$, $\beta>0$ can be an
arbitrary number.

Proof. We need to compare two functions (14) and (17), more exactly their
inverse discrete Fourier transform and inverse Fourier transform at points
$\tilde{x}\in K_{d}$. We have

$u_{d}(\tilde{x})-u(\tilde{x})=\frac{1}{4\pi^{2}}\left(\int\limits_{\hbar\mathbb{T}^{2}}e^{i\tilde{x}\cdot\xi}\tilde{u}_{d}(\xi)d\xi-\int\limits_{\mathbb{R}^{2}}e^{i\tilde{x}\cdot\xi}\tilde{u}(\xi)d\xi\right)=$
$=\frac{1}{4\pi^{2}}\int\limits_{\mathbb{R}^{2}\setminus\hbar\mathbb{T}^{2}}e^{i\tilde{x}\cdot\xi}A^{-1}_{\neq}(\xi)\left(A_{\neq}(\xi_{1},0)\tilde{g}(\xi_{1})+A_{\neq}(0,\xi_{2})\tilde{f}(\xi_{2})\right)d\xi,$

since according to our choice for $A_{d},f_{d},g_{d}$ the functions
$\tilde{u}_{d}$ and $\tilde{u}$ coincide in points
$\xi\in\hbar\mathbb{T}^{2}$.

We will estimate one summand.

$\left|\frac{1}{4\pi^{2}}\int\limits_{\mathbb{R}^{2}\setminus\hbar\mathbb{T}^{2}}e^{i\tilde{x}\cdot\xi}A^{-1}_{\neq}(\xi)A_{\neq}(\xi_{1},0)\tilde{g}(\xi_{1})d\xi\right|\leq$
$\leq\leavevmode\nobreak\
C\int\limits_{\hbar\pi}^{+\infty}\frac{d\xi_{2}}{(1+|\xi_{1}|+|\xi_{2}|)^{\ae}}\int\limits_{\hbar\pi}^{+\infty}|\xi_{1}|^{-\gamma}d\xi_{1},$

since $\tilde{g}\in S(\mathbb{R})$. It implies the required estimate.
$\blacksquare$

## Conclusion

In this paper we have considered two-dimensional cone only, but the authors
continue to work in multidimensional situations and we hope to obtain results
similar to a discrete half-space [21, 15].

As first practical applications the authors plan to study discrete variant of
a quarter-plane problem in diffraction theory [14] and elasticity theory [18].
We hope it will useful application of the developed theory.

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