# Shadow, ISCO, Quasinormal modes, Hawking spectrum, Weak Gravitational
lensing, and parameter estimation of a Schwarzschild Black Hole Surrounded by
a Dehnen Type Dark Matter Halo

Sohan Kumar Jha<EMAIL_ADDRESS>Chandernagore College, Chandernagore,
Hooghly, West Bengal, India

###### Abstract

Abstract

We consider Schwarzschildblack hole (BH) embedded in a Dehnen-$(1,4,0)$ type
dark matter halo (DDM) with two additional parameters - core radius $r_{s}$
and core density $\rho_{s}$ apart from mass $M$. We analyze the event horizon,
photon orbits, and ISCO around DDM BHs and emphasize the impact of DDM
parameters on them. Our study reveals the adverse effect of dark matter (DM)
on the radii of photon orbits, the innermost stable circular orbit (ISCO), and
the event horizon. We find the expressions for specific energy and angular
momentum for massive particles in time-like geodesics around DDM BH and
investigate their dependence on DDM parameters. We display BH shadows for
various values of core density and radius that reveal smaller shadows cast by
a SchwarzschildBH surrounded by DDM (SDDM) than a SchwarzschildBH in vacuum
(SV). We then move on to study quasinormal modes (QNMs) with the help of the
$6th$ order WKB method, the greybody factor using the semi-analytic bounds
method, and the Hawking spectrum for scalar and electromagnetic perturbations.
Core density and radius are found to have a significant impact on QNMs. Since
QNMs for scalar and electromagnetic perturbations differ significantly, we can
differentiate the two based on QNM observation. The greybody factor and the
power emitted as Hawking radiation are adversely impacted by the presence of
DM. We then study the weak gravitational lensing using the Gauss-Bonnet
theorem and obtain a deflection angle with higher-order correction terms.
Here, we see the deflection angle gets attenuated due to DM. Finally, we use
bounds on the deviation from Schwarzschild, $\delta$, reported by EHT for
$M87^{*}$, Keck, and VLTI observatories for $SgrA^{*}$ to gauge the viability
of our model. Our model is found to be concordant with observations for finite
parameter space. This leads to the possibility of our galactic center being
surrounded by DDM.

##  I Introduction

The existence of DM is one of the most sought-after topics in physics. Our
understanding of various astrophysical scenarios posits that BHs are
surrounded by matter fields. DM qualifies as one of the plausible candidates
for the matter field that may surround BHs. Observations related to giant
elliptical and spiral galaxies have provided the first breakthrough in the
search for DM rubin . One recent study has pegged the contribution of DM in a
galaxy’s mass up to $90\%$ persic . There exists strong astrophysical evidence
that points towards DM halos embedding supermassive BHs (SMBHs) that reside in
galactic centers akiyamal1 ; akiyamal6 . This makes it imperative to reckon
the contribution of DM near galactic center sofue ; boshkaye . Different DM
profiles can be considered to incorporate the effect of DM [7 \- 16]. Dehnen
density profile is commonly considered for dwarf galaxies dehnen ; mo . This
manuscript considers the static and spherically symmetric solution reported in
kalyan  where the Dehnen-$(1,4,0)$ type DM profile is considered. Particle
motion around BHs encodes information not only of the intrinsic space-time
geometry but also of the surrounding matter field. This makes it imperative to
study the BH shadow and ISCO to apprehend the effect of DM. BH shadow is
essentially a dark region in the celestial plane circumferenced by a bright
emission ring, whereas ISCO is a stable orbit closest to a BH for a test
particle. The dependence of BH shadow and ISCO on the BH parameters makes them
a productive field of study. Please see [20-29] for more details regarding the
application of BH shadow in the detection of DM and [30-36] for the effect of
DM on ISCO.

Normal modes are characteristics of closed systems, whereas QNMs are related
to open systems. When a BH is perturbed, after the initial outburst of
radiation comes the ringdown phase where QNMs arise stemming from the
dissipative oscillation of space-time KONOR . In the ringdown phase,
oscillations do not depend on initial conditions. As such, QNMs are the sole
function of intrinsic parameters of BH and the type of perturbation
responsible for QNMs. During the oscillations, the whole system loses energy
through the emission of gravitational waves (GWs). QNMs are complex-valued
frequencies where the real part gives the frequency of the emitted GWs, and
the imaginary part provides information regarding the stability of the
underlying space-time: a negative value signifies stability against
perturbation. They represent the decay rate of GWs. This makes the study of
QNMs a potent tool not only to gauge the effect of DM but also to probe the
stability of the BH-matter system. Studies related to QNMs for various BHs
have been carried out in [38-81].

Hawking radiation arises due to the unification of GR and quantum mechanics.
Hawking, with the help of quantum field theory, showed that BHs emit radiation
as a consequence of particles escaping BH HAWKING . The emitted radiation,
called Hawking radiation, is constituted by one of the particles created in a
pair production near the event horizon that can escape BH gravitational field,
and the second particle gets annihilated [83-85]. A temperature is associated
with a BH to be consistent with thermodynamics BEK ; KEIF . Various techniques
are available to obtain Hawking temperature [88-90]. The emission spectrum of
a BH differs from a blackbody spectrum. This is because, in the background of
a BH, particles move in a potential. While some particles get transmitted
toward spatial infinity, the rest are reflected back toward BH and consumed.
The greybody factor is associated with the quantum nature of a BH that
provides the transmission probability of radiation and corrects the blackbody
spectrum. Different strategies are there which provide greybody factor
[91-106]. Here, we employ semi-analytic bounds method GB ; GB1 ; GB2 .

When a light ray passes near a massive object such as SMBH, its path bends.
This way, gravitational lensing provides an excellent avenue to detect massive
astronomical objects. GL is also employed to probe the Universe’s expansion. A
significant number of studies have been devoted to studying GL for BHs, naked
singularities, and wormholes [107-129]. This article deals with GL in the weak
field limit known as weak lensing. We use the Gauss-Bonnet theorem first
proposed by Gibbons and Werner in GW . It was later extended to various cases
WERNER ; ISHIHARA1 ; ISHIHARA2 ; ONO1 ; ONO2 ; ONO3 . We obtain higher order
correction terms in the deflection angle using CRISNEJO . Theoretical
predictions of multifarious astrophysical phenomena provide information
regarding BH and its surrounding environment, whereas experimental
observations present an outstanding opportunity to probe the efficacy of our
models. Observations related to the shadow of $M87^{*}$ and $SgrA^{*}$
reported by the Event Horizon Telescope, Keck, and VLTI observatories provide
an unparalleled opportunity to put our model to the test. We use bounds on the
deviation from Schwarzschildfor $M87^{*}$ M871 ; M872  and $SgrA^{*}$ keck ;
vlti1 ; vlti2  to constrain DM parameters.

We organize our article as follows: Section II is where we introduce the BH
solution and discuss the impact of DM on the event horizon. We discuss QNMs in
section III, and section IV is devoted to studying the greybody factor and
Hawking spectrum. Section V deals with weak gravitational lensing and we try
to constrain core density and radius in section VI. This article ends with
conclusions in section VII.

##  II BH surrounded by Dehnen-type DM

Authors in article kalyan  have reported a static and spherically symmetric
metric where the SchwarzschildBH is embedded in a Dehnen-$(1,4,0)$ type DM
halo. The metric is

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (1)

where
$f(r)=1-\frac{2M}{r}+\frac{4\pi\rho_{s}r_{s}^{4}(3r+2r_{s})}{3r(r+r_{s})^{2}}$,
$\rho_{s}$ and $r_{s}$ being core density and core radius respectively. The
above solution reduces to a Schwarzschildmetric in vacuum in the limit
$\rho_{s}\rightarrow 0$ or $r_{s}\rightarrow 0$. The metric [1] has two
singularities: one at $r=0$ and another where $f(r)=0$. Kretschmann scalar for
the metric [1] is

$\displaystyle K=$
$\displaystyle\frac{16}{9r^{6}\left(r_{s}+r\right){}^{8}}(27M^{2}r^{8}+12r^{6}r_{s}^{2}\left(63M^{2}-104\pi
Mr_{s}^{3}\rho_{s}+42\pi^{2}r_{s}^{6}\rho_{s}^{2}\right)+12r^{5}r_{s}^{3}\left(126M^{2}-259\pi
Mr_{s}^{3}\rho_{s}+136\pi^{2}r_{s}^{6}\rho_{s}^{2}\right)$
$\displaystyle+2r^{4}r_{s}^{4}\left(945M^{2}-2172\pi
Mr_{s}^{3}\rho_{s}+1268\pi^{2}r_{s}^{6}\rho_{s}^{2}\right)+8r^{3}r_{s}^{5}\left(189M^{2}-462\pi
Mr_{s}^{3}\rho_{s}+284\pi^{2}r_{s}^{6}\rho_{s}^{2}\right)+4r^{2}r_{s}^{6}\left(189M^{2}\right.$
$\displaystyle\left.-480\pi
Mr_{s}^{3}\rho_{s}+305\pi^{2}r_{s}^{6}\rho_{s}^{2}\right)+216Mr^{7}r_{s}\left(M-\pi
r_{s}^{3}\rho_{s}\right)+4rr_{s}^{7}\left(4\pi
r_{s}^{3}\rho_{s}-3M\right)\left(23\pi
r_{s}^{3}\rho_{s}-18M\right)+3r_{s}^{8}\left(3M-4\pi
r_{s}^{3}\rho_{s}\right){}^{2}).$

The Kretschmann scalar has one singularity at $r=0$. Thus, the singularity of
the BH solution [1] at $r=0$ is an essential singularity that cannot be
removed by any coordinate transformation, whereas the singularity at $f(r)=0$
is a removable singularity. The largest solution of the equation $f(r)=0$
provides the position of the event horizon $r_{h}$. However, this equation for
the metric under consideration has no analytical solution. Thus, we will
resort to a numerical solution to obtain the event horizon. Qualitative
variations of the event horizon with core radius and density are shown below.

|
---|---
Figure 1: Variation of event horizon with $\rho_{s}$ kepping $r_{s}=0.5M$
shown in left panel and with $r_{s}$ kepping $\rho_{s}=1.0/M^{2}$ shown in the
right panel.

Fig. [1] shows that the event horizon decreases with core density and radius.
One can see that a pair $(\rho_{s},r_{s})$ provides a BH solution when we have
$r_{h}>0$. This renders a parameter space $(\rho_{s}M^{2},r_{s}/M)$ for which
we have a BH solution. It is displayed in Fig. [2].

---
Figure 2: Parameter space $(\rho_{s}M^{2},r_{s}/M)$ for existence of BH. The
red-colored region is where we have BH.

We can observe from Fig. [2] that for a finite parameter space, we have a BH
solution.

##  III Shadow radius and inner-most circular orbit

We, in this section, consider null and time-like orbits on the equatorial
plan, i.e., $\theta=\frac{\pi}{2}$. The Lagrangian in the background of the BH
under consideration is

$\mathscr{L}=\frac{1}{2}(-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}\dot{\phi}^{2}),$
(2)

where over-dot is differentiation with respect to affine parameter $\tau$.
Since the metric is independent of $t$ and $\phi$, we have two conserved
quantities of motion - energy $E$ and angular momentum $L$. For mass-less
particles, $E$ and $L$ are total energy and angular momentum, respectively,
whereas, for massive particles, they are specific energy and angular momentum.
They are given by

$E=-\frac{\partial\mathscr{L}}{\partial\dot{t}}=f(r)\dot{t}\quad\text{and}\quad
L=\frac{\partial\mathscr{L}}{\partial\dot{\phi}}=r^{2}\dot{\phi}$ (3)

Four-velocity of particles (massive and mass-less) follow the relation
$\dot{x_{\mu}}\dot{x^{\mu}}=\epsilon$ where $\epsilon$ is zero for mass-less
particles and $-1$ for massive particles. With this, we obtain the following
equation for radial coordinates:

$\dot{r}^{2}=E^{2}-f(r)\left(-\epsilon+\frac{L^{2}}{r^{2}}\right)=E^{2}-V(r),$
(4)

where $V(r)$ is the potential for the particle. Circular orbits are attributed
by $\dot{r}=\ddot{r}=0$. $V(r)=E^{2}$ follows from the first condition whereas
the second condition yields $\frac{dV(r)}{dr}=0$. From the second condition,
we obtain the radius of the photon orbit $(r_{ph})$ as the solution of the
equation

$r_{ph}f^{\prime}(r_{ph})=2f(r_{p}h),$ (5)

where ′ in the above equation is differentiation with respect to $r$. The
critical impact parameter $b_{ph}$ corresponding to the photon orbit, which is
also the shadow radius $(R_{s})$ of the BH, is

$b_{ph}=R_{s}=\frac{L}{E}=\frac{r_{ph}}{\sqrt{f(r_{ph})}.}$ (6)

It is not possible to obtain analytical expressions for either photon radius
or critical impact parameter. We will, therefore, obtain their values
numerically. We display variation of photon radius and shadow radius with core
density and radius in Fig. [3]. It shows that both the radii decrease with
core density as well as radius. Both the radii decrease almost linearly with
the core density. The change in the radii with core radius is very small
initially, and then it starts decreasing sharply.

|
---|---
Figure 3: Variation of photon and shadow radii with $\rho_{s}$ kepping
$r_{s}=0.5M$ shown in left panel and with $r_{s}$ kepping $\rho_{s}=1.0/M^{2}$
shown in the right panel.

We display the variation of shadow with core radius and density. It is evident
from Fig. [4] that the size of the shadow decreases due to the presence of DM.

|
---|---
Figure 4: BH shadows for different values of $\rho_{s}$ kepping $r_{s}=0.5M$
shown in left panel and for different values of $r_{s}$ kepping
$\rho_{s}=1.0/M^{2}$ shown in the right panel.

Next, we turn our attention to time-like geodesics where two conditions
$\dot{r}=\ddot{r}=0$ provide the following expressions for specific energy and
angular momentum:

$\displaystyle E^{2}$ $\displaystyle=$
$\displaystyle\frac{2f(r)^{2}}{2f(r)-rf^{\prime}(r)}$ $\displaystyle=$
$\displaystyle\frac{\left(3r^{2}(r-2M)+(3r-6M)r_{s}^{2}+6r(r-2M)r_{s}+8\pi
r_{s}^{5}\rho_{s}+12\pi
rr_{s}^{4}\rho_{s}\right){}^{2}}{3r\left(r_{s}+r\right)\left(3r^{3}(r-3M)+9r^{2}(r-3M)r_{s}+(3r-9M)r_{s}^{3}+9r(r-3M)r_{s}^{2}+24\pi
r^{2}r_{s}^{4}\rho_{s}+12\pi r_{s}^{6}\rho_{s}+32\pi
rr_{s}^{5}\rho_{s}\right)},$ $\displaystyle L^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{3}f^{\prime}(r)}{2f(r)-rf^{\prime}(r)}$
$\displaystyle=$
$\displaystyle\frac{r^{2}\left(3Mr^{3}+9Mr^{2}r_{s}+3Mr_{s}^{3}+l9Mrr_{s}^{2}-12\pi
r^{2}r_{s}^{4}\rho_{s}-4\pi r_{s}^{6}\rho_{s}-12\pi
rr_{s}^{5}\rho_{s}\right)}{3r^{3}(r-3M)+9r^{2}(r-3M)r_{s}+(3r-9M)r_{s}^{3}+9r(r-3M)r_{s}^{2}+24\pi
r^{2}r_{s}^{4}\rho_{s}+12\pi r_{s}^{6}\rho_{s}+32\pi rr_{s}^{5}\rho_{s}}.$

Fig. [5] demonstrates the behavior of $E$ vs $r/M$ for various values of core
radius and density. It shows that the specific energy of a test particle in
the background of a SchwarzschildBH surrounded by Dehnen-type DM is always
less than that for a SchwarzschildBH in vacuum. The minimum of the specific
energy occurs at the ISCO radius. It is also observed that the specific energy
tends to have a constant value far from BH. The impact of DM on the specific
angular momentum is illustrated in Fig. [6], where a comparison is also made
with SchwarzschildBH in vacuum. It is evident from Figs. [6, 6] that both the
parameters adversely impact the specific angular momentum. Here, we can
observe that the specific energy and angular momentum for a test particle in
the spacetime considered are less than that in the spacetime of a
SchwarzschildBH in vacuum.

Figure 5: Variation of $E$ with $r/M$ for different values of core density
keeping $r_{s}=0.5M$ shown in the left panel and for different values of core
radius keeping $\rho_{s}=1.0/M^{2}$ shown in the right panel.

Figure 6: Variation of $L$ with $r/M$ for different values of core density
keeping $r_{s}=0.5M$ shown in the left panel and for different values of core
radius keeping $\rho_{s}=1.0/M^{2}$ shown in the right panel.

An additional condition to be imposed on the potential for the inner-most
circular orbit is $\frac{d^{2}V(r)}{dr^{2}}|_{r=r_{isco}}=0$. This yields the
following equation whose solution gives the ISCO radius:

$\displaystyle
r_{isco}f(r_{isco})f^{\prime\prime}(r_{isco})-2r_{isco}f^{\prime}(r_{isco})^{2}+3f(r_{isco})f^{\prime}(r_{isco})=0$
which gives $\displaystyle 3r_{s}^{6}\left(-3M\left(280\pi
r_{\text{isco}}^{3}\rho_{s}+r_{\text{isco}}\right)+64\pi
r_{\text{isco}}^{4}\rho_{s}+18M^{2}\right)+6r_{\text{isco}}r_{s}^{5}\left(-Mr_{\text{isco}}\left(244\pi
r_{\text{isco}}^{2}\rho_{s}+9\right)+12\pi
r_{\text{isco}}^{4}\rho_{s}+54M^{2}\right)$ $\displaystyle+12\pi
r_{s}^{9}\rho_{s}\left(120\pi
r_{\text{isco}}^{3}\rho_{s}+r_{\text{isco}}-12M\right)+12\pi
r_{\text{isco}}r_{s}^{8}\rho_{s}\left(6\left(8\pi
r_{\text{isco}}^{3}\rho_{s}+r_{\text{isco}}\right)-71M\right)-36\pi
r_{\text{isco}}^{2}r_{s}^{7}\rho_{s}\left(58M-5r_{\text{isco}}\right)$
$\displaystyle+27Mr_{\text{isco}}^{2}r_{s}^{4}\left(-12\pi
r_{\text{isco}}^{3}\rho_{s}-5r_{\text{isco}}+30M\right)+135Mr_{\text{isco}}^{4}r_{s}^{2}\left(6M-r_{\text{isco}}\right)+54Mr_{\text{isco}}^{5}r_{s}\left(6M-r_{\text{isco}}\right)$
$\displaystyle+180Mr_{\text{isco}}^{3}r_{s}^{3}\left(6M-r_{\text{isco}}\right)+9Mr_{\text{isco}}^{6}\left(6M-r_{\text{isco}}\right)+560\pi^{2}r_{\text{isco}}r_{s}^{11}\rho_{s}^{2}+1344\pi^{2}r_{\text{isco}}^{2}r_{s}^{10}\rho_{s}^{2}+96\pi^{2}r_{s}^{12}\rho_{s}^{2}=0.$

The above equation does not have any analytical solution. As such, it is
numerically solved to study the impact of DM on the motion of massive
particles. We elucidate the impact of DM parameters on the ISCO radius
graphically in Fig. [7], which shows that similar to the cases of event
horizon, photon, and shadow radii, the ISCO radius decreases with core radius
and density. ISCO radius decreases with $\rho_{s}$ almost linearly. However,
initially, when we increase the core radius, the ISCO radius decreases by a
very small amount and then starts decreasing sharply.

Figure 7: Variation of ISCO radius with core density kepping $r_{s}=0.5M$
shown in left panel and with core radius kepping $\rho_{s}=1.0/M^{2}$ shown in
the right panel.

While Figs. [1 \- 7] provide a qualitative view of the impact of DM parameters
on various aspects of null and time-like geodesics, quantitative view will
provide deeper insight into the effect of DM on the motion of mass-less and
massive particles. Table [1] enlists numerical values of various quantities
for different values of core density, whereas Table [2] enlists those values
for different core radius.

$\rho_{s}M^{2}$ | $r_{h}\text{/M}$ | $r_{p}\text{/M}$ | $R_{s}\text{/M}$ | $r_{\text{isco}}\text{/M}$ | $E_{\text{isco}}$ | $L_{\text{isco}}\text{/M}$
---|---|---|---|---|---|---
0. | 2. | 3. | 5.19615 | 6. | 0.942809 | 3.4641
0.4 | 1.87711 | 2.82947 | 4.9645 | 5.60775 | 0.939139 | 3.32785
0.8 | 1.74042 | 2.64066 | 4.7098 | 5.18113 | 0.934583 | 3.17763
1.2 | 1.58391 | 2.42605 | 4.42331 | 4.70701 | 0.928666 | 3.0082
1.6 | 1.39527 | 2.17098 | 4.08858 | 4.16071 | 0.920417 | 2.81001
2.0 | 1.13913 | 1.83735 | 3.6662 | 3.48289 | 0.907278 | 2.56112

Table 1: Numerical values of various quatities with several values of
$\rho_{s}$ keeping $r_{s}=0.5M$.

$r_{s}\text{/M}$ | $r_{h}\text{/M}$ | $r_{p}\text{/M}$ | $R_{s}\text{/M}$ | $r_{\text{isco}}\text{/M}$ | $E_{\text{isco}}$ | $L_{\text{isco}}\text{/M}$
---|---|---|---|---|---|---
0. | 2. | 3. | 5.19615 | 6. | 0.942809 | 3.4641
0.12 | 1.99879 | 2.99837 | 5.19401 | 5.9963 | 0.942777 | 3.46289
0.24 | 1.98191 | 2.97533 | 5.16333 | 5.9436 | 0.94231 | 3.44533
0.36 | 1.91201 | 2.87906 | 5.03379 | 5.72302 | 0.940269 | 3.36999
0.48 | 1.71868 | 2.61137 | 4.67155 | 5.11665 | 0.933838 | 3.1557
0.6 | 1.19301 | 1.88899 | 3.70613 | 3.56044 | 0.908784 | 2.57035

Table 2: Numerical values of various quatities with several values of $r_{s}$
keeping $\rho_{s}=1.0/M^{2}$.

From Table [1], we observe that up to a certain value of $\rho_{s}$, we have
$R_{s}\leq r_{isco}$, and beyond that value, ISCO radius becomes smaller than
shadow radius. At a fixed value of $r_{s}=0.5M$, the value of the core density
where the ISCO and shadow radii coincide is $1.72272/M^{2}$. A similar
conclusion can be drawn from Table [2] where at the fixed value of
$\rho_{s}M^{2}=1.0$, the ISCO and shadow radii coincide at $r_{s}=0.579691M$.
This concludes our discussion on the null and time-like geodesics in the
background of the BH surrounded by Dehnen-type DM.

##  IV Quasinormal frequencies

QNMs encode the inherent characteristics of the underlying geometry and are
independent of any specific initial conditions. They also provide information
regarding the stability of the black hole system against perturbations. To
compute QNMs, we first take into account field equations given by

$\displaystyle\frac{1}{\sqrt{-g}}{\partial\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\chi)=0,$
(10)
$\displaystyle\frac{1}{\sqrt{-g}}{\partial\nu}(F_{\rho\sigma}g^{\rho\mu}g^{\sigma\nu}\sqrt{-g})=0,$
(11)

where $F_{\rho\sigma}={\partial\rho}A^{\sigma}-{\partial\sigma}A^{\rho}$,
$A_{\nu}$ being the electromagnetic four-potential. Owing to the spherical
symmetry in the existing system, we can decompose the field with the
introduction of the tortoise coordinate defined by
$\text{d}r_{\ast}=\frac{\text{d}r}{f(r)}$. This enables us to get a
Schr$\ddot{o}$dinger-like equation

$-\frac{\partial^{2}\Phi}{\partial t^{2}}+\frac{\partial^{2}\Phi}{\partial
r^{*2}}+V_{eff}(r^{*})\Phi=0,$ (12)

where $V_{eff}(r)$ is the effective potential or the Regge–Wheeler potential
given by

$\displaystyle V_{eff}(r)$ $\displaystyle=$ $\displaystyle
f(r)\left(\frac{\ell\left(\ell+1\right)}{r^{2}}+f(r)^{\prime}\frac{(1-s^{2})}{r}\right)$
(14) $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{r}+\frac{4\pi\left(2r_{s}+3r\right)r_{s}^{4}\rho_{s}}{3r\left(r_{s}+r\right){}^{2}}\right)$
$\displaystyle\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{\left(1-s^{2}\right)}{r}\left(\frac{2M}{r^{2}}-\frac{4\pi\left(2r_{s}+3r\right)r_{s}^{4}\rho_{s}}{3r^{2}\left(r_{s}+r\right){}^{2}}-\frac{8\pi\left(2r_{s}+3r\right)r_{s}^{4}\rho_{s}}{3r\left(r_{s}+r\right){}^{3}}+\frac{4\pi
r_{s}^{4}\rho_{s}}{r\left(r_{s}+r\right){}^{2}}\right)\right].$

Here $\ell$ is the multipole number. For $s=0$, we obtain potential for the
scalar field and $s=1$ yields potential for the vector field. To obtain QNMs
which are stationary solutions of Eq. [12], we write $\Phi(t,r)=e^{-i\omega
t}\Phi(r)$ where $\omega$ represents quasinormal frequency. It yields the
time-independent Schr$\ddot{o}$dinger-like equation

$\frac{\partial^{2}\Phi}{\partial
r^{*2}}-\left[\omega^{2}-V_{eff}(r^{*})\right]\Phi=0.$ (15)

We impose the boundary conditions that the waves are purely outgoing at
spatial infinity, i.e., $\Phi\sim e^{i\omega r_{\ast}}$ and purely ingoing at
the event horizon, i.e., $\Phi\sim e^{-i\omega r_{\ast}}$. It implies that
QNMs can only be detected when the source of perturbation is no longer
present.
The WKB approximation method is widely used to calculate QNMs. It was first
presented in Iyer  and was later extended to higher order in Konoplya . The
WKB method is effective for low overtone numbers $n$, especially for $n<\ell$,
which is the case in this paper. The $6th$ order WKB formula for QNMs is
Konoplya

$\frac{\text{i}(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{{}^{\prime\prime}}}}-\sum^{6}_{\text{i}=2}\Omega_{\text{i}}=n+\frac{1}{2},$
(16)

where $V_{0}$ is the maximum value of the Regge–Wheeler potential obtained at
$r_{*}=r_{0}$,
$V^{\prime\prime}_{0}=\frac{d^{2}V(r_{*})}{dr_{*}^{2}}|_{r_{*}=r_{0}}$, and
$\Omega_{\text{i}}$ are WKB corrections Konoplya . We display variation of GW
frequency and decay rate with core density $\rho_{s}$, core radius $r_{s}$,
and multipole number $\ell$ for $n=0$ in Fig. [8, 9, 10, 11]. We observe from
Fig. [8, 10] that the GW frequency increases with $\rho_{s}$, $\ell$, and
$r_{s}$ for both the perturbations. The nature of variation of decay rate with
multipole number for scalar and electromagnetic perturbations, however, are
opposite to each other. The decay rate for scalar perturbations decreases with
$\ell$ but increases for electromagnetic perturbations. Thus, GWs with higher
multipole numbers emitted due to scalar perturbation will propagate further
than those caused by electromagnetic perturbation. Fig. [9] shows that the
decay rate initially increases with core density reaching a maximum value at
around $\rho_{s}=1.6/M$ and then decreases for both perturbations. The decay
rate always increases with $r_{s}$ evident from Fig. [11].

Figure 8: Variations of gravitational wave frequency with DM density for
different $\ell$ keeping $r_{s}=0.5M$. The left one is for scalar
perturbation, and the right one is for electromagnetic perturbation.

Figure 9: Variation of decay rate with DM density for different $\ell$ keeping
$r_{s}=0.5M$. The left one is for scalar perturbation, and the right one is
for electromagnetic perturbation.

Figure 10: Variation of gravitational wave frequency with core radius for
different $\ell$ keeping $\rho_{s}=1.0/M^{2}$. The left one is for scalar
perturbation, and the right one is for electromagnetic perturbation.

Figure 11: Variation of decay rate with core radius for different $\ell$
keeping $\rho_{s}=1.0/M^{2}$. The left one is for scalar perturbation, and the
right one is for electromagnetic perturbation.

We further tabulate numerical values of QNMs for scalar perturbations in Table
III for various values of core density with $r_{s}=0.5M$ and in Table V for
various values of core radius with $\rho_{s}=1.0/M^{2}$. Tables IV and VI
provide QNMs for electromagnetic perturbation for various values of $\rho_{s}$
and $r_{s}$ with the same fixed values of $r_{s}$ and $\rho_{s}$,
respectively. The negative values of the imaginary part show that the BH
system is stable against perturbation. Since the frequency and decay rate
increase with $r_{s}$ for both perturbations, GWs emitted from BHs surrounded
by DM halo of smaller core radius will propagate further. The same conclusion
with respect to core density can be drawn when the density is
$\rho_{s}\lesssim 1.6/M^{2}$. Beyond that value, if we increase the core
density, the decay rate decreases, and thus, GWs emitted will propagate
further for denser DM halo.

$\rho_{s}M^{2}$ | $\ell$=1 | $\ell$=2 | $\ell$=3
---|---|---|---
0. | 0.29291 -0.0977616 i | 0.483642 -0.0967661 i | 0.675366 -0.0965006 i
0.4 | 0.306639 -0.09983 i | 0.506242 -0.0988966 i | 0.706904 -0.0986405 i
0.8 | 0.323268 -0.101961 i | 0.533643 -0.101103 i | 0.74515 -0.100859 i
1.2 | 0.344212 -0.104012 i | 0.568211 -0.103247 i | 0.793418 -0.103022 i
1.6 | 0.372269 -0.105508 i | 0.614665 -0.10486 i | 0.85833 -0.104661 i
2.0 | 0.414478 -0.104312 i | 0.685097 -0.103816 i | 0.956949 -0.103659 i

Table 3: The QNM frequencies of scalar perturbation with several values of
$\rho_{s}$.

$\rho_{s}M^{2}$ | $\ell$=1 | $\ell$=2 | $\ell$=3
---|---|---|---
0. | 0.248191 -0.092637 i | 0.457593 -0.0950112 i | 0.656898 -0.0956171 i
0.4 | 0.261054 -0.0949288 i | 0.479679 -0.0972084 i | 0.68807 -0.0977897 i
0.8 | 0.276809 -0.0973515 i | 0.506556 -0.0995055 i | 0.725941 -0.100054 i
1.2 | 0.296938 -0.099796 i | 0.540623 -0.101777 i | 0.773849 -0.102279 i
1.6 | 0.324449 -0.101841 i | 0.58671 -0.103575 i | 0.838495 -0.104011 i
2.0 | 0.367219 -0.101408 i | 0.65737 -0.102794 i | 0.937259 -0.103142 i

Table 4: The QNM frequencies of electromagnetic perturbation with several
values of $\rho_{s}$.

$r_{s}\text{/M }$ | $\ell$=1 | $\ell$=2 | $\ell$=3
---|---|---|---
0. | 0.29291 -0.0977616 i | 0.483642 -0.0967661 i | 0.675366 -0.0965006 i
0.1 | 0.292969 -0.0977684 i | 0.483739 -0.0967732 i | 0.675502 -0.0965078 i
0.2 | 0.293826 -0.0978778 i | 0.485144 -0.0968871 i | 0.67746 -0.0966223 i
0.3 | 0.297464 -0.0983785 i | 0.491121 -0.0974061 i | 0.685796 -0.097144 i
0.4 | 0.307654 -0.0998034 i | 0.507888 -0.0988789 i | 0.709191 -0.0986241 i
0.5 | 0.333074 -0.103013 i | 0.549819 -0.102199 i | 0.767733 -0.101964 i

Table 5: The QNM frequencies of scalar perturbation with several values of
$r_{s}$.

$r_{s}\text{/M }$ | $\ell$=1 | $\ell$=2 | $\ell$=3
---|---|---|---
0. | 0.248191 -0.092637 i | 0.457593 -0.0950112 i | 0.656898 -0.0956171 i
0.1 | 0.248248 -0.0926453 i | 0.457689 -0.0950187 i | 0.657033 -0.0956245 i
0.2 | 0.249057 -0.0927734 i | 0.459065 -0.0951375 i | 0.658971 -0.0957414 i
0.3 | 0.252481 -0.0933448 i | 0.464915 -0.0956762 i | 0.667216 -0.0962726 i
0.4 | 0.262072 -0.0949485 i | 0.481325 -0.0972021 i | 0.690356 -0.0977787 i
0.5 | 0.286193 -0.0985848 i | 0.522474 -0.10066 i | 0.74834 -0.101187 i

Table 6: The QNM frequencies of scalar perturbation with several values of
$r_{s}$.

It is evident from our analysis that the signature of Dehnen-type DM can be
gauged from the observation of QNMs. We can also distinguish QNMs for scalar
and electromagnetic perturbations based on observation. Additionally, we would
like to emphasize the efficacy of using the $6th$ order WKB method. Fig. [12]
shows that QNMs converge quickly for pair $n=0,\ell=1$ but fluctuate for
$n=3,\ell=0$.

Figure 12: Variations of the real and imaginary part of quasinormal
frequencies with respect to WKB order for various values of $(n,\ell)$ pair.
The left one is for (0,1), and the right one is for (3,0). The upper line in
each plot is for the real part of quasinormal modes, and the lower line is for
the imaginary part of quasinormal modes. We have taken $\rho_{s}=1.0/M^{2}$
and $r_{s}=0.5M$.

##  V Greybody factor and Hawking spectrum

Radiation emitted by a BH HAWKING ; BEK ; KEIF  as observed locally differs
from that observed at spatial infinity due to the redshift factor. Greybody
distribution provides the Hawking radiation received by an asymptotic
observer. The transmission probability of emitted radiation is given by GF.
Several different methods can be employed to obtain GF qn32 ; qn36 ; qn37 ;
qn38 ; qn39 ; qn40 ; qn41 ; qn42 ; qn43 ; qn44 ; qn45 ; qn46 . We will be
using the semi-analytic bounds method to obtain lower bounds on GF as GB ; GB1
; GB2

$T\left(\omega\right)\geq\sec
h^{2}\left(\frac{1}{2\omega}\int_{-\infty}^{+\infty}V_{eff}(r_{\ast})dr_{\ast}\right)\Rightarrow
T\left(\omega\right)\geq\sec
h^{2}\left(\frac{1}{2\omega}\int_{r_{eh}}^{+\infty}V_{eff}(r)\frac{dr}{f(r)}\right).$
(17)

In Fig. [13], we exhibit variation of GF with frequency $\omega$ for various
values of core density, and Fig. [14] illustrates the dependence of GF on core
radius. As evident from these figures, the presence of DM adversely impacts
the transmission probability of Hawking radiation. GF asymptotically
approaches unity, which is the value for a blackbody.

Figure 13: Greybody bounds for various values of core density. The left one is
for the scalar field, and the right one is for the electromagnetic field. We
have taken $\ell=1$.

Figure 14: Greybody bounds for BM BH. The left one is for the scalar field,
and the right one is for the electromagnetic field. We have taken $\ell=1$.

The power emitted by a BH in thermal equilibrium with its surrounding in the
$\ell th$ mode is yg2017 ; fg2016

$P_{\ell}\left(\omega\right)=\frac{A}{8\pi^{2}}T(\omega)\frac{\omega^{3}}{e^{\omega/T_{H}}-1}.$
(18)

Here, $A$ is taken to be the horizon area yg2017 , and $T_{H}$ is the Hawking
temperature kalyan .

Figure 15: Variation of power emitted in the $\ell$th mode with $\omega$ for
different values of $\rho_{s}$ keeping $r_{s}$ fixed at $0.5M$. The left one
is for the electromagnetic field, and the right one is for the scalar field.
We have taken $\ell=1$.

Figure 16: Variation of power emitted in the $\ell$th mode with $\omega$ for
different values of $r_{s}$ keeping $\rho_{s}$ fixed at $1.0/M^{2}$. The left
one is for the electromagnetic field, and the right one is for the scalar
field. We have taken $\ell=1$.

Fig. [15] illustrates the impact of core density on the power emitted, and
Fig. [16] shows the dependence of power emitted on core radius. It exhibits
the adverse impact of DM on thermal radiation. The power emitted decreases,
and the peak shifts towards the right with an increase in core density or
radius for both the perturbations. We can, therefore, conclude that thermal
radiation received from a Schwarzschildin the vacuum will be greater than that
received from a Schwarzschildsurrounded by Dehnen-type DM.

##  VI weak gravitational lensing

The motivation behind studying gravitational lensing is the dependence of
lensing angle on the inherent characteristics of the underlying spacetime. It
is for this reason that gravitational lensing is such a widely researched
topic. We intend to analyze the effect of DM on lensing. We will be using the
Gauss-Bonnet theorem to study weak gravitational lensing GW . The formula for
the gravitational lensing is ISHIHARA1 ; CARMO

$\gamma_{D}=-\int\int_{{}_{R}^{\infty}\Box_{S}^{\infty}}KdS,$ (19)

where ${}_{O}^{\infty}\Box_{S}^{\infty}$ is the quadrilateral (please see Fig.
(17)) and K is the Gaussian curvature. Considering null geodesics, one obtains
the Gaussian curvature as WERNER

$\displaystyle K$ $\displaystyle=$
$\displaystyle-\frac{f(r)^{3/2}}{r}\frac{d}{d\text{r}}\left(f(r)\frac{d}{d\text{r}}(\frac{r}{\sqrt{f(r)}})\right),$
$\displaystyle=$
$\displaystyle\frac{3M^{2}}{r^{4}}-\frac{2M}{r^{3}}+\left(\frac{12\pi}{r^{4}}-\frac{24\pi
M}{r^{5}}\right)r_{s}^{4}\rho_{s}+\mathcal{O}\left(\frac{\rho_{s}r_{s}^{5}}{r^{5}},\frac{M\rho_{s}r_{s}^{5}}{r^{6}}\right).$

Figure 17: Schematic diagram of the quadilateral
${}_{O}^{\infty}\Box_{S}^{\infty}$.

We rewrite Eq. (19) as ONO1

$\int\int_{{}_{O}^{\infty}\Box_{S}^{\infty}}KdS=\int_{\phi_{S}}^{\phi_{O}}\int_{\infty}^{r_{0}}K\sqrt{\zeta}drd\phi,$
(20)

where $r_{0}$ is the least distance from BH. We first consider a straight-line
trajectory and obtain the deflection angle. Here, the path is given by
$r=\frac{b}{sin\phi}$. Our initial deflection angle using straight-line
trajectory is $\gamma_{D}^{0}=\frac{4M}{b}+\frac{3\pi
M^{2}}{4b^{2}}-\left(\frac{45\pi^{2}M^{2}}{8b^{4}}+\frac{32\pi
M}{3b^{3}}+\frac{3\pi^{2}}{b^{2}}\right)r_{s}^{4}\rho_{s}+\mathcal{O}\left(\frac{M^{3}}{b^{3}},\frac{M^{3}\rho_{s}r_{s}^{4}}{b^{5}}\right)$.
To obtain higher order correction terms, we will use the trajectory CRISNEJO

$u=\frac{1}{r}=\frac{\sin\phi}{b}+\frac{M(1-\cos\phi)^{2}}{b^{2}}-\frac{M^{2}(60\phi\,\cos\phi+3\sin
3\phi-5\sin\phi)}{16b^{3}}+\mathcal{O}\left(\frac{M^{2}\alpha}{b^{5}}\right),$
(21)

where $u=1/r$. The deflection for the above trajectory is

$\gamma_{D}=\frac{4M}{b}+\frac{15\pi
M^{2}}{4b^{2}}+\frac{128M^{3}}{3b^{3}}-\left(\frac{\left(6272\pi+405\pi^{3}\right)M^{3}}{10b^{5}}+\frac{60\pi^{2}M^{2}}{b^{4}}+\frac{128\pi
M}{3b^{3}}+\frac{3\pi^{2}}{b^{2}}\right)r_{s}^{4}\rho_{s}+\mathcal{O}\left(\frac{M^{4}}{b^{4}},\frac{M^{4}\rho_{s}r_{s}^{4}}{b^{6}}\right),$
(22)

where we have integrated $\phi$ from $0$ to $\pi+\gamma_{D}^{0}$. Fig. [18]
illustrates the effect of DM parameters on the deflection angle. As we can see
from the figure, the deflection angle initially increases with the impact
parameter $b$ reaching a maximum value and then starts decreasing. Some such
maximum positions are $b=1.23095M$ for $\rho_{s}=0.2/M^{2},r_{s}=0.5M$,
$b=0.867132M$ for $\rho_{s}=1.0/M^{2}$, $r_{s}=0.28M$. As we increase either
$\rho_{s}$ or $r_{s}$, the position of the maximum value shifts towards the
right. It is evident from Fig. [18] that the deflection decreases due to the
presence of DM, i.e., a SchwarzschildBH in the vacuum will deflect a light ray
more than a SchwarzschildBH embedded in a Dehnen-type DM halo.

Figure 18: Deflection angle for weak gravitational lensing. The left the panel
is for different values of core density keeping $r_{s}=0.5M$, and the right
one is for different values of core radius keeping $\rho_{s}=1.0/M^{2}$ .

Fig. [18] shows that for definite values of core density and radius, there
exists an impact parameter that results in null deflection. Few such values
are $b=0.92628M$ for $\rho_{s}M^{2}=0.2$, $r_{s}=0.5M$ and $b=0.752447M$ for
$\rho_{s}M^{2}=1.0$, $r_{s}=0.3M$. Null deflection indicates that the light
rays with those specific impact parameters do not deviate from their original
path. This occurs because the gravitational field of BH gets canceled by that
of DM. Our analysis in this section makes it clear that we can gauge the
impact of Dehnen-type DM from lensing observation, as the impact of DM on the
angle of deflection is significant.

##  VII parameter estimation using shadow observable

This section will utilize recent images and corresponding observables related
to SMBHs $M87^{*}$ and $SgrA^{*}$ to constrain two parameters - core radius
$r_{s}$ and core density $\rho_{s}$. Constraints regarding deviation from
Schwarzschild, $\delta$, for $M87^{*}$ obtained from EHT M871 ; M872  and
$SgrA^{*}$ obtained from Keck and VLTI observatories keck ; vlti1 ; vlti2
will be used here. We first define the observable $\delta$ as del

$\delta=\frac{R_{s}}{3\sqrt{3}M}-1.$ (23)

$R_{s}$ is the shadow radius for the BH under consideration. $3\sqrt{3}M$ is
the shadow radius for SchwarzschildBH. A positive value of $\delta$ will
indicate a larger shadow radius for our BH than Schwarzschild, whereas
negative values will infer a smaller shadow. Bounds on deviation parameter
$\delta$ obtained for $M87^{*}$ and $SgrA^{*}$ are tabulated in Table [7]. Our
metric will be put to test using these bounds.

BH | Observatory | $\delta$ | 1$\sigma$ bounds | 2$\sigma$ bounds
---|---|---|---|---
$M87^{*}$ | EHT | $-0.01^{+0.17}_{-0.17}$ | $4.26\leq\frac{R_{s}}{M}\leq 6.03$ | $3.38\leq\frac{R_{s}}{M}\leq 6.91$
$SgrA^{*}$ | VLTI | $-0.08^{+0.09}_{-0.09}$ | $4.31\leq\frac{R_{s}}{M}\leq 5.25$ | $3.85\leq\frac{R_{s}}{M}\leq 5.72$
Keck | $-0.04^{+0.09}_{-0.10}$ | $4.47\leq\frac{R_{s}}{M}\leq 5.46$ | $3.95\leq\frac{R_{s}}{M}\leq 5.92$

Table 7: Bounds on $\delta$ from different observatories.

We display variations of deviation parameter with core density in Fig. [19]
and with core radius in Fig. [19] along with lower $1\sigma$ and $2\sigma$
bounds obtained from EHT, Keck, and VLTI observations. $\delta$ decreases
almost linearly with increasing $\rho_{s}$ whereas the change in $\delta$ with
$r_{s}$ is much steeper. To have a qualitative idea of the impact changes in
core density and radius have on $\delta$, we numerically calculate the rate of
change of deviation parameter with respect to $\rho_{s}$ and $r_{s}$, i.e.,
$\frac{\partial\delta}{\partial\rho_{s}}$ and $\frac{\partial\delta}{\partial
r_{s}}$ at point $(1.0/M^{2},0.5M)$ which come out to be $-0.137559$ and
$-1.04576$, respectively. Thus, a change in core radius brings a larger change
in deviation parameter than a similar amount of change in core density.

Figure 19: Variations of deviation parameter shown with respect to core
density keeping $r_{s}=0.5M$ in the left panel and with respect to core radius
keeping $\rho_{s}M^{2}=1.0$ in the right panel. Horizontal solid lines are
1-$\sigma$ lower bound, and dashed lines are 2-$\sigma$ lower bounds. The
color code for horizontal lines is: Black is for EHT, Blue is for Keck, and
Red is for VLTI.

Horizontal lines in Fig. [19] provide lower $1\sigma$ and $2\sigma$ bounds on
$\delta$. Their intersection with the variation curve indicates the existence
of upper bounds of $\rho_{s}$ and $r_{s}$. We have obtained upper bounds on
core radius and core density for different observations. Our study shows that
keeping $\rho_{s}M^{2}=1.0$, $r_{s}\leq 0.547123M$ within $1\sigma$ confidence
level and $r_{s}\leq 0.6200394M$ within $2\sigma$ for $M87^{*}$. Similarly,
upper bounds on core radius consistent with Keck and VLTI observations are
$0.517663M$ and $0.5404195M$ within $1\sigma$ confidence level and $0.580395M$
and $0.589354M$ within $2\sigma$ confidence level. Upper bounds on core
density are obtained, keeping the core radius fixed at $0.5M$. Upper bounds on
$\rho_{s}$ are as follows:
From EHT observation $\rho_{s}M^{2}\leq 1.403235$ within $1\sigma$ and
$\rho_{s}M^{2}\leq 2.204515$ within $2\sigma$.
From Keck observation, $\rho_{s}M^{2}\leq 1.14017$ within $1\sigma$ bounds and
$\rho_{s}M^{2}\leq 1.74513$ within $2\sigma$ bounds.
VLTI observation yields $\rho_{s}M^{2}\leq 1.340095$ within $1\sigma$ and
$\rho_{s}M^{2}\leq 1.84493$ within $2\sigma$.
These results are tabulated in [8].

| bounds on $r_{s}/M$ | bounds on $\rho_{s}M^{2}$
---|---|---
BH | Observatory | 1$\sigma$ bounds | 2$\sigma$ bounds | 1$\sigma$ bounds | 2$\sigma$ bounds
$M87^{*}$ | EHT | $0.547123$ | $0.6200394$ | $1.403235$ | $2.204515$
$SgrA^{*}$ | VLTI | $0.5404195$ | $0.589354$ | $1.340095$ | $1.84493$
Keck | $0.517663$ | $0.580395$ | $1.14017$ | $1.74513$

Table 8: Bounds on $r_{s}/M$ and $\rho_{s}M^{2}$ from different observatories.

Table [8] shows the upper bounds of core radius and density for fixed values
of core density and radius. To have a better idea of the viability of our
model, we exhibit variation of deviation parameter $\delta$ with core radius
and density for $M87^{*}$ and $SgrA^{*}$.

|  |  |
---|---|---|---
Figure 20: Variation of deviation parameter $\delta$ with core $\rho_{s}$ and $r_{s}$. The left one is for $M87^{*}$ with EHT constraints, and the right one is for $SgrA^{*}$ with Keck bounds. In each plot, the upper solid black line corresponds to the lower $2\sigma$ bound, and the lower one is for $1\sigma$ bounds.  |
---|---
Figure 21: Variation of deviation parameter $\delta$ with core $\rho_{s}$ and
$r_{s}$ for $SgrA^{*}$ with VLTI bounds. The upper solid black line
corresponds to the lower $2\sigma$ bound, and the lower one is for $1\sigma$
bounds.

White regions in the above figures provide the parameter space where we do not
have black holes. Figs. [20, 21] show that for finite parameter space
$(\rho_{s}M^{2}-r_{s}/M)$, our BH under consideration is congruent with
experimental data from EHT, Keck, and VLTI observatories. We can, therefore,
conclude that a SchwarzschildBH embedded in a Dehnen-type DM halo is a viable
candidate for SMBH.

##  VIII conclusions

We have extensively studied the impact of Dehnen-type DM on null and time-like
geodesics, QNMs, greybody factor, Hawking temperature, and weak GL. We have
also tried to constrain DM parameters, namely, core density and radius, using
$M87^{*}$ and $SgrA^{*}$ data. Utilizing Lagrangian formulation, we first
obtain the potential for the motion of mass-less and massive particles in the
background of our BH. Then, imposing appropriate conditions on the potential,
we obtain equations whose solutions yield photon radius and ISCO radius.
Photon radius, along with imposed conditions on the potential, provides a
critical impact parameter, which is also the shadow radius for an asymptotic
observer. We observe that a SchwarzschildBH embedded in a DDM halo cast a
smaller shadow than a SchwarzschildBH in the vacuum. The ISCO radius, too, is
found to be smaller for SDDM. Even though the extent of the impact of core
radius and density on these observables are different, their nature of effect
is the same, i.e., increasing core radius or density decreases photon and ISCO
radii. We have studied the qualitative and quantitative nature of variation of
observables with respect to DM parameters. Our quantitative inquisition shows
that even though for a wide range of values, shadow radius $R_{s}$ lies
between event horizon $r_{h}$ and ISCO radius $r_{isco}$, there exist values
of core radius and density for which $R_{s}\geq r_{isco}$. For example,
keeping core radius fixed at $r_{s}=0.5M$, $r_{h}<R_{s}<r_{isco}$ when
$\rho_{s}M^{2}<1.72272$ and $R_{s}\geq r_{isco}$ for $\rho_{s}M^{2}\geq
1.72272$. Similarly, $r_{h}<R_{s}<r_{isco}$ when $r_{s}<0.579691M$ and
$R_{s}\geq r_{isco}$ for $r_{s}\geq 0.579691M$ where core density is fixed at
$\rho_{s}M^{2}=1.0$.

Next, QNMs, with the help of the $6th$ order WKB method, are studied. Since we
have confined ourselves to the case of $n=0$, the $6th$ order WKB method
provides accurate results. We have considered scalar and electromagnetic
perturbations. The frequency of GWs emitted is found to be increasing with DM
parameters for both the perturbations. The imaginary part of QNMs is always
negative, indicating the stability of the BH-matter combination against
perturbation. Although GWs with higher multipole numbers always have higher
frequency irrespective of the type of perturbation, the situation is not
similar with regard to the decay rate. While for scalar perturbation, emitted
GWs decay faster for smaller multipole numbers, the situation is reversed in
the case of electromagnetic perturbation. This makes GWs with higher multipole
numbers originating from BHs perturbed due to scalar field and GWs with lower
multipole numbers originating from BHs perturbed due to electromagnetic field
move further and live longer. The impact of core radius on the decay rate is
always adverse. On the other hand, the decay rate initially increases with
core density reaching a maximum value and then starts decreasing. Since the
frequency and decay rate of GWs emitted due to scalar and electromagnetic
perturbations are significantly different, we can differentiate the two
perturbations based on QNM observation.

We then move on to study the impact of DM on the greybody factor and Hawking
spectrum for scalar and electromagnetic radiation. The greybody factor
provides information regarding the transmission probability of radiation to be
observed at spatial infinity. Since the greybody factor decreases with core
density and radius, the probability of radiation getting transmitted to an
asymptotic observer decreases. It will reduce the amount of thermal radiation
received. It is evident from Fig. [15, 16]. Here, we see a reduction in the
Hawking radiation received by an asymptotic observer due to an increase in
core radius and density for both types of perturbation. We apprehend the
impact of DM parameters on weak GL using the Gauss-Bonnet theorem. We have
obtained higher-order correction terms in the deflection angle where terms up
to the cubic order in $M$ are retained. The deflection angle gets attenuated
due to the presence of DM. The deflection angle initially increases with
impact parameter $b$ reaching a maximum value and then starts decreasing. Some
such maximum positions are $b=1.23095M$ for $\rho_{s}M^{2}=0.2,r_{s}=0.5M$,
$b=0.867132M$ for $\rho_{s}M^{2}=1.0$, $r_{s}=0.28M$. Interestingly, we find
values of impact parameters for which the deflection angle is zero, indicating
no deflection of light rays. Few such values are $b=0.92628M$ for
$\rho_{s}M^{2}=0.2$, $r_{s}=0.5M$ and $b=0.752447M$ for $\rho_{s}M^{2}=1.0$,
$r_{s}=0.3M$.

Finally, we have used dounds on deviation from Schwarzschild$\delta$ for
$M87^{*}$ and $SgrA^{*}$ to probe the viability of our model and put constrain
on DM parameters- core radius and density. The parameter $\delta$ for our BH
never reaches its upper bounds since the deviation parameter is always
negative, indicating a smaller BH shadow. The deviation parameter is a
decreasing function of core radius and density. We have displayed parameter
space $(\rho_{s}M^{2}-r_{s}/M)$ for $\delta$ with regard to $M87^{*}$ and
$SgrA^{*}$. It clearly shows that for finite parameter space, our model
satisfies experimental observations. We have also obtained $1\sigma$ and
$2\sigma$ upper bounds on core radius and core density for a fixed value of
core density and core radius, respectively. To be consistent with all three
observations, we must have $r_{s}\leq 0.517663M$ within $1\sigma$ and
$r_{s}\leq 0.580395M$ within $2\sigma$ for $\rho_{s}M^{2}=1.0$. Similarly,
upper bounds on the core density are $1.14017/M^{2}$ within $1\sigma$ and
$1.74513/M^{2}$ within $2\sigma$ for $r_{s}=0.5M$. These observations make
SDDM a feasible candidate for SMBH. We, in our next endeavor, intend to study
the rotating counterpart of SDDM and apprehend the DM effect when an
additional parameter, i.e., the BH spin comes into play.

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# On a General Discrete Boundary Value Problem

Vladimir Vasilyev  Chair of Applied Mathematics and Computer Modeling
Belgorod State National Research University
Pobedy street 85, Belgorod 308015, Russia<EMAIL_ADDRESS>,  Alexander
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 study a general discrete boundary value problem in Sobolev–Slobodetskii
spaces in a plane quadrant and reduce it to a system of integral equations. We
show a solvability of the system for a small size of discreteness starting
from a solvability of its continuous analogue.

###### Key words and phrases:

elliptic symbol, invertibility, digital pseudo-differential operator, discrete
equation, periodic wave factorization, system of integral equation

###### 2010 Mathematics Subject Classification:

Primary: 35S15; Secondary: 65T50

##  1\. Introduction

The theory of pseudo-differential operators [1, 2] and related equations and
boundary value problems [3] exists more than a half-century, and up to now it
takes attention of scholars. Also there are some discrete theories of boundary
value problems for partial differential equations [6, 7], but these studies
are not applicable for pseudo-differential equations. According to this fact
the first author has initiated a studying discrete theory of pseudo-
differential equations [12, 4, 13, 14] having in mind forthcoming studies
their approximation properties and applications to computational algorithms
[8, 11].

Since model equations in [3] were studied in a half-space, it is a canonical
domain for manifold with smooth boundary, next step was done with a cone, it
is a canonical domain for manifold with conical points at boundary [10]. At
this step one needs a special factorization for an elliptic symbol, it was
transferred to the discrete case [15]using analogies with Fourier series [5].

This paper is devoted to a model discrete pseudo-differential equation and
discrete boundary value problem in a quadrant in the plane and its solvability
starting from solvability their continuous analogues under small parameter of
discreteness.

##  2\. Digital pseudo-differential operators and discrete equations

Here we introduce some starting concepts and results which will help us moving
to statement of a general boundary value problem.

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

We use also notations
$\zeta^{2}=h^{-2}((e^{ih\cdot\xi_{1}}-1)^{2}+(e^{ih\cdot\xi_{2}}-1)^{2})$ and
$S(h\mathbb{Z}^{2})$ for the discrete analogue of the Schwartz space of
infinitely differentiable rapidly decreasing at infinity functions.

Definition 1. The space $H^{s}(h\mathbb{Z}^{2})$ consists of discrete
functions and it is a closure of the space $S(h\mathbb{Z}^{2})$ with respect
to the norm

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

where $\tilde{u}_{d}(\xi)$ denotes the discrete Fourier transform

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

Let $A_{d}(\xi)$ be a measurable periodic function defined in $\mathbb{R}^{2}$
with the basic cube of periods $\hbar\mathbb{T}^{2}$.

Definition 2. A digital pseudo-differential operator $A_{d}$ with the symbol
$A_{d}(\xi)$ in discrete quadrant $K_{d}$ is called the following operator

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

Here we will consider symbols satisfying the condition

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

with positive constants $c_{1},c_{2}$ non-depending on $h$. This class of
symbols satisfying this condition will be denoted by $E_{\alpha}$. The number
$\alpha\in\mathbb{R}$ is called an order of the digital pseudo-differential
operator $A_{d}$.

We study solvability of the discrete equation

(2) $(A_{d}u_{d})(\tilde{x})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{x}\in K_{d},$

in the space $H^{s}(K_{d})$, and for this purpose we need certain specific
domains of 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 $K$. We will work with holomorphic functions $f(x+i\tau)$ in
such domains ${\mathcal{T}}_{h}(K)$.

Definition 3. Periodic wave factorization of the 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 holomorphic
continuation into tube domains ${\mathcal{T}}_{h}(K),{\mathcal{T}}_{h}(-K)$
respectively satisfying the estimates

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

with positive constants $c_{1},c^{\prime}_{1},c_{2},c^{\prime}_{2}$ non-
depending on $h$;

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

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

Everywhere below we assume that we have this periodic wave factorization of
the symbol $A_{d}(\xi)$ with the index $\ae$.

Using methods developed in [12] we can prove the following result.

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

(3)
$\tilde{u}_{d}(\xi)=A^{-1}_{d,\neq}(\xi)\left(\sum\limits_{k=0}^{n-1}\left(\tilde{c}_{k}(\xi_{1})\zeta_{2}^{k}+\tilde{d}_{k}(\xi_{2})\zeta_{1}^{k}\right)\right),$

where $\tilde{c}_{k}(\xi_{1}),\tilde{d}_{k}(\xi_{2}),k=0,1,\cdots,n-1,$ are
arbitrary functions from–
$\widetilde{H}^{s_{k}}(h\mathbb{T}),s_{k}=s-\ae+k-1/2.$

The a priori estimate

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

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

##  3\. Discrete boundary value problem

###  3.1. Statement and solvability

Starting from Theorem 1 we introduce the following boundary conditions

(4) $\displaystyle(B_{d,j}u_{d})(\tilde{x}_{1},0)=b_{d,j}(\tilde{x}_{1}),$
$\displaystyle(G_{d,j}u_{d})(0,\tilde{x}_{2})=g_{d,j}(\tilde{x}_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ j=0,1.\cdots,n-1,$

where $B_{d,j},G_{d,j}$ are digital pseudo-differential operators of order
$\beta_{j},\gamma_{j}\in{\mathbb{R}}$ with symbols
$\widetilde{B}_{d,j}(\xi)\in E_{\beta_{j}},\widetilde{G}_{d,j}(\xi)\in
E_{\gamma_{j}}$

$\displaystyle(B_{d,j}u_{d})(\tilde{x})=\frac{1}{(2\pi)^{2}}\int\limits_{\hbar{\bf
T}^{2}}\sum\limits_{\tilde{y}\in h{\bf
Z}^{2}}e^{i\xi\cdot(\tilde{x}-\tilde{y})}\widetilde{B}_{d,j}(\xi)\tilde{u}_{d}(\xi)d\xi,$
$\displaystyle(G_{d,j}u_{d})(\tilde{x})=\frac{1}{(2\pi)^{2}}\int\limits_{\hbar{\bf
T}^{2}}\sum\limits_{\tilde{y}\in h{\bf
Z}^{2}}e^{i\xi\cdot(\tilde{x}-\tilde{y})}\widetilde{G}_{d,j}(\xi)\tilde{u}_{d}(\xi)d\xi.$

One can rewrite boundary conditions (4) in Fourier images

(5)
$\displaystyle\int\limits_{-\hbar\pi}^{\hbar\pi}\widetilde{B}_{d,j}(\xi_{1},\xi_{2})\tilde{u}_{d}(\xi_{1},\xi_{2})d\xi_{2}=\tilde{b}_{d,j}(\xi_{1}),$
$\displaystyle\int\limits_{-\hbar\pi}^{\hbar\pi}\widetilde{G}_{d,j}(\xi_{1},\xi_{2})\tilde{u}_{d}(\xi_{1},\xi_{2})d\xi_{1}=\tilde{g}_{d,j}(\xi_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ j=0,1.\cdots,n-1,$

so that according to properties of digital pseudo-differential operators and
trace properties we need to require $b_{d,j}(\tilde{x}_{1})\in
H^{s-\beta_{j}-1/2}(h{\mathbb{Z}}),g_{d,j}(\tilde{x}_{2})\in
H^{s-\gamma_{j}-1/2}(h{\mathbb{Z}})$.

Multiplying the equality (3) by $\widetilde{B}_{d,j}(\xi_{1},\xi_{2})$ and
$\widetilde{G}_{d,j}(\xi_{1},\xi_{2})$, integrating over
$[-\hbar\pi,\hbar\pi]$ on $\xi_{2}$ and $\xi_{1}$, taking into account the
conditions (5) we obtain the following $(2n\times 2n)$-system of linear
integral equations

(6)
$\displaystyle\sum\limits_{k=0}^{n-1}\left(r_{jk}(\xi_{1})\tilde{c}_{k}(\xi_{1})+\int\limits_{-\hbar\pi}^{\hbar\pi}l_{jk}(\xi_{1},\xi_{2})\tilde{d}_{k}(\xi_{2})d\xi_{2}\right)=\tilde{b}_{d,j}(\xi_{1})$
$\displaystyle\sum\limits_{k=0}^{n-1}\left(\int\limits_{-\hbar\pi}^{\hbar\pi}m_{jk}(\xi_{1},\xi_{2})\tilde{c}_{k}(\xi_{1})d\xi_{1}+p_{jk}(\xi_{2})\tilde{d}_{k}(\xi_{2})\right)=\tilde{g}_{d,j}(\xi_{2}),$
$\displaystyle j=0,1,\dots,n-1,$

with unknowns functions $\tilde{c}_{k},\tilde{d}_{k},k=0,1,\dots,n-1$. We have
used the following notations

$r_{jk}(\xi_{1})=\int\limits_{-\hbar\pi}^{\hbar\pi}\tilde{B}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)\zeta_{2}^{k}d\xi_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
p_{jk}(\xi_{2})=\int\limits_{-\hbar\pi}^{\hbar\pi}\tilde{G}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)\zeta_{1}^{k}d\xi_{1},$
$l_{jk}(\xi_{1},\xi_{2})=\tilde{B}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)\zeta_{1}^{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
m_{jk}(\xi_{1},\xi_{2})=\tilde{G}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)\zeta_{2}^{k},$

$j,k=0,1,\dots,n-1$.

Thus, we can formulate the following assertion.

Theorem 2. The boundary value problem (2),(4) is uniquely solvable in the
space $H^{s}(K_{d})$ with data $b_{d,j}\in
H^{s-\beta_{j}-1/2}(h\mathbb{Z}_{+},g_{d,j}\in
H^{s-\gamma_{j}-1/2}(h\mathbb{Z}_{+}$ if and only if the system (6) has the
unique solution
$\tilde{c}_{k},\tilde{d}_{k}\in\tilde{H}^{s_{k}}(\hbar\mathbb{T})$,
$j,k=0,1,\dots,n-1$.

###  3.2. Continuous case

Here we will describe continuous boundary value problem which is related to
considered discrete boundary value problem (2),(4).

Let $A$ be a pseudo-differential operator

$(Au)(x)=\int\limits_{\mathbb{R}^{2}}\int\limits_{\mathbb{R}^{2}}\tilde{A}(\xi)e^{i\xi(y-x)}u(y)dyd\xi$

with symbol $\tilde{A}(\xi)$ satisfying the condition

(7) $|\tilde{A}(\xi)|\sim(1+|\xi|)^{\alpha}$

and admitting the wave factorization with respect to $K$

$\tilde{A}(\xi)=A_{\neq}(\xi)\cdot A_{=}(\xi).$

with index $\ae$ such that $\ae-s=n+\delta,n\in\mathbb{N},|\delta|<1/2$.

Further, let $B_{j},G_{j},j=0,1,\dots,n-1$ be pseudo-differential operators
with symbols $\tilde{B}_{j}(\xi),\tilde{G}_{j}(\xi)$ satisfying the condition
(7) with $\beta_{j},\gamma_{j}$ instead of $\alpha$.

The following boundary value problem

(8) $\displaystyle(Au)(x)=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x\in K,$ $\displaystyle(B_{j}u)(x_{1},0)=b_{j}(x_{1}),$
$\displaystyle(G_{j}u)(0,x_{2})=g_{j}(x_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ j=0,1,\dots,n-1$

is a continuous analogue of the discrete boundary value problem (2),(4). It
was shown in [10] the problem (8) is equivalent to the following system of
integral equations

(9)
$\displaystyle\sum\limits_{k=0}^{n-1}\left(R_{jk}(\xi_{1})\tilde{C}_{k}(\xi_{1})+\int\limits_{-\infty}^{+\infty}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}\right)=\tilde{b}_{j}(\xi_{1})$
$\displaystyle\sum\limits_{k=0}^{n-1}\left(\int\limits_{-\infty}^{+\infty}M_{jk}(\xi_{1},\xi_{2})\tilde{C}_{k}(\xi_{1})d\xi_{1}+P_{jk}(\xi_{2})\tilde{D}_{k}(\xi_{2})\right)=\tilde{g}_{j}(\xi_{2}),$
$\displaystyle j=0,1,\dots,n-1,$

with unknowns functions $\tilde{C}_{k},\tilde{D}_{k},k=0,1,\dots,n-1$. The
following notations are used

$R_{jk}(\xi_{1})=\int\limits_{-\infty}^{+\infty}\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)(i\xi_{2})^{k}d\xi_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
P_{jk}(\xi_{2})=\int\limits_{-\infty}^{+\infty}\tilde{G}_{j}(\xi)A_{\neq}^{-1}(\xi)(i\xi_{1})^{k}d\xi_{1},$
$L_{jk}(\xi_{1},\xi_{2})=\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)(i\xi_{1})^{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
M_{jk}(\xi_{1},\xi_{2})=\tilde{G}_{j}(\xi)A_{\neq}^{-1}(\xi)(i\xi_{2})^{k},$

$j,k=0,1,\dots,n-1$. If we can solve the system (9) and find
$\tilde{C}_{k},\tilde{D}_{k},k=0,1,\dots,n-1$ the the solution of the boundary
value problem (9) can be constructed by the formula [10]

(10)
$\tilde{u}(\xi)=A^{-1}_{\neq}(\xi)\left(\sum\limits_{k=0}^{n-1}\left(\tilde{C}_{k}(\xi_{1})(i\xi_{2})^{k}+\tilde{D}_{k}(\xi_{2})(i\xi_{1})^{k}\right)\right),$

where $\tilde{C}_{k}(\xi_{1}),\tilde{D}_{k}(\xi_{2}),k=0,1,\cdots,n-1,$ are
arbitrary functions from
$\widetilde{H}^{s_{k}}(\mathbb{R}),s_{k}=s-\ae+k-1/2$.

Our next problems are the following. Given operator $A$ and boundary operators
$B_{j},G_{j}$ how to choice the digital operators $A_{d}$ and
$B_{d,j},G_{d,j}$ to obtain the implication: the unique solvability of the
system (9) gives the unique solvability of the system (6) for enough small
$h$. This question will be discussed in the next section.

##  4\. Comparison theorems

###  4.1. Projection method

Let us introduce the following space of vector-functions:

$\tilde{\bf H}^{\Lambda}(\mathbb{R})=\tilde{\bf
H}^{S}(\mathbb{R})\oplus\tilde{\bf H}^{S}(\mathbb{R}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{\bf
H}^{S}(\mathbb{R})=\oplus\sum\limits_{k=0}^{n-1}\tilde{H}^{s_{k}}(\mathbb{R}),$

Norms in these spaces will be defined in the following way. For
$f\in\tilde{\bf
H}^{S}(\mathbb{R}),f=(f_{0},\dots,f_{n-1}),f_{k}\in\tilde{H}^{s_{k}}(\mathbb{R}),g\in\tilde{\bf
H}^{S}(\mathbb{R}),g=(g_{0},\dots,g_{n-1}),g_{k}\in\tilde{H}^{s_{k}}(\mathbb{R})$
we put

$||f||_{S}=\sum\limits_{k=0}^{n-1}||f_{k}||_{s_{k}}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
||g||_{S}=\sum\limits_{k=0}^{n-1}||g_{k}||_{s_{k}},$

and if $F\in\tilde{\bf H}^{\Lambda}(\mathbb{R}),F=(f,g),f\in\tilde{\bf
H}^{S}(\mathbb{R}),g\in\tilde{\bf H}^{S}(\mathbb{R})$ we put

$||F||_{\Lambda}=||f||_{S}+||g||_{S}.$

Let us introduce the following notations. We denote the system (9) in the
following way

$\begin{pmatrix}R&L\\\ M&P\end{pmatrix}\begin{pmatrix}C\\\
D\end{pmatrix}=\begin{pmatrix}B\\\ G\end{pmatrix},$

where
$C=(\tilde{c}_{0},\dots,\tilde{c}_{n-1})^{T},D=(\tilde{d}_{0},\dots,\tilde{d}_{n-1})^{T},B=(\tilde{b}_{0},\dots,\tilde{b}_{n-1})^{T},G=(\tilde{g}_{0},\dots,\tilde{g}_{n-1})^{T}$;
operators $R,L,M,P$ acting in the space $\tilde{\bf H}^{S}(\mathbb{R})$ are
the following: $R$ is multiplier by the matrix-function
$(r_{jk})_{j,k=0}^{n-1}$, $P$ is multiplier by the matrix-function
$(p_{jk})_{j,k=0}^{n-1}$, $L$, $M$ are matrix integral operators with kernels
$L_{jk},M_{jk}$ respectively.

Further, we will denote $\Xi_{h}$ the restriction operator on the segment
$\hbar\mathbb{T}$ so that for $f\in\tilde{\bf
H}^{S}(\mathbb{R}),f=(f_{0},\dots,f_{n-1})$ the notation $\Xi_{h}f$ means the
following

$\Xi_{h}f=(\chi_{h}f_{0},\dots,\chi_{h}f_{n-1}),$

where $\chi_{h}$ is an indicator of $\hbar\mathbb{T}$.

We denote by $Q$ the operator

$Q=\begin{pmatrix}R&L\\\ M&P\end{pmatrix}$

Theorem 3. Let $s-\beta_{j}>1,s-\gamma_{j}>2,j=0,1,\dots,n-1$. We have the
estimate

$||\Xi_{h}Q-Q\Xi_{h}||_{\tilde{\bf
H}^{\Lambda}(\mathbb{R})\rightarrow\tilde{\bf
H}^{\Lambda}(\mathbb{R})}\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h^{\varepsilon},$

where

$\varepsilon=\min\limits_{0\leq j\leq n-1}\\{s-\beta_{j}-1,s-\gamma_{j}-1\\},$

const does not depend on $h,s_{k}=s-\ae+k-1/2,k=0,1,\dots,n-1$.

###### Proof.

Obviously, the matrices $R,P$ give vanishing result in the norm, and we need
to work with integral operators only. Let us consider the operator $L$, and
extract one its component $L_{jk}$,

$\int\limits_{-\infty}^{+\infty}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
L_{jk}(\xi_{1},\xi_{2})=\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)\xi_{1}^{k}.$

We have

$\chi_{h}(\xi_{1})\int\limits_{-\infty}^{+\infty}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}-\int\limits_{-\bar{\pi}}^{+\hbar\pi}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}$
$=\begin{cases}\left(\int\limits_{-\infty}^{-\hbar\pi}+\int\limits_{\hbar\pi}^{+\infty}\right)L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \xi_{1}\in\hbar\mathbb{T},\\\
-\int\limits_{-\hbar\pi}^{\hbar\pi}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\xi_{1}\notin\hbar\mathbb{T}.\end{cases}$

Let us consider the first case and estimate

$\left|\int\limits_{\hbar\pi}^{+\infty}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}\right|\leq\leavevmode\nobreak\
const\int\limits_{\hbar\pi}^{+\infty}(1+|\xi|)^{\beta_{j}-\ae}|\xi_{1}|^{k}|\tilde{D}_{k}(\xi_{2})|d\xi_{2}$
$\leq\leavevmode\nobreak\
const\int\limits_{\hbar\pi}^{+\infty}(1+|\xi|)^{\beta_{j}-s+1/2}|\tilde{D}_{k}(\xi_{2})|(1+|\xi_{2}|)^{s_{k}}d\xi_{2}$

(we have taken into account $s_{k}=s-\ae+k-1/2$ and now we apply the
Cauchy–Schwartz inequality)

$\leq\leavevmode\nobreak\
const(1+|\xi_{1}|+\hbar)^{\beta_{j}-s+1}||\tilde{D}_{k}||_{s_{k}}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h^{s-\beta_{j}-1}||D_{k}||_{s_{k}}.$

Squaring the latter inequality, multiplying by $(1+|\xi_{|})^{2s_{k}}$ and
integrating over $\hbar\mathbb{T}$ we obtain

$\int\limits_{-\hbar\pi}^{\hbar\pi}(1+|\xi_{|})^{2s_{k}}\left|\int\limits_{\hbar\pi}^{+\infty}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}\right|^{2}d\xi_{1}$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h^{2(s-\beta_{j}-1)}||D_{k}||^{2}_{s_{k}}\int\limits_{0}^{+\infty}(1+|\xi_{|})^{2s_{k}}d\xi_{1}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h^{2(s-\beta_{j}-1)}||D_{k}||^{2}_{s_{k}}$

For the second case ($|\xi_{1}|>\hbar\pi$) we obtain

$\left|\int\limits_{-\hbar\pi}^{+\hbar\pi}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}\right|\leq\leavevmode\nobreak\
const\int\limits_{-\hbar\pi}^{+\hbar\pi}(1+|\xi|)^{\beta_{j}-\ae}|\xi_{1}|^{k}|\tilde{D}_{k}(\xi_{2})|d\xi_{2}$
$\leq\leavevmode\nobreak\
const\int\limits_{-\hbar\pi}^{+\hbar\pi}(1+|\xi|)^{\beta_{j}-\ae}|\xi_{1}|^{n-1}(1+|\xi_{2}|)^{-s_{k}}|\tilde{D}_{k}(\xi_{2})|(1+|\xi_{2}|)^{s_{k}}d\xi_{2}$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
|\xi_{1}|^{n-1}(1+|\xi_{1}|)^{-s_{k}}\int\limits_{-\hbar\pi}^{+\hbar\pi}(1+|\xi|)^{\beta_{j}-\ae}|\tilde{D}_{k}(\xi_{2})|(1+|\xi_{2}|)^{s_{k}}d\xi_{2}$

(we apply the Cauchy–Schwartz inequality in the integral)

$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
(1+|\xi_{1}|)^{n-s_{k}-1}(1+|\xi_{1}|)^{\beta_{j}-\ae+1/2k}||D_{k}||_{s_{k}}$

Squaring the latter inequality, multiplying by $(1+|\xi_{|})^{2s_{k}}$ and
integrating over $\mathbb{R}\setminus\hbar\mathbb{T}$ we obtain

$\left(\int\limits_{-\infty}^{-\hbar\pi}+\int\limits_{\hbar\pi}^{+\infty}\right)(1+|\xi_{|})^{2s_{k}}\left|\int\limits_{\hbar\pi}^{+\infty}L_{jk}(\xi_{1},\xi_{2})\tilde{D}_{k}(\xi_{2})d\xi_{2}\right|^{2}d\xi_{1}$
$\leq\leavevmode\nobreak\
const||D_{k}||^{2}_{s_{k}}\int\limits_{\hbar\pi}^{+\infty}(1+\xi_{1})^{2n-2+2\beta_{j}+1-2\ae}d\xi_{1}\leq\leavevmode\nobreak\
const||D_{k}||^{2}_{s_{k}}h^{2s-2\beta_{j}+2\delta},$

since
$2n+2\beta_{j}-2\ae=2n+2\beta_{j}-2(s+n+\delta)=2\beta_{j}-2s-2\delta<0$.

Thus, we have proved that

$||\chi_{h}L_{jk}-L_{jk}\chi_{h}||_{H^{s_{k}}(\mathbb{R})\rightarrow
H^{s_{k}}(\mathbb{R})}\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h^{s-\beta_{j}-1},$

since $s-\beta_{j}-1<s-\beta_{j}+\delta$.

Almost the same inequality can be obtained for $M_{jk}$

$||\chi_{h}M_{jk}-M_{jk}\chi_{h}||_{H^{s_{k}}(\mathbb{R})\rightarrow
H^{s_{k}}(\mathbb{R})}\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h^{s-\gamma_{j}-1},$

These estimates complete the proof. ∎

Corollary 1. Under conditions of Theorem 3 the invertibility of the operator
$Q$ in the space $\tilde{\bf H}^{\Lambda}(\mathbb{R})$ implies the
invertibility of the operator $\Xi_{h}Q\Xi_{h}$ in the space $\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})$ for enough small $h$.

###### Proof.

We apply the results of the paper [9] which imply the following. If

$||\Xi_{h}Q-Q\Xi_{h}||_{\tilde{\bf
H}^{\Lambda}(\mathbb{R})\rightarrow\tilde{\bf
H}^{\Lambda}(\mathbb{R})}\rightarrow 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ h\to 0$

then the equation in the space $\tilde{\bf H}^{\Lambda}(\mathbb{R})$

(11) $Qu=v$

admits applying so called projection method. In other words it means that
unique solvability of the equation (11) in the space $\tilde{\bf
H}^{\Lambda}(\mathbb{R})$ implies unique solvability of the equation

(12) $\Xi_{h}Q\Xi_{h}u=\Xi_{h}v$

in the space $\tilde{\bf H}^{\Lambda}(\hbar\mathbb{T})$ for enough small $h$.
Moreover, if there is bounded operator $Q^{-1}$ in the space $\tilde{\bf
H}^{\Lambda}(\mathbb{R})$ then there is bounded operator
$(\Xi_{h}Q\Xi_{h})^{-1}$ for enough small $h$ and

$||(\Xi_{h}Q\Xi_{h})^{-1}||_{\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})\rightarrow\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})}\leq\leavevmode\nobreak\ const,$

where const does not depend on $h$.

Indeed, a reader can easily verify that

$||(\Xi_{h}Q\Xi_{h})^{-1}-\Xi_{h}Q^{-1}\Xi_{h}||_{\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})\rightarrow\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})}\rightarrow 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ h\to 0.$

∎

###  4.2. Discrete and continuous

To compare discrete and continuous operators we need a special choice of
discrete operators. We will do it in the following way.

The symbol $A_{d}(\xi)$ of the discrete operator $A_{d}$ will be constructed
as follows. Given wave factorization for $\tilde{A}(\xi)$

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

we take restrictions of factors $A_{\neq}(\xi),A_{=}(\xi)$ on
$\hbar\mathbb{T}^{2}$ and periodically continue them into $\mathbb{R}^{2}$. We
denote these elements by $A_{d,\neq}(\xi),A_{d,=}(\xi)$ and construct the
periodic symbol $A_{d}(\xi)$ which admits periodic wave factorization with
respect to $K$

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

with the same index $\ae$. We construct discrete pseudo-differential operators
$B_{d,j},G_{d,j}$ taking their symbol as restrictions of symbols
$\tilde{B}_{j}(\xi),\tilde{G}_{j}(\xi)$ on $\hbar\mathbb{T}^{2}$ with
periodical continuations into $\mathbb{R}^{2},,j=0,1,\dots,n-1$. The discrete
boundary functions $b_{d,j}$, $g_{d,j}$ are constructed in the same way. Thus,
we have the corresponding discrete boundary value problem (2),(4).

Lemma 1. The estimate

$|(i\xi_{m})^{k}-\zeta_{m}^{k}|\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h|\xi_{m}|^{k+1}$

holds for $\xi_{m}\in\hbar\mathbb{T},m=1,2$ const does not depend on $h$.

###### Proof.

First, we estimate

$|\zeta_{1}|=\left|\sum\limits_{\nu=1}^{\infty}\frac{(i\xi_{1})^{\nu+1}h^{\nu}}{(\nu+1)!}\right|=|\xi_{1}|\left|\sum\limits_{\nu=0}^{\infty}\frac{(i\xi_{1})^{\nu}h^{m}}{(\nu+1)!}\right|\leq|\xi_{1}|\sum\limits_{\nu=0}^{\infty}\frac{(|\xi_{1}|h)^{\nu}}{\nu!}$
$=|\xi_{1}|e^{|\xi_{1}|h}\leq|\xi_{1}|e^{\pi}.$

Second,

$|\zeta_{1}-i\xi_{1}|=|\hbar(e^{i\xi_{1}h}-1)-i\xi_{1}|=\left|\sum\limits_{\nu=1}^{\infty}\frac{(i\xi_{1})^{\nu+1}h^{\nu}}{(\nu+1)!}\right|$
$\leq|\xi_{1}|^{2}h\sum\limits_{\nu=0}^{\infty}\frac{|\xi_{1}|^{\nu}h^{\nu}}{\nu!}=|\xi_{1}|^{2}he^{|\xi_{1}|h}\leq|\xi_{1}|^{2}he^{\pi}.$

We have

$\zeta_{1}^{k}-(i\xi_{1})^{k}=(\zeta_{1}-i\xi_{1})\left(\sum\limits_{\nu=0}^{k-1}\zeta_{1}^{\nu}(i\xi_{1})^{k-1-\nu}\right),$

and thus

$|\zeta_{1}^{k}-(i\xi_{1})^{k}|\leq|\zeta_{1}-i\xi_{1}|\sum\limits_{\nu=0}^{k-1}|\zeta_{1}|^{\nu}|\xi_{1}|^{k-1-\nu}$

Applying above estimates we obtain required inequality.

∎

Lemma 2. Let $s-\beta_{j}>2,s-\gamma_{j}>2,j=0,1,\dots,n-1$.The following
estimates

$\displaystyle|L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2})|\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi|)^{\beta_{j}-\ae+k+1},$
$\displaystyle|M_{jk}(\xi_{1},\xi_{2})-m_{jk}(\xi_{1},\xi_{2})|\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi|)^{\gamma_{j}-\ae+k+1},$
$\displaystyle|R_{jk}(\xi_{1})-r_{jk}(\xi_{1})|\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi_{1}|)^{\beta_{j}-\ae+k+2},$
$\displaystyle|P_{jk}(\xi_{2})-p_{jk}(\xi_{2})|\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi_{1}|)^{\gamma_{j}-\ae+k+2}$

hold for $\xi_{1},\xi_{2}\in\hbar\mathbb{T}$.

###### Proof.

According to above conventions we have for $\xi\in\hbar\mathbb{T}^{2}$ and
using Lemma 1 we have

$|L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2})|=|\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)-\tilde{B}_{d,j}\xi)A_{d,\neq}^{-1}(\xi)||B_{j}(\xi)||\xi_{1}^{k}-\zeta_{1}^{k}|$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
(1+|\xi|)^{\beta_{j}-\ae}h|\xi_{1}|^{k+1}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi|)^{\beta_{j}-\ae+k+1}.$

Further,

$|R_{jk}(\xi_{1})-r_{jk}(\xi_{1})|=\left|\int\limits_{-\infty}^{+\infty}\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)\xi_{2}^{k}d\xi_{2}-\int\limits_{-\hbar\pi}^{+\hbar\pi}\tilde{B}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)\zeta_{2}^{k}d\xi_{2}\right|$
$\leq\int\limits_{-\hbar\pi}^{+\hbar\pi}|\tilde{B}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)||\xi_{2}^{k}-\zeta_{2}^{k}|d\xi_{2}+\left(\int\limits_{-\infty}^{-\hbar\pi}+\int\limits_{\hbar\pi}^{+\infty}\right)|\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)\xi_{2}^{k}|d\xi_{2}.$

For the first integral we have

$\int\limits_{-\hbar\pi}^{+\hbar\pi}|\tilde{B}_{d,j}(\xi)A_{d,\neq}^{-1}(\xi)||\xi_{2}^{k}-\zeta_{2}^{k}|d\xi_{2}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\
h\int\limits_{-\hbar\pi}^{+\hbar\pi}(1+|\xi|)^{\beta_{j}-\ae+k+1}d\xi_{2}$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h(1+|\xi_{1}|)^{\beta_{j}-\ae+k+2},$

since $\beta_{j}-\ae+k+2<0,s-\beta_{j}>2.$

The second summand

$\left|\left(\int\limits_{-\infty}^{-\hbar\pi}+\int\limits_{\hbar\pi}^{+\infty}\right)|\tilde{B}_{j}(\xi)A_{\neq}^{-1}(\xi)\xi_{2}^{k}|d\xi_{2}\right|\leq\leavevmode\nobreak\
const\int\limits_{\hbar\pi}^{+\infty}(1+|\xi_{1}|+|\xi_{2}|)^{\beta_{j}-\ae+k}d\xi_{2}$
$\leq\leavevmode\nobreak\
const(1+|\xi_{1}|+\hbar)^{\beta_{j}-\ae+k+1}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi_{1}|)^{\beta_{j}-\ae+k+2}.$

The same estimates are valid for $M_{jk}-m_{jk}$ and $P_{jk}$ with
$\gamma_{j}$ instead of $\beta_{j}$. ∎

We introduce similar notations for the system (6) so that this system takes
the form

$\begin{pmatrix}r&l\\\ m&p\end{pmatrix}\begin{pmatrix}c\\\
d\end{pmatrix}=\begin{pmatrix}B_{d}\\\ G_{d}\end{pmatrix},$

where

$q=\begin{pmatrix}r&l\\\ m&p\end{pmatrix}$

is linear bounded operator acting in the space $\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})$.

Theorem 4. Let $s-\beta_{j}>3,s-\gamma_{j}>3,j=0,1,\dots,n-1$. A comparison
between operators $Q$ and $q$ is given by the estimate

$||\Xi_{h}Q\Xi_{h}-q||_{\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})\rightarrow\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})}\leq\leavevmode\nobreak\ const!h,$

where const does not depend on $h$.

###### Proof.

We need to estimate $H^{s_{k}}(\hbar\mathbb{T}$-norms of the following
elements

$(R_{jk}(\xi_{1})-r_{jk}(\xi_{1}))f(\xi_{1}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
(R_{jk}(\xi_{2})-p_{jk}(\xi_{2}))f(\xi_{2}),$
$\int\limits_{-\hbar\pi}^{\hbar\pi}(L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2}))f(\xi_{2})d\xi_{2},$
$\int\limits_{-\hbar\pi}^{\hbar\pi}(M_{jk}(\xi_{1},\xi_{2})-m_{jk}(\xi_{1},\xi_{2}))f(\xi_{1}d\xi_{1}.$

We have according to Lemma 2

$|(R_{jk}(\xi_{1})-r_{jk}(\xi_{1}))f(\xi_{1})|\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h(1+|\xi_{1}|)^{\beta_{j}-\ae+k+2}|f(\xi_{1})|.$

Multiplying the latter inequality by $(1+|\xi_{1}|)^{s_{k}}$, squaring,
integrating over $\hbar\mathbb{T}$ and applying the Cauchy–Schwartz inequality
we obtain

$\int\limits_{-\hbar\pi}^{+\hbar\pi}(1+|\xi_{1}|)^{2s_{k}}|R_{jk}(\xi_{1})-r_{jk}(\xi_{1})|^{2}|f(\xi_{1})|^{2}d\xi_{1}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h^{2}||f||^{2}_{s_{k}}$

since $\beta_{j}-\ae+k+2<0$.

Let us consider

$\int\limits_{-\hbar\pi}^{\hbar\pi}(L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2}))f(\xi_{2})d\xi_{2}.$

Using Lemma 2 we have

$\left|\int\limits_{-\hbar\pi}^{\hbar\pi}(L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2}))f(\xi_{2})d\xi_{2}\right|$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h\int\limits_{-\hbar\pi}^{\hbar\pi}(1+|\xi|)^{\beta_{j}-\ae+k+1}|f(\xi_{2})|d\xi_{2}$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h\int\limits_{-\hbar\pi}^{\hbar\pi}(1+|\xi|)^{\beta_{j}-s+5/2}|f(\xi_{2})|(1+|\xi_{2}|)^{s_{k}}d\xi_{2}$

since $\beta_{j}-\ae+k+2-s_{k}=-\beta_{j}-s+5/2$. Now applying Cauchy–Schwartz
inequality we find

$\left|\int\limits_{-\hbar\pi}^{\hbar\pi}(L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2}))f(\xi_{2})d\xi_{2}\right|$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h||f||_{s_{k}}\left(\int\limits_{-\hbar\pi}^{\hbar\pi}(1+|\xi_{1}|+|\xi_{2}|)^{2\beta_{j}-2s+5}d\xi_{2}\right)^{1/2}$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h||f||_{s_{k}}(1+|\xi_{1}|)^{\beta_{j}-s+3}$

according to the condition $s-\beta_{j}>3$. Squaring, multiplying by
$(1+|\xi_{1}|)^{2s_{k}}$ and integrating over $\hbar\mathbb{T}$ we conclude

$\int\limits_{-\hbar\pi}^{\hbar\pi}(1+|\xi_{1}|)^{2s_{k}}\left|\int\limits_{-\hbar\pi}^{\hbar\pi}(L_{jk}(\xi_{1},\xi_{2})-l_{jk}(\xi_{1},\xi_{2}))f(\xi_{2})d\xi_{2}\right|^{2}d\xi_{1}$
$\leq\leavevmode\nobreak\ const\leavevmode\nobreak\
h^{2}||f||^{2}_{s_{k}}\int\limits_{-\hbar\pi}^{\hbar\pi}(1+|\xi_{1}|)^{2(\beta_{j}-s+3+s_{k})}d\xi_{1}\leq\leavevmode\nobreak\
const\leavevmode\nobreak\ h^{2}||f||^{2}_{s_{k}}$

since $2(\beta_{j}-s+3+s_{k})<-1$. Indeed,
$2(\beta_{j}-s+3+s_{k})=2(\beta_{j}-s+3+s-\ae+k-1/2)=2(\beta_{j}-s-\delta).$
Obviously, the inequality $2(\beta_{j}-s-\delta)<-1$ is equivalent to
$s-\beta_{j}>-1-\delta$.

∎

Corollary 2. Under conditions of Theorem 4 the invertibility of the operator
$Q$ in the space $\tilde{\bf H}^{\Lambda}(\mathbb{R})$ implies the
invertibility of the operator $q$ in the space $\tilde{\bf
H}^{\Lambda}(\hbar\mathbb{T})$ for enough small $h$.

###### Proof.

Indeed, we have the invertibility of $\Xi_{h}Q\Xi_{h}$ by Corollary 1 and the
invertibility of $q$ is obtained by Theorem 4.

∎

## Conclusion

Main goal of the paper was to prove unique solvability of discrete boundary
value problem having in main unique solvability of its continuous analogue. It
was done by a special choice od discrete operator and discrete boundary
conditions. We hope that estimates of Theorem 3,4 will help us to obtain some
estimates for discrete and continuous solutions.

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