# Telescope Array Bursts, Radio Pulses and Axion Quark Nuggets

Xunyu Liang and Ariel Zhitnitsky  Department of Physics and Astronomy,
University of British Columbia, Vancouver, V6T 1Z1, BC, Canada

###### Abstract

Telescope Array (TA) experiment has recorded [1, 2] several short time bursts
of air shower like events. These bursts are very distinct from conventional
single showers, and are found to be strongly correlated with lightnings. In
our previous work [3] we proposed that these bursts represent the direct
manifestation of the dark matter annihilation events within the so-called
axion quark nugget (AQN) model. In the present work we suggest to test this
proposal to search for the radio signals in frequency band $\nu\in(0.5-200)$
MHz which must be synchronized with the TA bursts.

##  I Introduction

This work is tightly linked to our recent proposal [3] interpreting the
mysterious bursts observed by Telescope Array (TA) experiment [1, 2] in terms
of the AQN annihilation events under the thunderstorm. These events are very
unusual and cannot be interpreted in terms of conventional single showers as
reviewed below. In the present work we shall argue that the proposed mechanism
[3] in terms of the AQN annihilation events inevitably predict the radio wave
pulses which must be synchronized with TA bursts. Based on this prediction we
suggest to test our proposal by searching for the radio signals in frequency
band $\nu\in(0.5-200)$ MHz which must be synchronized with the TA bursts.

Such test would unambiguously support or refute the proposal because the main
arguments of the present work based on synchronization which represents pure
geometrical property of the system. This is because the radio pulse and the TA
burst are originated from the same location, emitted at the same instant, and
propagate with the speed of light. This feature of synchronization is not
sensitive to many assumptions and uncertainties which were inevitably present
in estimates [3] interpreting the TA bursts as the AQN annihilation events.

Our presentation is organized as follows. In next section II we overview the
TA bursts observations [1, 2] and the basic ideas of the proposal [3] with the
resolution of these puzzling bursts in terms of the AQN annihilation events.
In sections III and IV we argue that the emission of the radio pulse is
inevitable consequence of the proposed mechanism, and we estimate its
numerical characteristics. In sections V and VI we estimate the radio pulse
due to the Earth’s magnetic field. Finally, section VII is our conclusion
where we suggest to search for a synchronized radio pulse with TA burst.

##  II TA bursts as the AQN annihilation events under thunderclouds

The main goal of this section is to overview the basic idea of the proposal
[3]. We organize this section as follows: first, in subsection II.1 we
overview the TA bursts observations [1, 2] emphasizing on the very distinct
features of the events. In subsection II.3 we explain the resolution of these
puzzling bursts in terms of the AQN annihilation events. In subsection II.2 we
highlight the very few key elements of the AQN construction relevant for this
work by referring to original paper [3] for all less important elements of the
proposal. In other words, in the subsection II.2 we shall only mention the
crucial ingredients related to the physics of emission of the radio pulses,
which represent the topic of the present work. Such organization of this work
helps us to avoid an unavoidable repetition of the background material which
can be found in [3].

###  II.1 Mysterious bursts observed by [1, 2]

The unusual features of the bursts recorded by [1, 2] can be briefly
formulated as follows:

1. All reconstructed air shower fronts for the burst events are much more curved than usual cosmic ray (CR) air showers. This feature is expressed in terms of the time- spreading versus spatial-spreading of the particles in the bursts. The corresponding “curvature” is much more pronounced for the bursts in comparison with conventional CR air showers, see see Fig. 3 and Fig. 4 in [1]. Furthermore, the bursts events do not have sharp edges in waveforms in comparison with conventional CR events;

2. The events are temporally clustered within 1 ms, which would be a highly unlikely occurrence for three consecutive conventional HECR hits in the same area within a radius of approximately 1 km. The total 10 burst events have been observed during 5 years of observations. The authors of ref. [1] estimate the expectation of chance coincidence is less than $10^{-4}$ for five years of observations. If one tries to fit the observed bursts with conventional code, the energy for HECR events should be in $10^{13}$ eV energy range, while the observed bursts correspond to $(10^{18}-10^{19})$ eV energy range as estimated by signal amplitude and distribution. Therefore, the estimated energy from individual events within the bursts is five to six orders of magnitude higher than the energy estimated by event rate.

3. Most of the observed bursts are “synchronized” (difference between burst and lightning is less than 1 ms) or “related” (difference between burst and lightning is less than 200 ms) with the lightning events. Some of them bursts are not related to any lightnings. However, all 10 recorded bursts occur under thunderstorm.

It is very hard to understand all these features in terms of conventional CR
physics as the bursts cannot be reconciled with conventional CR physics. At
the same time all unusual features (including the energetics, the flux
estimates, the time and spatial spreading of each events within the bursts) as
listed in items 1-3  above can be naturally explained within the AQN
framework. Before we proceed with corresponding explanation we have to briefly
overview the basic features of the AQN model to explain the nature and the
source of the TA bursts. This is the topic of the next subsection II.2.

###  II.2 The AQNs under the thunderclouds

The axion quark nugget (AQN) dark matter model [4] was invented long ago with
a single motivation to naturally explain the observed similarity between the
dark matter and the visible densities in the Universe, i.e. $\Omega_{\rm
DM}\sim\Omega_{\rm visible}$ without any fitting parameters. The AQN
construction in many respects is similar to the Witten’s quark nuggets, see
[5, 6, 7], and review [8]. This type of DM is “cosmologically dark” not
because of the weakness of the AQN interactions, but due to their small cross-
section-to-mass ratio, which scales down many observable consequences of an
otherwise strongly-interacting DM candidate.

There are two additional elements in the AQN model compared to the original
models [5, 6, 7, 8]. First new element is the presence of the axion domain
walls which are copiously produced during the QCD transition111The axion field
had been introduced into the theory to resolve the so-called the strong ${\cal
CP}$ problem which is related to the fundamental initial parameter
$\theta_{0}\neq 0$. This source of ${\cal CP}$ violation is no longer
available at the present time as a result of the axion’s dynamics in early
Universe. One should mention that the axion remains the most compelling
resolution of the strong ${\cal CP}$ problem, see original papers on the axion
[9, 10, 11, 12, 13, 14, 15], and recent reviews [16, 17, 18, 19, 20, 21, 22,
23, 24].. This domain wall plays a dual role: first it serves as an additional
stabilization factor for the nuggets, which helps to alleviate a number of
problems with the original nugget construction [5, 6, 7, 8]. Secondly, the
same axion field $\theta(x)$ generates the strong and coherent $\cal{CP}$
violation in the entire visible Universe.

Another feature of the AQN model which plays absolutely crucial role for the
present work is that nuggets can be made of matter as well as antimatter
during the QCD transition. Precisely the coherence of the $\cal{CP}$ violating
field on large scale mentioned above provides a preferential production of one
species of nuggets made of antimatter over another species made of matter. The
preference is determined by the initial sign of the $\theta$ field when the
formation of the AQN nuggets starts. The direct consequence of this feature
along with coherent $\cal{CP}$ violation in entire Universe is that the DM
density, $\Omega_{\rm DM}$, and the visible density, $\Omega_{\rm visible}$,
will automatically assume the same order of magnitude densities $\Omega_{\rm
DM}\sim\Omega_{\rm visible}$ without any fine tuning. We refer to the original
papers [25, 26, 27, 28] devoted to the specific questions related to the
nugget’s formation, generation of the baryon asymmetry, and survival pattern
of the nuggets during the evolution in early Universe with its unfriendly
environment, see also independent analysis [29] confirming these basic
results.

Precisely the AQNs made of antimatter are capable to release a significant
amount of energy when they enter the Earth’s atmosphere and annihilation
processes start to occur between antimatter hidden in form of the AQNs and the
atmospheric material. The emission of positrons from the nuggets made of
antimatter during the thunderstorms plays the crucial role in the proposal
[3]. This is because the thunderclouds are characterized by large preexisting
electric field $\cal{E}$ which serves as a trigger and accelerator of the
liberated positrons.

We refer to review papers [30, 31] devoted to study of the lightning where
pre-existing electric field $\cal{E}$ plays a crucial role in dynamics of the
lightning processes. While there is a consensus on typical parameters of the
electric field which are important for the lightning dynamics, the physics of
the of the initial moment of lightning remains a matter of debate, and refs
[30, 31] represent different views on this matter, see also references [32,
33, 34] where some specific elements of existing disagreement have been
explicitly formulated and debated.

For our purposes, however, the disputable elements do not play any role in our
studies. Important elements for the present work, which are not controversial,
are the temporal and spatial characteristics of the electric field $\cal{E}$
and their values under thunderclouds. These parameters are well established
and are not part of the debates, and we quote these parameters below. The
electric field $\cal{E}$ in the thunderclouds is characterized by the
following parameters [30, 31]

$\displaystyle{\cal{E}}\simeq{\rm\frac{kV}{cm}},~{}~{}~{}~{}l_{a}\simeq
100~{}{\rm m},~{}~{}~{}~{}\tau_{\cal{E}}\simeq\frac{l_{a}}{c}\simeq
0.3{\rm\,\mu s}$ (1)

where $l_{a}$ is the so-called avalanche length.

If the AQN enters the electric field (1) along its path it may liberate the
positrons from AQNs as the additional energy $\Delta E$ assumes the same order
of magnitude as the binding energy $E_{\rm bound}\sim\rm keV$ of the
positrons, i.e.

$\displaystyle\Delta E\simeq[e{\cal{E}}\cdot R_{\rm cap}]\sim
2\rm~{}keV\gtrsim E_{\rm bound}.$ (2)

In this formula $R_{\rm cap}$ is a typical distance (from the nugget’s core)
where positrons reside. These excited positrons localized sufficiently far
away from the nugget’s core due to the large rate of elastic collisions with
air molecules which kick them off from the unperturbed ground state positions.
The $R_{\rm cap}$ which enters (2) can be estimated in terms of the ionization
charge $Q$ and internal temperature of the nuggets $T\simeq 10\rm~{}keV$,

$\displaystyle R_{\rm cap}\simeq\frac{\alpha Q}{T}\sim 2~{}{\rm cm},$ (3)

see [3] with proper estimates. This additional energy (2) of order of several
keV could liberate the weakly coupled positrons from the nuggets. These
liberated positrons find themselves in the background of strong electric field
characterized by typical length scale $l_{a}\simeq 100$ m according to (1).
This pre-existing field will accelerate them to MeV energies on length scale
(1). Indeed,

$\displaystyle E_{\rm exit}\simeq[e{\cal{E}}\cdot l_{a}]\sim 10\rm~{}MeV$ (4)

after the positrons exit the region of strongly fluctuating electric field
which is known to be present under thunderclouds.

###  II.3 “Mysterious bursts” proposal [3] confronts the observations [1, 2]

The goal of this subsection is to explain the puzzling bursts within AQN
framework as proposed in [3]. It will be briefly explained how the unusual
features 1-3 listed above in subsection II.1 can be naturally understood
within the AQN framework.

####  II.3.1 “curvature” puzzle

We start our discussions with item 1 from the list. In the AQN framework this
“curved” feature can be easily understood by noticing that essential parameter
in this proposal is the initial spread of the particles determined by angle
$\Delta\alpha\simeq\left({v_{\perp}}/{c}\right)\in(0-0.1)$. As a result of
this spread the typical time scale is determined by $\Delta\alpha$ and it
could naturally assume relatively large values up to $(7-8){\rm\,\mu s}$ as
estimated in (6) and consistent with observations, see Fig. 3 and Fig. 4 in
[1].

Indeed, all released positrons will obviously move in the same direction which
is entirely determined by the direction of the electric field ${\cal E}$ at
the moment of exit. Small angle
$\Delta\alpha\simeq\left({v_{\perp}}/{c}\right)\in(0-0.1)$ in the velocity
distribution at the exit point is determined by initial energy (2) which
itself is determined by transverse component perpendicular to electric field,
$v_{\perp}\simeq\sqrt{2\Delta E/m}\simeq 0.1c$.

Therefore, after travelling the distance $r$ the spatially spread range
$\Delta s$ is estimated as

$\displaystyle\Delta s\simeq
r\left(\frac{\Delta\alpha}{\cos\alpha}\right)\simeq\frac{1~{}{\rm
km}}{\cos\alpha}\left(\frac{r}{10\rm~{}km}\right)\left(\frac{\Delta\alpha}{0.1}\right),$
(5)

see Fig. 1 for precise definitions of the parameters.

Figure 1: The positrons move along the cone with angle $\Delta\alpha$ and
inclination angle $\alpha$ with respect to the vertical direction. The angular
spread $\Delta\alpha\ll\alpha$ is assumed to be small. The spatial spread on
the surface is determined by $\Delta s$, while the additional travelling path
is determined by $\Delta r$, see estimates in the text. The altitude is
assumed to be within conventional range $h\simeq(4-12)$ km. Instant direction
of the electric field $\mathbf{\cal{E}}$ at the moment of exit of the
positrons is also shown.

At the same time, the parameter $\Delta t$ can be estimated as follows:

$\displaystyle\Delta t\simeq\frac{\Delta r}{c}\simeq 3{\rm\,\mu
s}\cdot(\tan\alpha)\cdot\left(\frac{r}{10\rm~{}km}\right)\left(\frac{\Delta\alpha}{0.1}\right)$
(6)

where $\Delta r\simeq r\tan\alpha\Delta\alpha$ see Fig. 1. One can infer from
(6) that time spread $(2\Delta t)$ indeed could be relatively large, up to
$(7-8){\rm\,\mu s}$, in contrast with conventional CR air shower when this
timing spread is always below $2{\rm\,\mu s}$, see Fig. 5 in [1]. We use
$(2\Delta t)$ in our estimates with extra factor 2 as the angle
$\Delta\alpha=(v_{\perp}/c)$ determining the particle distribution, could
assume the positive or negative value, depending on sign of $v_{\perp}$ with
respect to instant direction of the electric field $\mathbf{{\cal{E}}}$ as
shown on Fig. 1.

Another distinct characteristic of the the AQN framework is much stronger
localization of the particles around the axis when it is typically below 2 km,
see Fig. 3 and Fig. 4 in [1], in contrast with conventional CR air shower when
it normally assumes much greater values around 3.5 km, Fig. 5 in [1]. This
feature also finds its natural explanation within AQN framework. Indeed, the
spread of the particles on the plane determined by $\Delta s$, while the
timing is characterized by $\Delta t\simeq{\Delta r}/{c}$ estimated above (6).
These parameters are linearly proportional to each other and assume proper
values consistent with observations. Indeed,

$\displaystyle\Delta r\simeq\Delta
s\sin\alpha,~{}~{}~{}\Delta\alpha\simeq\left(\frac{v_{\perp}}{c}\right)\in(0-0.1)$
(7)

such that $2\Delta t$ may vary between $(0.5-8){\rm\,\mu s}$ when $2\Delta s$
changes between $(0.5-2)$ km with approximately linear slope determined by
electric field direction $\sin\alpha$. This behaviour222One should comment
here that the actual trajectories of the positrons are not the straight lines
due to the Earth’s magnetic field ${\cal{B}}\sim 0.5$ Gauss as we discuss in
section V. However, the main point here is that the emitted positrons are
characterized by the same velocity and localization at the instant of exit.
Therefore the presence of the magnetic field ${\cal{B}}$ does not spoil our
estimates for the temporal and spatial spreads given by (7) which must hold
irrespectively to the presence of ${\cal{B}}$. is consistent with observed
events presented on Fig. 3 and Fig. 4 in [1].

This behaviour in terms of the timing and spatial spreads for the bursts
should be contrasted with conventional CR distribution when the timing spread
is much shorter and always below $2{\rm\,\mu s}$ while the spatial spread is
much longer, up to 3.5 km, see Fig. 5 in [1]. This difference between the
distributions in bursts and conventional CR air showers was coined as the
“curvature” puzzle. As we mentioned above, this puzzle is naturally resolved
within the AQN framework.

Similar arguments also explain why the observed events do not have sharp edges
in waveforms (see Fig. 6 in [1]). The point is that conventional CR air
showers typically have a single ultra-relativistic particle which generates a
very sharp edge in waveforms. It should be contrasted with large number of
positrons which produce the non-sharp edges in waveforms in the AQN-based
proposal.

What is important here is that these features are not sensitive to many
uncertainties which inevitably present in estimates of [3]. This is because
the “curvature” feature discussed above has pure kinematical (geometrical)
nature. It is not related to any complicated problems such as a large Mach
number, turbulence, the AQN internal structure, the dynamics of the electric
field under the thunderstorm, etc. One can say that the “curvature” property
observed in bursts is an almost model-independent consequence of the AQN
framework, not sensitive to many uncertainties of the absolute normalization
factors in the estimates presented in [3]. The same comment also applies to
the property of the non-sharp edges in waveforms as this feature is also very
generic consequence of the AQN framework, not related to any uncertainties in
the estimates [3].

####  II.3.2 “clustering” puzzle

The item 2 shows a dramatic inconsistency (when interpreted in terms of the CR
air showers) for the bursts events when the event rate suggests that the
energy should be in $10^{13}$ eV range, while the intensity of the events
suggests $(10^{18}-10^{19})$ eV range. In the AQN framework the burst
represents the cluster of events related to one and the same AQN which
traverses the distance $L_{\rm burst}\simeq 0.25$ km during $\Delta t=1$ ms.
In different words, the burst is not a collection of independent events.
Conventional Poissonian distribution does not apply here. Instead, the bursts
represent the clustering events when the intensities are determined by the
numbers of liberated positrons when the AQNs enter strong electric field under
the thunderstorm. Furthermore, the spatial spread for each individual event
within the same cluster also lies within the same range $\sim$ 1 km according
to (5). On the one hand, the observations are extremely hard to explain in
terms of conventional CR air showers. On the other hand, the same observations
are perfectly consistent with the AQN based estimates.

####  II.3.3 “synchronization” puzzle

According to item 3 most of the observed bursts are “synchronized” or
“related” to the lightning events. However, some of the bursts are not related
to any lightnings. Therefore, one cannot assume that the bursts are initiated
by the lightning events. However, all 10 recorded bursts occur under
thunderstorm. This is very puzzling property of the bursts if interpreted in
terms of the conventional CR air showers because it is very hard to understand
how the thunderstorms may dramatically modify CR features as discussed above.

At the same time the “synchronization” puzzle is perfectly consistent with AQN
based proposal [3] because the thunderstorm with its pre-existing electric
field (1) plays a crucial role in the mechanism as the electric field
instantaneously liberates the positrons and also accelerates them up to 10 MeV
energies. The mean free path of such energetic positrons is sufficiently long
such that these positrons can easily reach the TA surface detectors, and can
produce the signals consistent with the bursts.

To summarize this section: the mysterious bursts (with highly unusual features
as listed by items 1-3 in Section II.1) of shower-like events observed by TASD
[1, 2] are naturally interpreted as the cluster events generated by the AQNs
propagating in thunderstorm environment. Some of the features of these events
(such as intensity and the basic normalization factors) suffer inevitable
uncertainties due to very complex dynamical properties of the system (the AQN
propagation under thunderstorm with large Mach number). However, there are
some features of the system such as given by eqs. (5), (6), (7) which are not
sensitive to these uncertainties and represent almost model-independent
consequences of the proposal [3].

##  III Radio pulse and TA burst as synchronized events

This section is devoted to studies of the radio signals which always accompany
TA bursts when interpreted in terms of the AQN annihilation events under the
thunderstorm as presented in previous section II. We shall argue below that
the emission of the radio pulse is inevitable consequence of the proposed
mechanism. Furthermore, the radio pulse must be synchronized with TA burst
irrespectively whether the bursts are related or unrelated to the lightning
events. This synchronization must be within $10{\rm\,\mu s}$ from from every
TA event as the positrons and radio waves originated from the same location at
the same instant and both propagate with the speed of light. As the typical
time and spatial spreads of the TA events do not exceed $10{\rm\,\mu s}$ the
synchronization must be within the same temporal and spatial range.

This section is organized as follows. In next subsection III.1 we study the
trajectory of the liberated positrons in a typical electric field
characterized by parameters (1). We analyze the total intensity and the
angular distribution of the radio emission in subsection III.2, while the
spectral properties of the pulse are studied in subsection III.3. Finally, in
subsection III.4 we study the properties of the electric field’s pulse which
can be detected at the surface detector area.

###  III.1  Positron’s acceleration in electric field under thunderstorm

Our goal here is to describe the dynamics of the liberated positrons which are
emitted with initial energies in keV range according to (2). To simplify
things we assume that the electric field ${\cal{E}}$ directed along
$\mathbf{z}$-axis is uniform and characterized by parameters (1). In this case
the solution is well known [35] and can be described as follows. At the very
initial moment of acceleration at $t\ll t_{0}$ the particle moves with non-
relativistic velocities according to conventional formulae:

$\displaystyle
z(t)\simeq\frac{p_{z}^{0}}{m}t+\frac{|e|{\cal{E}}}{2m}t^{2},~{}~{}~{}~{}t\ll
t_{0}\equiv\frac{mc}{|e|{\cal{E}}}$ $\displaystyle
x_{\perp}(t)\simeq\frac{p_{\perp}^{0}}{m}t,~{}~{}~{}\frac{p_{\perp}^{0}}{m}\simeq\frac{p_{z}^{0}}{m}\simeq\frac{v_{\perp}}{c},$
(8)

where $p_{z}^{0}\sim p_{\perp}^{0}$ is the initial positron’s momentum at the
moment of liberation from AQN. Numerically, this non-relativistic portion of
the positron’s journey is very short as $t_{0}\sim 10^{-8}\rm s$. It will be
ignored in all our discussions which follow.

The portion of the positron’s journey which plays the key role in our
discussions is the motion with constant acceleration $e{\cal{E}}/m$ and the
velocity close to $c$. The corresponding motion is determined by the formulae:

$\displaystyle
z(t)=\frac{c}{|e|{\cal{E}}}\left[\sqrt{(p_{\perp}^{0})^{2}+m^{2}c^{2}+(p_{z}^{0}+|e|{\cal{E}}t)^{2}}-\frac{E^{0}}{c}\right],$
$\displaystyle
x_{\perp}(t)\simeq\frac{cp_{\perp}^{0}}{|e|{\cal{E}}}\ln\frac{2|e|{\cal{E}}t}{mc},~{}~{}~{}~{}t\gg
t_{0}\equiv\frac{mc}{|e|{\cal{E}}}$ (9)

where $E^{0}\simeq(mc^{2}+\Delta E)$ is the total initial energy of positrons
at the moment of liberation.

The most important observation here is that the positrons move with ultra-
relativistic velocities for most of the time, and the displacement in
transverse direction (with respect to the orientation of the electric field)
is very modest such that the majority of the particles remain in the system
and continue their acceleration for the entire time interval $\tau_{\cal
E}\sim 0.3\,\mu\rm s$ determined by parameter $l_{a}$ according to (1). At
this point most of the particles assume very high energy close to 10 MeV
according to (4) while transverse momentum distribution is chracterized by
$(v_{\perp}/c)\lesssim 0.1$. The elastic scattering of the positrons off the
atmospheric molecules during this short journey plays very minor role as the
kinetic energy of positrons being in keV range at the initial instant assumes
MeV energies very quickly such that the corresponding cross section never
becomes sufficiently large to modify the positron’s dynamics as computed above
in vacuum.

###  III.2 Intensity and the angular distribution

Our next task is to present conventional formulae for the intensity and the
angular distribution of the radio emission due to the relativistic positrons
(III.1) accelerating in electric field $\mathbf{{\cal{E}}}$ with parameters
(1). As we discussed above the velocity $\mathbf{v}$ of the positrons is
mostly oriented along the electric field while $v_{\perp}$ is small.
Therefore, we assume that $\mathbf{{\cal{E}}}\parallel\mathbf{v}$ to simplify
our formulae which follow. In this simplified geometry the angular
distribution for the intensity assumes the following form, see e.g. [35, 36]:

$\displaystyle\frac{dI(t)}{d\Omega}\simeq\frac{N^{2}e^{2}\mathbf{a}^{2}(t)}{4\pi
c^{3}}\frac{\sin^{2}\theta}{(1-\frac{v}{c}\cos\theta)^{5}},~{}~{}~{}\mathbf{a}\equiv\frac{e\mathbf{{\cal{E}}}}{m\gamma^{3}},$
(10)

where $N$ is the number of coherent positrons participating in the radio wave
emission, to be estimated below. The angle $\theta$ in this expression is
defined as usual as the angle between the velocity $\mathbf{v}$ of the charged
particle at the moment of emission and the direction $\mathbf{n}$ of the
observer, i.e. $\mathbf{v}\cdot\mathbf{n}=v\cos\theta$. Formula (10)
explicitly shows that the emission mostly occurs along $\mathbf{{\cal{E}}}$
direction as $\mathbf{{\cal{E}}}\parallel\mathbf{v}$. This is precisely the
same direction where burst events are recorded as shown on Fig. 1.

Integration over all angles leads to well known expression for the total
intensity:

$\displaystyle
I(t)\simeq\frac{2N^{2}e^{2}}{3c^{3}}\left[\mathbf{a}^{2}(t)\gamma^{6}\right],~{}~{}~{}\gamma\equiv\frac{1}{\sqrt{(1-\frac{v^{2}}{c^{2}})}}$
(11)

Formula (11) represents the well known feature of the emission that the
intensity is strongly enhanced for ultra-relativistic particles, which is
obviously the case for positrons accelerated up to 10 MeV energies according
to (4). This implies that the relativistic enhancement factor entering (11) is
enormous: $(E_{\rm exit}/m)^{6}\sim 10^{8}$.

###  III.3 Spectral characteristics of the radio emission

The main goal of this subsection is to understand the spectral characteristics
of the radio emission. It will also allow to estimate the coherence factor $N$
entering (10) and (11) as this factor obviously depends on the frequency of
radiation and corresponding wave length.

The starting point for these studies is spectral property of the electric
field $\mathbf{E}_{\omega}$ emitted by the accelerating positrons, see e.g.
[35, 36]:

$\displaystyle\mathbf{E}_{\omega}=\int_{-\infty}^{+\infty}\mathbf{E}e^{i\omega
t}dt,~{}~{}\mathbf{E}=\frac{Ne}{c^{2}R}\frac{\mathbf{n}\times\left((\mathbf{n}-\frac{\mathbf{v}}{c}\right)\times\mathbf{a})}{(1-\frac{\mathbf{n}\cdot\mathbf{v}}{c})^{3}},~{}~{}~{}$
(12)

where all quantities at the right hand side of eq. (12) must be computed as
the retarded times $t^{\prime}$:

$\displaystyle t^{\prime}\approx
t-\frac{R_{0}}{c}+\frac{\mathbf{n}\cdot\mathbf{v}t^{\prime}}{c}~{}~{}\rightarrow~{}~{}t=t^{\prime}\left(1-\frac{\mathbf{n}\cdot\mathbf{v}}{c}\right)+\frac{R_{0}}{c},~{}~{}~{}$
(13)

where we assumed that the positrons move with approximately constant time-
independent velocity $|\mathbf{v}|\approx c$. This assumption allows us to
represent the Fourier component $\mathbf{E}_{\omega}$ in the following form

$\displaystyle\mathbf{E}_{\omega}=\frac{e^{ikR_{0}}}{R_{0}}\left(\frac{Ne}{c^{2}}\right)\left(\frac{\omega}{\omega^{\prime}}\right)^{2}\left[\mathbf{n}\times\left((\mathbf{n}-\frac{\mathbf{v}}{c})\times\mathbf{a}_{\omega^{\prime}}\right)\right],~{}~{}~{}$
(14)

where $\omega^{\prime}$ and $\mathbf{a}_{\omega^{\prime}}$ are defined as
follows

$\displaystyle\omega^{\prime}\equiv\omega\left(1-\frac{\mathbf{n}\cdot\mathbf{v}}{c}\right),~{}~{}~{}\mathbf{a}_{\omega^{\prime}}=\int_{-\infty}^{+\infty}\mathbf{a(t^{\prime})}e^{i\omega^{\prime}t^{\prime}}dt^{\prime}$
(15)

which precisely the combination entering the Fourier transform (12).

Using Fourier component for $\mathbf{E}_{\omega}$ as given by (14) and
magnetic component
$\mathbf{B}_{\omega}=i(\mathbf{k}_{\omega}\times\mathbf{E}_{\omega})$ one can
compute the energy $dE_{\mathbf{n}\omega}$ emitted into solid angle $d\Omega$
with frequency interval $d\omega$ [35, 36]:

$\displaystyle\frac{dE_{\mathbf{n}\omega}}{d\Omega
d\omega}=\left(\frac{N^{2}e^{2}}{4\pi^{2}c^{3}}\right)\left(\frac{\omega}{\omega^{\prime}}\right)^{4}\left|\mathbf{n}\times\left((\mathbf{n}-\frac{\mathbf{v}}{c})\times\mathbf{a}_{\omega^{\prime}}\right)\right|^{2}.~{}~{}~{}$
(16)

One can simplify this expression by assuming that the
$\mathbf{a}\parallel\mathbf{v}$ as we previously discussed. Furthermore, one
can carry out the integration over $d\Omega$ by integrating over
$d\omega^{\prime}$ for a given $\omega$ as these two variables are related
according to (15). Indeed, the emission is mostly concentrated along the cone
with $\theta^{2}\approx(1-v^{2}/c^{2})$. Furthermore,

$\displaystyle\omega^{\prime}\equiv\omega\left(1-\frac{\mathbf{n}\cdot\mathbf{v}}{c}\right)\approx\frac{\omega}{2}\left[\frac{1}{\gamma^{2}}+{\theta^{2}}\right],$
(17)

where we expanded $\cos\theta\approx(1-\theta^{2}/2)$ and represented small
factor $(1-v/c)\approx{1}/{(2\gamma^{2})}$ in terms of the conventional
combination $\gamma$ which is valid approximation for ultra-relativistic
positrons with $|\mathbf{v}|\simeq c$. Now, integration over angles $d\Omega$
can be replaced by integration over $d\omega^{\prime}$ as these two variables
are related (17). Therefore,

$\displaystyle d\Omega=2\pi\sin\theta d\theta\approx
2\pi\left(\frac{d\theta^{2}}{2}\right)\approx
2\pi\left(\frac{d\omega^{\prime}}{\omega}\right)$ (18)

such that the energy $dE_{\omega}$ emitted within the frequency interval
$d\omega$ assumes the form

$\displaystyle\frac{dE_{\omega}}{d\omega}$ $\displaystyle=$
$\displaystyle\left(\frac{N^{2}e^{2}\omega^{2}}{2\pi
c^{3}}\right)\int_{\frac{\omega}{2\gamma^{2}}}^{\infty}\frac{d\omega^{\prime}|\mathbf{a}_{\omega^{\prime}}|^{2}}{(\omega^{\prime})^{3}}\left[2-\frac{\omega}{\omega^{\prime}\gamma^{2}}\right],$
(19)

where the low limit of integration is determined by $\theta=0$ in Eq. (17).

The same expression can be thought as the angular distribution of the emission

$\displaystyle\frac{dE_{\omega}}{d\omega}$ $\displaystyle=$
$\displaystyle\left(\frac{N^{2}e^{2}\gamma^{6}}{2\pi
c^{3}}\right)\frac{16|\mathbf{a}_{\omega^{\prime}}|^{2}}{(1+\gamma^{2}\theta^{2})^{3}}\left[\frac{\gamma^{2}\theta^{2}}{1+\gamma^{2}\theta^{2}}\right]\cdot\frac{d\Omega}{2\pi},~{}~{}~{}$
(20)

where $|\mathbf{a}_{\omega^{\prime}}|$ should be expressed in terms of
$\theta$ according to (17).

Our next task is to model the acceleration $\mathbf{a}(t)$ for a typical
thunderstorm electric field with parameters (1). The corresponding Fourier
transform (15) determines $\mathbf{a}_{\omega^{\prime}}$ which enters
expression for ${dE_{\omega}}/{d\omega}$ as given by (19). Our simplest
possible choice is as follows:

$\displaystyle\mathbf{a}(t)\approx\frac{e\mathbf{{\cal{E}}}}{\gamma^{3}m},~{}~{}~{}t\in(0,\tau_{{\cal{E}}}),$
(21)

while $\mathbf{a}(t)\approx 0$ for $t$ being outside of this interval. The
corresponding expression for $\mathbf{a}_{\omega^{\prime}}$ defined by (15)
assumes the form

$\displaystyle|\mathbf{a}_{\omega^{\prime}}|^{2}=\frac{4e^{2}{\cal{E}}^{2}}{m^{2}\gamma^{6}\omega^{\prime
2}}\sin^{2}(\frac{\omega^{\prime}\tau_{{\cal{E}}}}{2}).$ (22)

Now we can perform the integration $d\omega^{\prime}$ in (19). For our simple
model for acceleration (21) the integral can be approximately computed for
small $(\omega^{\prime}\tau_{{\cal{E}}})\ll 1$. This leads us to the order of
magnitude estimate for ${dE_{\omega}}/{d\omega}$:

$\displaystyle\frac{dE_{\omega}}{d\omega}\approx\left(\frac{N^{2}e^{2}}{2\pi
c^{3}}\right)\cdot\left(\frac{e^{2}{\cal{E}}^{2}\tau^{2}_{{\cal{E}}}}{m^{2}\gamma^{6}}\right)\cdot\left(\frac{4\gamma^{4}}{3}\right).$
(23)

The same expression can be derived from (20) by integrating over the angles
$d\Omega$.

This spectral density holds as long as $\omega$ is sufficiently small. To be
more precise:

$\displaystyle\omega\lesssim{\gamma^{2}}{\tau^{-1}_{{\cal{E}}}},$ (24)

which represents the dominant contribution to the integral (19) for small
$\omega$. The emission with higher frequencies will be power-suppressed as
$\omega^{-2}$.

Few comments are in order. The spectral density (23) integrated over all
angles $d\Omega$ approximately constant and does not depend on $\omega$ as
long as condition (24) is satisfied. This is of course a well-known feature of
the constant acceleration (21) which we used as a simplified model for the
electric field. In reality, the electric field under thunderstorm obviously
fluctuates in time and space. This will obviously modify the spectral features
(23). However, we expect that a typical frequency of the emission as given by
(24) will hold because it is basically determined by the typical time scale of
the problem $\tau_{{\cal{E}}}$ and expected relativistic factor $\gamma$ which
is not very sensitive to the details of the fluctuating electric field
${\cal{E}}$.

The crucial observation here is that the typical frequency of emission is not
simply given by the scale $\tau^{-1}_{{\cal{E}}}$ as one could naively expect
based on dimensional arguments. Rather, the emission extends to much broader
region due to the large relativistic factor $\gamma^{2}$ as eq. (24) states.

We postpone for the detail numerical estimates to section IV. Now we want to
make few simple numerical estimates supporting the main claim of this work
that the frequency of emission is in the radio band $\nu\in(0.5-200)$ MHz.
Furthermore, the emission is mostly oriented along the same direction where TA
bursts are recorded. Therefore, one should anticipate a strong synchronization
between the TA bursts and the radio pulses as both emission occurs at the same
location at the same instant, and propagate to the corresponding detectors
with the same speed of light. These features are not very sensitive to details
of the dynamics of the AQNs, nor specific features of electric field under
thunderstorm. Rather, all these features are inevitable consequences of the
basic AQN framework along with pure geometrical properties of these
observables.

With these comments in mind we can represent the typical band as follows

$\displaystyle\nu\lesssim\frac{{\gamma^{2}}}{2\pi{\tau_{{\cal{E}}}}}\approx
200~{}{\rm
MHz},~{}~{}~{}~{}\nu\equiv\frac{\omega}{2\pi},~{}~{}~{}\gamma\approx 20,$ (25)

which we expect to hold irrespectively to specific features of the AQN model
as it is entirely determined by well established typical time scales of the
electric field under the thunderstorm (1). One could naively think that the
typical frequency of emission could be extrapolated to very low $\nu$ as the
spectral density (23) apparently does not depend on frequency. This is not
quite correct conclusion though as we shall discuss in next section IV.

###  III.4 Radio pulse of the electric field

The goal of this section is to estimate the intensity of the electric field
(12) at very large distances $R$ where it can be potentially detected. The
orientation of $\mathbf{E}$ field is determined by cross product (12) where
one can assume (for the simplicity of the numerical estimates) that
$\mathbf{v}\parallel\mathbf{a}$, similar to our previous estimates for the
spectral intensity of the emission (19). The absolute value for $|\mathbf{E}|$
at large distances can be estimated as follows

$\displaystyle|\mathbf{E}|\approx\frac{Ne|\mathbf{a}|\theta}{c^{2}R}\left(\frac{\omega}{\omega^{\prime}}\right)^{3}\approx\frac{Ne|\mathbf{a}|\theta}{c^{2}R}\left(\frac{2\gamma^{2}}{1+\gamma^{2}\theta^{2}}\right)^{3},$
(26)

where $R$ is the distance from the emission site to the area where the
electric field $|\mathbf{E}|$ could be recorded. It should not be confused
with parameter $r$ which enters all formulae from section II.3 and describes
the distance from the emission site to the SD site where the energetic
positrons could be detected. Numerically these parameters are similar, of
course. However, the radio signal can be observed in a different location from
the local area where the positrons hit the TASD.

This formula explicitly shows that the time duration of the radio pulse and
its properties are unambiguously determined by the features of the electric
field $\cal{E}$ under thunderstorm. For simple model (21) this implies that
the radio pulse lasts $\tau_{\cal{E}}$ while the absolute value of the pulse
is estimated as

$\displaystyle|\mathbf{E}(\theta,t)|\approx\frac{Ne\theta}{c^{2}R}\left(\frac{2\gamma^{2}}{1+\gamma^{2}\theta^{2}}\right)^{3}\left(\frac{e\mathbf{{\cal{E}}}(t)}{m\gamma^{3}}\right),~{}t\in(0,\tau_{{\cal{E}}}).~{}~{}~{}~{}~{}$
(27)

It is important to emphasize that the duration of the pulse $\tau_{{\cal{E}}}$
is not very sensitive to the details of the AQN model as it is entirely
determined by well established typical time scales of the electric field under
the thunderstorm (1). This feature is very similar to our previous arguments
regarding a typical frequency of the radio emission (25). At the same time the
intensity of the pulse is highly sensitive to the details of the AQN model as
it depends on the number of positrons $N$ participating in the emission. In
this respect this feature of insensitivity to any specific details of the AQN
model for the pulse duration $\tau_{{\cal{E}}}$ and the frequency $\nu$ as
given by (25) is similar to our studies of the TA burst events when such
features as the “curvature”, the timing and the spatial spread of the bursts
are not sensitive to the details of the AQN model, but entire determined by
the geometry as reviewed in section II.3. This should be contrasted with
estimations [3] of the intensity of the TA bursts which are highly sensitive
to specific features of the model, which are hard to carry out in a
quantitative manner.

##  IV Numerical Estimates

Our goal here is to present some numerical results using parameters which had
been used in our previous estimations [3] related to puzzling TASD bursts as
reviewed in section II.3. One should emphasize from the very beginning that
the corresponding estimates suffer from huge uncertainties, and should be
considered as the order of magnitude estimates, at the very best. Our main
arguments of the present work are not based on these highly model dependent
estimates. Rather, our main arguments are based on essentially model-
independent features of the radio emission such as typical frequency band (25)
and strong synchronization between TASD burst events and the radio emission.
Nevertheless, we think that an order of magnitude estimates for the amplitude
of the electric field as presented below could be useful as they demonstrate
the consistency of the proposal. The same estimates also demonstrate the
principle feasibility to detect such radio signals.

We start with numerical estimation of the strength of the electric field of
the pulse (27) in conventional units $\rm(mV/m)$:

$\displaystyle|\mathbf{E}(\theta,t)|\approx
90{\rm\frac{mV}{m}}\cdot\left[\frac{(\gamma\theta)}{(1+\gamma^{2}\theta^{2})^{3}}\right]$
(28)
$\displaystyle\qquad\qquad\cdot\left(\frac{\gamma}{20}\right)^{2}\cdot\left(\frac{N}{10^{9}}\right)\cdot\left(\frac{\rm
10~{}km}{R}\right),~{}~{}~{}~{}t\in(0,\tau_{{\cal{E}}})$

where factor $N$ is the number of the coherent positrons participating in the
emission, to be estimated below. The orientation of the electric field is
determined by the cross product as given by (12), and it is approximately (up
to small angle $\theta$) points along $\mathbf{a}$ which represents the
direction of the electric field under thundercloud at the instant of emission.
The temporal shape of the pulse (bipolar, unimodal or even more complicated
form) is also determined by the same fluctuating electric field $\cal{E}$ at
the moment of emission. The intensity of the field is strongly peaks along
$\mathbf{n}$ with typical angle $\theta\gamma\lesssim 1$, see Fig. 2 for a
precise angular distribution of $|\mathbf{E}(\theta)|$.

Figure 2: Strength of electric field (28) induced under thunderstorm versus
observation angle $\theta$. The parameters are chosen to be $N=10^{9}$,
$\gamma=20$, ${\cal E}=1{\rm\,kV/cm}$.

Our next task is the estimation of parameter $N$ entering (28). Our estimates
will be entirely based on the observations, rather than on theoretical
computations which inevitably suffer from huge uncertainties as reviewed in
section II.3. Therefore, the logic of our estimation of parameter $N$ is as
follows. Let us assume that the TASD bursts which have been recorded are
indeed due to the positrons emitted by AQNs as proposed in [3]. In this case
the number of detected particles must be proportional to the density of
particle detectors. This number is known for presently operating 507 SD
detectors with area $3\rm m^{2}$ each. These detectors are covering $680~{}\rm
km^{2}$ total area. The number of detected particles must be also proportional
to the area $\sim\pi(r\Delta\alpha)^{2}$ where particles had been recorded by
TASD during a single event within the burst. Therefore, the number of
positrons $N$ at the emission site can be estimated from the following
relation:

$\displaystyle N_{\rm positrons}^{\rm detected}[\Delta s]\approx
N\left[\rm\frac{507\cdot 3~{}m^{2}}{680~{}km^{2}}\right]\left[\langle
e^{-\frac{r}{\lambda}}\rangle\right]\left[\pi(r\Delta\alpha)^{2}\right]~{}~{}~{}~{}~{}~{}$
(29)

In our estimate (29) the number of detected particles being recorded over the
surface area $\pi(\Delta s)^{2}$ (left hand side of eq. (29)) varies for
different events within a single burst, and it normally varies in the range
$(2-6)\cdot 10^{2}$. In our estimate (29) we inserted the suppression factor
$\langle\exp(-r/\lambda)\rangle\sim 0.1$ introduced in [3] to account for some
attenuation of the positrons travelling a distance $r\sim$ 10 km or so when
the mean free path $\lambda$ for positrons with few MeV energy is order of
kilometre at the sea level and several kilometres at higher altitudes. One
should also note that $\pi(r\Delta\alpha)^{2}$ on the right hand side of eq.
(29) is not identically the surface area $\sim\pi(\Delta s)^{2}$ where
particles are being recorded. These areas are directly related to each other
through angle $\alpha$ according to (5), see Fig. 1 for notations.

The relation (29) suggests that number $N$ entering (28) for
$(r\Delta\alpha)\sim 1~{}\rm km$ (which corresponds to a typically observed
spatial spread for the burst events) can be estimated as

$\displaystyle N\approx(0.3-1)\cdot 10^{9}~{}~{}~{}{\rm
for}~{}~{}~{}(r\Delta\alpha)\sim 1~{}\rm km.$ (30)

This estimate for $N$ corresponds to the amplitude of the electric field (28)
on the level $20\rm(mV/m)$ measured at distance $R\simeq 10$ km from the
source of the emission within the angle range $(\theta\gamma)\lesssim 1$ where
the most of the radio emission occurs.

One should comment here that such strength of the amplitude $|\mathbf{E}|$ on
the level $20\rm(mV/m)$ with the duration time of the radio pulse on the scale
$\sim 0.3\mu\rm s$ could be recorded with existing technology, see e.g. [37]
where wide-band radio interferometry (0.1-30 MHz) has been used to detect such
short signals with amplitudes of electric field on the level
$10^{2}\rm(mV/m)$, which is very close to what is required for the purposes of
the present work. Therefore, such signal (28) can be in principle measured if
proper instruments are designed and built at the TASD site.

Our last task in this section is establishing of the lower frequency bound for
the radio emission. The spectral density (23) which is approximately a
constant, naively suggests that the spectrum extends to arbitrary low
frequencies. In fact, it is a premature conclusion as formula (23) was derived
assuming a constant acceleration. In reality, a typical acceleration time is
determined by (1) such that one should expect that
$\nu\gtrsim(2\pi{\tau_{{\cal{E}}}})^{-1}\approx 0.5~{}{\rm MHz}$. Therefore,
the AQN-induced $\sim 0.3\mu$s pulse represents a radio emission in the
bandwidth $\nu\in(0.5-200)\rm~{}MHz$ with the amplitude of the corresponding
electric field of the order $|\mathbf{E}(\theta,t)|\sim 20\rm(mV/m)$ at
distance $R\sim 10\rm~{}km$ as estimated above. Important feature of this
spectrum is that it is very broad. Indeed, the intensity of the radiation
changes by factor two or so when the frequency varies by two orders of
magnitude. Another comment which follows from Fig. 3 is that the portion of
the positron’s energy being converted into the radio waves is very tiny as
initial positrons energy can be estimated as $10N~{}{\rm MeV}$ when $N$ is
estimated in (30). This implies that the initial energy is many orders of
magnitude greater than the radio wave energy shown on Fig. 3.

Figure 3: Spectrum of Eq. (19) in terms of frequency $\nu$, thunderstorm
radiation. The parameters are chosen to be $N=10^{9}$, $\gamma=20$, ${\cal
E}=1{\rm\,kV/cm}$.

The crucial point of our analysis can be formulated as follows. This radio
pulse must be synchronized with an event within TA burst within $10\,\mu$s
because the TA burst and the radio emission are originated from the same
location emitted at the same instant and both signals propagate with the speed
of light along the same direction to be detected within $\sim\rm km^{2}$ area.

It is important to emphasize that such strong synchronization must hold even
if the TA bursts are not related to any lightning events and can be easily
discriminated from much stronger radio emission which always accompany
lightnings. Therefore, such synchronization between the radio pulses and the
TA bursts, if observed, can be easily distinguished from any other possible
correlation between radio emission and conventional lightning events. The
correlation between flashes and radio emission during thunderstorms is well
known and well documented generic feature of the the lightning discharges, see
[37] with large number of references on the original literature.

##  V Geosynchrotron radiation

Another significant radio emission is the geosynchrotron radiation which is
due to presence of the Earth’s magnetic field ${\cal B}\sim 0.5{\rm\,gauss}$.
This effect becomes important near the end of radio emission under
thunderstorm. At this instant all the positrons are ultrarelativistic
($\gamma\sim 20$) but they are no longer accelerating as they alreday left the
region of the external electric field $\cal{E}$. Trajectory of positrons are
helical in presence of the magnetic field of Earth ${\cal B}\sim
0.5{\rm\,gauss}$. The acceleration in this case is perpendicular to the
direction of velocity $\mathbf{v}\perp\mathbf{a}_{\cal B}$, so it is
convenient to set up the following coordinate with respect to the direction of
observer $\mathbf{n}$:

$\displaystyle\hat{\mathbf{v}}\cdot\mathbf{n}=\cos\theta\,,\quad\hat{\mathbf{a}}_{\cal
B}\cdot\mathbf{n}=\sin\theta\cos\phi$ (31)

Radius of the helical trajectory can be estimated as

$\displaystyle\rho\approx\frac{c^{2}}{|\mathbf{a}_{\cal
B}|}\approx\frac{0.7{\rm\,km}}{\sin\theta_{\cal
B}}\left(\frac{\gamma}{20}\right)\left(\frac{0.5{\rm\,gauss}}{{\cal
B}}\right)\,,$ (32) $\displaystyle|\mathbf{a}_{\cal B}|\approx\frac{e{\cal
B}c\sin\theta_{\cal B}}{\gamma m}\,,$

where $\theta_{\cal B}$ is defined to be the angle between direction of ${\cal
B}$ and the velocity of positrons $\mathbf{v}$.

Radiation from an ultrarelativistic particle has a narrow emission angle
$\theta\lesssim\gamma^{-1}$, similar to our previous discussions. However, the
pulse of radiation for geosynchrotron radiation is much shorter [35, 36]:

$\displaystyle\tau_{\cal
B}\approx\frac{\rho}{2\gamma^{3}c}=\frac{0.14{\rm\,ns}}{\sin\theta_{\cal
B}}\left(\frac{20}{\gamma}\right)^{2}\left(\frac{0.5{\rm\,gauss}}{{\cal
B}}\right)$ (33)

This interval is much shorter than the time scale $\tau_{\cal{E}}\sim
0.3{\rm\,\mu s}$ discussed previously due to extra suppression factor
$\sim\gamma^{-3}$. The energy spectrum integrated over angle $d\Omega$ is well
known [38, 35, 36]:

$\displaystyle\frac{dE_{\omega}}{d\omega}$ $\displaystyle=\frac{c\tau_{\cal
B}}{2\pi\rho}\frac{\sqrt{3}N^{2}e^{2}}{c}\gamma\frac{\omega}{\omega_{c}}\int_{\omega/\omega_{c}}^{\infty}K_{5/3}(x)dx\,,$
(34) $\displaystyle\omega_{c}$
$\displaystyle=\frac{3}{2}\gamma^{3}\left(\frac{c}{\rho}\right)\simeq\frac{3}{4\tau_{\cal
B}}\,,$

where $K_{5/3}(x)$ is the modified Bessel function, and $\omega_{c}$ is the
critical angular frequency beyond which radiation becomes negligible.

The electric field of the geosynchrotron radiation follows from (12):

$\displaystyle|\mathbf{E}|$ $\displaystyle=\frac{Ne|\mathbf{a}_{\cal
B}|}{c^{2}R}\left(\frac{\omega}{\omega^{\prime}}\right)^{2}\sqrt{1-\frac{(\mathbf{n}\cdot\hat{\mathbf{a}}_{\cal
B})^{2}}{\gamma^{2}}\left(\frac{\omega}{\omega^{\prime}}\right)^{2}}$ (35)
$\displaystyle\approx\frac{Ne}{c^{2}R}\left(\frac{e{\cal B}c\sin\theta_{\cal
B}}{\gamma
m}\right)\left(\frac{2\gamma^{2}}{1+\gamma^{2}\theta^{2}}\right)^{2}$
$\displaystyle\quad\times\sqrt{1-\left(\frac{2\gamma\theta\cos\phi}{1+\gamma^{2}\theta^{2}}\right)^{2}}\,,$

The estimation of electric field strength holds as long as the positron
current remains coherent. The coherent length $l_{\rm coh}$ can be estimated
in what follows. The dispersion in velocity is $\delta v\sim v_{\perp}\lesssim
0.1c$ from Eq. (7). The condition of coherence requires the dispersion length
$\delta l$ to be much less than the radius $\rho$ of trajectory:

$\delta l\approx\frac{\delta v}{c}l_{\rm coh}\sim\frac{v_{\perp}}{c}l_{\rm
coh}\ll\rho\,.$ (36)

It implies $l_{\rm coh}\sim\rho$ that is insensitive to details of local
parameters such as $\gamma$ and ${\cal B}$, and the geosynchrotron radiation
is only significant within the first cycle of rotation.

##  VI Numerical estimates on geosynchrontron radiation

Similar to Sec. IV, we make numerical estimation for the electric field
strength and the frequency bandwidth. Numerical estimation for the strength of
the electric field of pulse (35) in conventional units (mV/m) is given as:

$\displaystyle|\mathbf{E}|$ $\displaystyle\approx
100{\rm\,\frac{mV}{m}}\cdot\frac{\sin\theta_{\cal
B}}{(1+\gamma^{2}\theta^{2})^{2}}\sqrt{1-\left(\frac{2\gamma\theta\cos\phi}{1+\gamma^{2}\theta^{2}}\right)^{2}}$
(37)
$\displaystyle\quad\times\left(\frac{\gamma}{10}\right)^{3}\left(\frac{N}{10^{9}}\right)\left(\frac{10{\rm\,km}}{R}\right)\left(\frac{{\cal
B}}{0.5{\rm\,gauss}}\right)\,,$

see Fig. 4 for a precise angular distribution with specific parameters.

Figure 4: Strength of electric field (35) induced by geosynchrotron radiation
versus obervation angle $\theta$. The parameters are chosen to be $N=10^{9}$,
$\gamma=20$, ${\cal B}=0.5{\rm\,gauss}$, $\theta_{\cal B}=\phi=\pi/4$.

The upper limit of frequency band is determined by the critical frequency
$\omega_{c}$:

$\nu\lesssim\frac{\omega_{c}}{2\pi}=\frac{3}{8\pi\tau_{\cal B}}\approx
600{\rm\,MHz}\,,\quad(\theta_{\cal{B}}=\frac{\pi}{4})\,,$ (38)

and the lower frequency cutoff is given by the fundamental frequency:

$\nu\gtrsim\frac{c}{2\pi\rho}\approx{50{\rm\,kHz}}\,,\quad(\theta_{\cal{B}}=\frac{\pi}{4}).$
(39)

Therefore, the geosynchrotron induces a radio pulse with time duration $\sim
0.14{\rm\,ns}$ in the bandwidth $\nu\in(0.05-600){\rm\,MHz}$ (see Fig. 5) and
the amplitude of the corresponding electric field is of order $\sim
50{\rm\,mV/m}$ at distance $R\sim 10{\rm\,km}$.

Figure 5: Spectrum of Eq. (34) in terms of frequency $\nu$, geosynchrotron
radiation. The parameters are chosen to be $N=10^{9}$, $\gamma=20$, ${\cal
B}=0.5{\rm\,gauss}$, $\theta_{\cal B}=\pi/4$

While numerical values of the frequency bandwidth $\nu\sim(0.1-10^{2})~{}\rm
MHz$ and the maximal strength of electric field of the radio wave
[$|\mathbf{E}|\sim(20-50)~{}\rm mV/m$] are similar in both cases of radio
pulses (which we term as $\cal{E}$-type and $\cal{B}$-type correspondingly)
there is an important distinct feature, which is manifested in dramatically
different time durations in these two cases. To be more precise, a
$\cal{B}$-type pulse induced by geosynchrotron radiation ($\tau_{\cal B}\sim
0.14{\rm\,ns}$) is much shorter than $\cal{E}$-type pulse with ($\tau_{\cal
E}\sim 0.3{\rm\,\mu s}$).

This distinct feature is in fact an absolutely crucial element for possible
potential observations of the emission. This difference between $\tau_{\cal
B}$ and $\tau_{\cal E}$ is the main reason for dramatically different energy
injection which is measured in units $\rm MeV\cdot MHz^{-1}$ in case of
$\cal{E}$-type emission to be contrasted with much lower $\rm keV\cdot
MHz^{-1}$ scale in case of $\cal{B}$-type emission, see Figs. 3 and 5
correspondingly.

Therefore, the total energy of the radio emission integrated over the pulse
time is much greater for the $\cal{E}$-type radiation in comparison with
$\cal{B}$-type radiation. Furthermore, the required resolution to detect the
very short $\cal{B}$-type pulse with $\tau_{\cal B}\sim\rm ns$ requires much
more demanding instruments than recording of the conventional $\cal{E}$-type
pulses with duration time of order $\tau_{\cal E}\sim{\rm\mu s}$. These
arguments strongly suggest that $\cal{E}$-type radiation plays the dominant
role in emission processes. Therefore, any future studies (including possible
observations) w should be the focused on $\cal{E}$-type radiation analyzed in
Section IV.

##  VII Conclusion

As we argued at the very end of previous Section VI the $\cal{E}$-type
radiation plays the dominant role in radio emission. The corresponding results
have been summarized at the very end of Section IV where it has been argued
that the radio pulse can be recorded if proper instruments are designed and
built at the TASD site. The pulse must be synchronized with TASD bursts. Such
a signal cannot be confused with any other spurious and noise signals as a
result of this synchronization. Therefore, observing (not observing) such
synchronized signals can confirm, substantiate or refute our proposal.

Our arguments are entirely based on the AQN framework. Why should we take this
model seriously? In section II.2 we mentioned that this model was originally
formulated to resolve simultaneously two fundamental puzzles of the modern
cosmology: the observed matter-antimatter asymmetry and the observed
similarity between the dark and visible components of the Universe, i.e. the
ratio $\Omega_{\rm DM}\sim\Omega_{\rm visible}$ holds irrespectively to any
specific features of the model.

The basic logic of the construction suggests that there must be many
observable consequences of this model. Indeed, the AQNs may offer a resolution
[39, 40] to some seemingly unrelated puzzles such as the “Solar Corona
Mystery” when the so-called “nanoflares” conjectured by Parker long ago [41]
are identified with the annihilation events in the AQN framework. The
luminosity of the Extreme UV (EUV) radiation from corona due to these
annihilation events is unambiguously determined by the DM density. It is very
nontrivial consistency check that the computed luminosity from the corona
nicely matches with observed EUV radiation. The same events of annihilation
are also manifested themselves as the radio impulsive events [42] in quiet
solar corona. The computed intensity and spectral features are consistent with
recently recorded radio events in quiet solar corona by Murchison Widefield
Array Observatory [43], which also represents a very nontrivial consistency
check of the proposal [39, 40] on the “Solar Corona Mystery” resolution.

The AQNs may also offer a resolution of the “Primordial Lithium Puzzle” [44]
and the longstanding puzzle with the DAMA/LIBRA observation of the annual
modulation at $9.5\sigma$ confidence level [45]. Furthermore, it may resolve
the observed (by XMM-Newton at $11\sigma$ confidence level [46]) puzzling
seasonal variation of the X-ray background in the near-Earth environment in
the 2-6 keV energy range as suggested in [47]. The AQN annihilation events in
the Earth’s atmosphere could produce infrasound and seismic acoustic waves as
discussed in [48] when the infrasound and seismic acoustic waves have been
recorded without any traces of accompanying meteor-like events. This list of
possible observational consequences of the AQN framework is already quite
long, but definitely this list is far from being complete.

Our present proposal suggests that the TA bursts with very unusual features
will be synchronized with radio pulses. If this synchronization is observed
and the interpretation of TA bursts as the AQN annihilation events is
confirmed by future studies it would be the first direct  (non-gravitational)
evidence which reveals the nature of the DM, in contrast with large number of
indirect hints listed above.

## Acknowledgements

This research was supported in part by the Natural Sciences and Engineering
Research Council of Canada, and X.L. also by the UBC four year doctoral
fellowship.

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# Criteria for algebraic operators to be unitary

Zenon Jan Jabłoński  Instytut Matematyki, Uniwersytet Jagielloński, ul.
Łojasiewicza 6, PL-30348 Kraków, Poland<EMAIL_ADDRESS>,  Il
Bong Jung  Department of Mathematics, Kyungpook National University, Daegu
41566, Korea<EMAIL_ADDRESS>and  Jan Stochel  Instytut Matematyki,
Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland
<EMAIL_ADDRESS>
###### Abstract.

Criteria for an algebraic operator $T$ on a complex Hilbert space
$\mathcal{H}$ to be unitary are established. The main one is written in terms
of the convergence of sequences of the form $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$
with $h\in\mathcal{H}$. Related questions are also discussed.

###### Key words and phrases:

Algebraic operator, unitary operator, normaloid operator, strong stability

###### 2020 Mathematics Subject Classification:

Primary 47B15; Secondary 47B20

The research of the first and third authors was supported by the NCN (National
Science Center), decision No. DEC-2021/43/B/ST1/01651. The second author was
supported by Basic Science Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Education
(NRF-2021R111A1A01043569).

##  1\. Introduction

By the spectral theorem, a unitary operator with a finite spectrum is
algebraic and its spectrum is contained in ${\mathbb{T}}$, the unit circle
centered at $0$. The most fundamental example of a unitary algebraic operator
is the Fourier transform. According to the famous theorem of Plancherel, the
Fourier transform extends uniquely to a unitary operator on
$L^{2}(\mathbb{R})$ (see e.g., [18, Theorem IX.6]). Denote it by
$\mathscr{F}$. The Fourier transform $\mathscr{F}$ has the following
properties:

$\mathscr{F}^{0}=I$, $\mathscr{F}^{1}=\mathscr{F}$, $\mathscr{F}^{2}=P$,
$\mathscr{F}^{3}=\mathscr{F}^{-1}$ and $\mathscr{F}^{4}=I$,

where $I$ is the identity operator and $P(f)(x)=f(-x)$. This implies that
$p(x)=x^{4}$ is the minimal polynomial of $\mathscr{F}$. As a consequence, the
Fourier transform is a unitary algebraic operator with (purely point) spectrum
$\sigma(\mathscr{F})=\\{1,-1,{\mathrm{i}\hskip 0.43057pt},-{\mathrm{i}\hskip
0.43057pt}\\}$ (see [18, Theorems IX.1 and IX.6]).

A natural question arises under what additional assumptions an algebraic
(bounded linear) operator $T$ on a complex Hilbert space $\mathcal{H}$ with
spectrum in ${\mathbb{T}}$ is unitary. To answer this question let us look at
some broader classes $\mathscr{A}$ of operators that can be characterized as
follows: an operator $T$ belongs to $\mathscr{A}$ if and only if the sequences
of the form $\\{\|T^{n}h\|^{2}\\}_{n=0}^{\infty}$ ($h\in\mathcal{H}$) belong
to the corresponding class $\mathscr{S}$ of scalar sequences; in most cases,
the class $\mathscr{S}$ appears naturally in harmonic analysis on
$*$-semigroups. In particular, the celebrated theorem of Lambert states that
the class of subnormal operators corresponds to Stieltjes moment sequences
(see [17]). In this line of correspondence, we can list the classes of
$m$-isometric operators [2, 3, 4, 13], completely hypercontractive operators
[1], completely hyperexpansive operators [5], alternatingly hyperexpansive
operators [20], conditionally positive definite operators [14], and so on. The
answer to our question (see Theorem 1.1 below) is written in terms of the
convergence of the sequences of the form $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$
($h\in\mathcal{H}$). The condition of their convergence seems to be optimal,
since in the light of Remark 3.3 the assumption of their boundedness ceases to
be sufficient. It is also worth mentioning that there are contractions (for
which the sequences $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$, $h\in\mathcal{H}$,
automatically converge) with spectrum in ${\mathbb{T}}$, called unimodular
contractions, which are not unitary (see [19]). Clearly, unimodular
contractions are normaloid. Let us further note that if we replace the class
of normaloid operators by a class of more regular operators, it may turn out
that members of the latter class with spectrum in ${\mathbb{T}}$ are unitary.
In particular, by Stampfli’s theorem (see [21, Corollary, p. 473]), every
hyponormal operator with spectrum in ${\mathbb{T}}$ is unitary.

Before formulating the main result, we establish some notation and
terminology. Denote by $\mathbb{C}$ the field of complex numbers. Set
${\mathbb{T}}=\\{z\in\mathbb{C}\colon|z|=1\\}$. Write $\mathbb{N}$,
$\mathbb{Z}_{+}$ and $\mathbb{R}_{+}$ for the sets of positive integers,
nonnegative integers and nonnegative real numbers, respectively. Let
$\boldsymbol{B}(\mathcal{H})$ stand for the $C^{*}$-algebra of all bounded
linear operators on a complex Hilbert space $\mathcal{H}$. For
$T\in\boldsymbol{B}(\mathcal{H})$, denote by $\mathscr{N}(T)$, $\sigma(T)$ and
$r(T)$ the kernel, the spectrum and the spectral radius of $T$, respectively.
An operator $T\in\boldsymbol{B}(\mathcal{H})$ is said to be normaloid if
$r(T)=\|T\|$, or equivalently, by Gelfand’s formula for spectral radius, if
and only if $\|T^{n}\|=\|T\|^{n}$ for all $n\in\mathbb{N}$. Call
$T\in\boldsymbol{B}(\mathcal{H})$ algebraic if there exists a nonzero
polynomial $p$ (in one indeterminate with complex coefficients) such that
$p(T)=0$; such a $p$ is said to be minimal if $p$ is the (unique) monic
polynomial of least degree among all nonzero polynomials $q$ such that
$q(T)=0$.

The following theorem, which is the main result of the paper, characterizes
unitary algebraic operators in terms of the convergence of the sequences of
the form $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$. Its proof is given in Section 3.

######  Theorem 1.1.

Suppose that $T\in\boldsymbol{B}(\mathcal{H})$ is algebraic and
$\sigma(T)\subseteq{\mathbb{T}}$. Then the following statements are
equivalent:

  1. (i)

$T$ is unitary,

  2. (ii)

$T$ is normaloid,

  3. (iii)

$\|T\|\leqslant 1$ $($or equivalently, $\|T\|=1$$)$,

  4. (iv)

the sequence $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$ is convergent in
$\mathbb{R}_{+}$ for every $h\in\mathcal{H}$.

##  2\. Preparatory facts

In this section we give some basic facts about algebraic operators needed in
this paper. We begin with a purely linear algebra result, the proof of which
is left to the reader. If $\mathscr{M}$ is a complex vector space, then the
identity transformation on $\mathscr{M}$ is denoted by $I_{\mathscr{M}}$ (or
simply by $I$ if no ambiguity arises). We write

$\displaystyle\mathscr{M}=\mathscr{M}_{1}\dotplus\ldots\dotplus\mathscr{M}_{m}$
(2.1)

in the case when $\mathscr{M}$ is a direct sum of (finitely many) vector
subspaces $\mathscr{M}_{1},\ldots,\mathscr{M}_{m}$.

######  Lemma 2.1.

Suppose that (2.1) holds. Let $A\colon\mathscr{M}\to\mathscr{M}$ be a linear
transformation such that $A(\mathscr{M}_{j})\subseteq\mathscr{M}_{j}$ for all
$j=1,\ldots,m$ and let $z\in\mathbb{C}$. Then $A-zI_{\mathscr{M}}$ is a
bijection if and only if $A|_{\mathscr{M}_{j}}-zI_{\mathscr{M}_{j}}$ is a
bijection for all $j=1,\ldots,m$. Moreover, if $A-zI_{\mathscr{M}}$ is a
bijection, then $(A-zI_{\mathscr{M}})^{-1}(\mathscr{M}_{j})=\mathscr{M}_{j}$
for all $j=1,\ldots,m$ and

$\displaystyle(A|_{\mathscr{M}_{j}}-zI_{\mathscr{M}_{j}})^{-1}=(A-zI_{\mathscr{M}})^{-1}|_{\mathscr{M}_{j}},\quad
j=1,\ldots,m.$

######  Corollary 2.2.

Suppose that $\mathcal{H}$ is a complex Hilbert space which is a direct sum of
finitely many nonzero closed vector subspaces
$\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$. Let
$T\in\boldsymbol{B}(\mathcal{H})$ be such that
$T(\mathcal{H}_{j})\subseteq\mathcal{H}_{j}$ for all $j=1,\ldots,m$. Then
$\sigma(T|_{\mathcal{H}_{j}})\subseteq\sigma(T)$ for all $j=1,\ldots,m$.

For the sake of self-containedness, we sketch the proof of the following lemma
that collects indispensable facts about algebraic operators.

######  Lemma 2.3.

Let $T\in\boldsymbol{B}(\mathcal{H})$. Then the following conditions are
equivalent:

  1. (i)

$T$ is algebraic,

  2. (ii)

there exist an integer $m\geqslant 1$, integers $i_{1},\ldots,i_{m}\geqslant
1$, distinct complex numbers $z_{1},\ldots,z_{m}$ and closed nonzero vector
subspaces $\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$ of $\mathcal{H}$ such that

    1. (ii-a)

$\mathcal{H}=\mathcal{H}_{1}\dotplus\ldots\dotplus\mathcal{H}_{m}$,

    2. (ii-b)

$T(\mathcal{H}_{j})\subseteq\mathcal{H}_{j}$ for all $j=1,\ldots,m$,

    3. (ii-c)

$(T_{j}-z_{j}I_{j})^{i_{j}}=0$ for all $j=1,\ldots,m$, where
$T_{j}:=T|_{\mathcal{H}_{j}}$ and $I_{j}:=I_{\mathcal{H}_{j}}$,

    4. (ii-d)

$\sigma(T)=\\{z_{1},\ldots,z_{m}\\}$ and $\sigma(T_{j})=\\{z_{j}\\}$ for all
$j=1,\ldots,m$,

    5. (ii-e)

there exists a constant $c\in(0,\infty)$ such that

$\displaystyle\|h_{j}\|\leqslant c\|h_{1}+\ldots+h_{m}\|,\quad
j=1,\ldots,m,\,h_{1}\in\mathcal{H}_{1},\ldots,h_{m}\in\mathcal{H}_{m}.$ (2.2)

###### Proof.

(i)$\Rightarrow$(ii) Let $T$ be an algebraic operator and $p$ be its minimal
polynomial. Clearly, $\deg p\geqslant 1$. It follows from the fundamental
theorem of algebra that

$\displaystyle p(x)=(x-z_{1})^{i_{1}}\cdots(x-z_{m})^{i_{m}}$

with unique integers $i_{1},\ldots,i_{m}\geqslant 1$ and distinct complex
numbers $z_{1},\ldots,z_{m}$. In view of [6, Lemma 6.1], the condition (ii-a)
holds with $\mathcal{H}_{j}:=\mathscr{N}((T-z_{j}I)^{i_{j}})\neq\\{0\\}$ for
$j=1,\dots,m$. This implies that $\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$ are
closed vector subspaces of $\mathcal{H}$ which are invariant for $T$. As a
consequence, (ii-b) and (ii-c) hold. By the spectral mapping theorem,
$\sigma(T)\subseteq\\{z_{1},\ldots,z_{m}\\}$. Since
$(T_{j}-z_{j}I_{j})^{i_{j}}=0$ and $\mathcal{H}_{j}\neq\\{0\\}$, we infer from
the spectral mapping theorem and Corollary 2.2 that

$\displaystyle\\{z_{j}\\}=\sigma(T_{j})\subseteq\sigma(T),\quad j=1,\ldots,m,$

which implies (ii-d).

Now, we proceed to the proof of (ii-e). Define for $j=1,\ldots,m$ the linear
projection $P_{j}\colon\mathcal{H}\to\mathcal{H}$ by

$\displaystyle P_{j}(h_{1}+\ldots+h_{m})=h_{j},\quad
h_{1}\in\mathcal{H}_{1},\ldots,h_{m}\in\mathcal{H}_{m}.$

By (ii-a), this definition is correct. Using [6, Lemma 6.1(iii)] we see that

$\displaystyle\mathcal{K}_{2}:=\mathcal{H}_{2}\dotplus\ldots\dotplus\mathcal{H}_{m}=\mathscr{N}\bigg{(}\prod_{j=2}^{m}(T-z_{j}I)^{i_{j}}\bigg{)},$

and so $\mathcal{K}_{2}$ is a closed vector subspace of $\mathcal{H}$. Since,
by (ii-a), $\mathcal{H}=\mathcal{H}_{1}\dotplus\mathcal{K}_{2}$, we infer from
[7, Theorem III.13.2] that $P_{1}\in\boldsymbol{B}(\mathcal{H})$. A similar
argument shows that $P_{j}\in\boldsymbol{B}(\mathcal{H})$ for all
$j\in\\{1,\ldots,m\\}$. This implies (2.2).

(ii)$\Rightarrow$(i) It is enough to note that $p(T)=0$ with
$p(x)=(x-z_{1})^{i_{1}}\cdots(x-z_{m})^{i_{m}}$. This completes the proof. ∎

##  3\. Proof of the main result

We begin by stating an auxiliary lemma. For the reader’s convenience, we give
its proof, which is essentially the same as that of Lemma 2.1 in the
unpublished paper [11].

######  Lemma 3.1.

Let $b,w\in\mathbb{C}$ be such that $|w|=1$ and $w\neq\pm 1$. Assume that the
sequence $\\{{\rm Re\,}(w^{n}b)\\}_{n=0}^{\infty}$ is convergent. Then $b=0$.

###### Proof.

Suppose to the contrary that $b\neq 0$.

Consider first the case when $w^{m}\neq 1$ for all $m\in\mathbb{N}$. Then, by
Jacobi’s theorem (see [10, Theorem I.3.13]), the sequence
$\\{w^{n}\\}_{n=0}^{\infty}$ is dense in ${\mathbb{T}}$. Hence, there exist
two subsequences $\\{w^{k_{n}}\\}_{n=0}^{\infty}$ and
$\\{w^{l_{n}}\\}_{n=0}^{\infty}$ of the sequence $\\{w^{n}\\}_{n=0}^{\infty}$
such that

$\lim_{n\to\infty}w^{k_{n}}=\frac{\bar{b}}{|b|}$ and
$\lim_{n\to\infty}w^{l_{n}}=-\frac{\bar{b}}{|b|}$.

As a consequence, we have

$\displaystyle|b|=\lim_{n\to\infty}{\rm
Re\,}(w^{k_{n}}b)=\lim_{n\to\infty}{\rm Re\,}(w^{l_{n}}b)=-|b|,$

which contradicts $b\neq 0$.

Suppose now that $w^{m}=1$ for some $m\in\mathbb{N}$. By our assumption on
$w$, $m$ must be greater than or equal to $3$. It is easily seen that ${\rm
Re\,}(b),{\rm Re\,}(wb),\ldots,{\rm Re\,}(w^{m-1}b)$ are the cluster points of
the sequence $\\{{\rm Re\,}(w^{n}b)\\}_{n=0}^{\infty}$. Hence, we have

$\displaystyle{\rm Re\,}(b)={\rm Re\,}(w^{j}b),\quad j\in\mathbb{N}.$ (3.1)

Observe that

$\displaystyle|b|=|w^{j}b|,\quad j\in\mathbb{N}.$ (3.2)

By (3.1) and (3.2) with $j=1$, either $b=wb$, which contradicts $w\neq 1$, or
$\bar{b}=wb$. Next, by (3.1) and (3.2) with $j=2$, either $b=w^{2}b$, a
contradiction, or $\bar{b}=w^{2}b$ which, together with $\bar{b}=wb$, leads to
a contradiction. This completes the proof. ∎

Before proving the main result of this paper, we characterize power bounded
algebraic operators with spectrum in the unit circle. Recall that
$T\in\boldsymbol{B}(\mathcal{H})$ is said to be power bounded if
$\sup_{n\in\mathbb{Z}_{+}}\|T^{n}\|<\infty$.

######  Lemma 3.2.

Let $T\in\boldsymbol{B}(\mathcal{H})$. Then the following conditions are
equivalent:

  1. (i)

$T$ is a power bounded algebraic operator such that
$\sigma(T)\subseteq{\mathbb{T}}$,

  2. (ii)

there exist an integer $m\geqslant 1$, closed nonzero vector subspaces
$\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$ of $\mathcal{H}$ and distinct complex
numbers $z_{1},\ldots,z_{m}$ such that

    1. (ii-a)

$\mathcal{H}=\mathcal{H}_{1}\dotplus\ldots\dotplus\mathcal{H}_{m}$,

    2. (ii-b)

$T(h_{1}+\ldots+h_{m})=z_{1}h_{1}+\ldots+z_{m}h_{m}$ for all
$h_{1}\in\mathcal{H}_{1},\ldots,h_{m}\in\mathcal{H}_{m}$,

    3. (ii-c)

$\\{z_{1},\ldots,z_{m}\\}\subseteq{\mathbb{T}}$.

###### Proof.

(i)$\Rightarrow$(ii) Assume that $T$ is a power bound algebraic operator such
that $\sigma(T)\subseteq{\mathbb{T}}$. By Lemma 2.3, there exist an integer
$m\geqslant 1$, integers $i_{1},\ldots,i_{m}\geqslant 1$, distinct complex
numbers $z_{1},\ldots,z_{m}$ and closed nonzero vector subspaces
$\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$ of $\mathcal{H}$ that satisfy the
conditions (ii-a)-(ii-d) of this lemma. Since
$\sigma(T)\subseteq{\mathbb{T}}$, we have

$\displaystyle|z_{l}|=1,\quad l=1,\ldots,m.$ (3.3)

Fix $k\in\\{1,\ldots,m\\}$. Take $h_{k}\in\mathcal{H}_{k}\setminus\\{0\\}$.
Using (3.3) , Lemma 2.3(ii-c) and [6, Sublemma 6.3], we deduce that

$\alpha_{n}(h_{k}):=\frac{1}{n^{N(h_{k})}}\|T_{k}^{n}h_{k}\|$ converges to a
positive real number as $n\to\infty$, (3.4)

where $N(h_{k})$ is the unique nonnegative integer such that

$(T_{k}-z_{k}I_{k})^{N(h_{k})}h_{k}\neq 0$ and
$(T_{k}-z_{k}I_{k})^{N(h_{k})+1}h_{k}=0$. (3.5)

In particular, we have

$\displaystyle\|T_{k}^{n}h_{k}\|=n^{N(h_{k})}\alpha_{n}(h_{k}),\quad
n\geqslant 1.$ (3.6)

Since $T_{k}$ is power bounded, the sequence
$\\{\|T_{k}^{n}h_{k}\|\\}_{n=0}^{\infty}$ is bounded. Hence, one can infer
from (3.4) and (3.6) that $N(h_{k})=0$. This, together with (3.5), implies
that $T_{k}h_{k}=z_{k}h_{k}$. As a consequence, the system $T$, $z_{1}$, …,
$z_{m}$, $\mathcal{H}_{1}$, …, $\mathcal{H}_{m}$ satisfies the conditions
(ii-a), (ii-b) and (ii-c).

(ii)$\Rightarrow$(i) By (ii-a) and (ii-b), $p(T)=0$ with
$p(x)=(x-z_{1})\cdots(x-z_{m})$, which means that $T$ is an algebraic
operator. According to the spectral mapping theorem,
$\sigma(T)\subseteq\\{z_{1},\ldots,z_{m}\\}$. In turn, by (ii-b) and the
assumption that each $\mathcal{H}_{i}$ is nonzero, we deduce that
$z_{1},\ldots,z_{m}$ are eigenvalues of $T$. Therefore, by (ii-c), we have

$\displaystyle\sigma(T)=\\{z_{1},\ldots,z_{m}\\}\subseteq{\mathbb{T}}.$

Now, using Lemma 2.3(ii-e) (or the uniform boundedness principle), we deduce
the power boundedness of $T$ from (ii-a), (ii-b) and (ii-c). This completes
the proof. ∎

######  Remark 3.3.

Regarding Theorem 1.1, we note that in view of Lemma 3.2 there exist algebraic
operators $T$ with spectrum in ${\mathbb{T}}$ that are not unitary but have
the property that each sequence $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$ is bounded
(or equivalently, $T$ is power bounded). $\diamondsuit$

Now, we are ready to prove the main result of this paper.

###### Proof of Theorem 1.1.

The implications (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv) are
obvious.

(iv)$\Rightarrow$(i) Assume that $T$ is algebraic,
$\sigma(T)\subseteq{\mathbb{T}}$ and the sequence
$\\{\|T^{n}h\|\\}_{n=0}^{\infty}$ is convergent in $\mathbb{R}_{+}$ for every
$h\in\mathcal{H}$. By the uniform boundedness principle, $T$ is power bounded.
In view of the implication (i)$\Rightarrow$(ii) of Lemma 3.2, there exist an
integer $m\geqslant 1$, distinct complex numbers $z_{1},\ldots,z_{m}$ and
closed nonzero vector subspaces $\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$ of
$\mathcal{H}$ that satisfy the conditions (ii-a), (ii-b) and (ii-c) of Lemma
3.2. Fix distinct $k,l\in\\{1,\ldots,m\\}$. Take $h_{k}\in\mathcal{H}_{k}$ and
$h_{l}\in\mathcal{H}_{l}$. Then, by the conditions (ii-a), (ii-b) and (ii-c)
of Lemma 3.2, we have

$\displaystyle\|T^{n}(h_{k}+h_{l})\|^{2}$
$\displaystyle=\|z_{k}^{n}h_{k}+z_{l}^{n}h_{l}\|^{2}$
$\displaystyle=\|h_{k}\|^{2}+2{\rm Re\,}\big{(}(z_{k}\bar{z}_{l})^{n}\langle
h_{k},h_{l}\rangle\big{)}+\|h_{l}\|^{2},\quad n\geqslant 0.$

Combined with (iv), this implies that the sequence $\big{\\{}{\rm
Re\,}\big{(}(z_{k}\bar{z}_{l})^{n}\langle
h_{k},h_{l}\rangle\big{)}\big{\\}}_{n=0}^{\infty}$ is convergent. If
$z_{k}\bar{z}_{l}=-1$, then we see that ${\rm Re\,}\langle
h_{k},h_{l}\rangle=0$. Substituting ${\mathrm{i}\hskip 0.43057pt}h_{k}$ in
place of $h_{k}$, we deduce that $\langle h_{k},h_{l}\rangle=0$. The only
possibility left is that $z_{k}\bar{z}_{l}\neq\pm 1$. Since, by the condition
(ii-c) of Lemma 3.2, $|z_{k}\bar{z}_{l}|=1$, we infer from Lemma 3.1 that
$\langle h_{k},h_{l}\rangle=0$. This shows that
$\mathcal{H}=\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{m}$. Hence, by the
conditions (ii-b) and (ii-c) of Lemma 3.2, we conclude that $T$ is a unitary
operator. This completes the proof. ∎

##  4\. Related results

For the reader’s convenience, we record here some useful facts related to the
main topic of this paper concerning certain classes of operators. We begin by
discussing the question of the existence of the limits
$\lim_{n\to\infty}\|T^{n}h\|$ in the case of normaloid operators and the issue
of strong stability111We refer the reader to [15, 16] for a discussion of the
different types of stability of operators. in the context of subnormal
operators. Both are intimately related to Theorem 1.1.

######  Proposition 4.1.

Let $T\in\boldsymbol{B}(\mathcal{H})$ be a normaloid operator. Then the
following conditions are equivalent:

  1. (i)

the sequence $\\{\|T^{n}h\|\\}_{n=0}^{\infty}$ is convergent in
$\mathbb{R}_{+}$ for every $h\in\mathcal{H}$,

  2. (ii)

$T$ is power bounded,

  3. (iii)

$T$ is a contraction.

###### Proof.

By the uniform boundedness principle and Gelfand’s formula for $r(T)$, (i)
implies (ii) and (ii) implies (iii). That (iii) implies (i) is obvious. ∎

Before formulating the next result, we give the necessary definitions and
facts related to the concept of subnormality. Recall that an operator
$T\in\boldsymbol{B}(\mathcal{H})$ is said to be subnormal if there exist a
complex Hilbert space $\mathcal{K}$ and a normal operator
$N\in\boldsymbol{B}(\mathcal{K})$ such that $\mathcal{H}\subseteq\mathcal{K}$
(isometric embedding) and $Th=Nh$ for all $h\in\mathcal{H}$. Such an $N$ is
called a normal extension of $T$; if $\mathcal{K}$ has no proper closed vector
subspace containing $\mathcal{H}$ and reducing $N$, then $N$ is called
minimal. By a semispectral measure of a subnormal operator
$T\in\boldsymbol{B}(\mathcal{H})$ we mean the Borel
$\boldsymbol{B}(\mathcal{H})$-valued measure $F$ on $\mathbb{C}$ defined by

$\displaystyle
F(\varDelta)=PE(\varDelta)|_{\mathcal{H}},\quad\text{$\varDelta$ - Borel
subset of $\mathbb{C}$},$ (4.1)

where $E$ is the spectral measure of a minimal normal extension
$N\in\boldsymbol{B}(\mathcal{K})$ of $T$ and $P\in\boldsymbol{B}(\mathcal{K})$
is the orthogonal projection of $\mathcal{K}$ onto $\mathcal{H}$. In view of
[12, Proposition 5], a subnormal operator has exactly one semispectral
measure. We also need the following fact (see [12, Proposition 4]).

(4.2)

According to [8, Proposition II.4.6]), the following holds.

Any subnormal operator is normaloid. (4.3)

We refer the reader to [8] for the foundations of the theory of subnormal
operators.

######  Proposition 4.2.

Let $S\in\boldsymbol{B}(\mathcal{H})$ be a subnormal operator,
$N\in\boldsymbol{B}(\mathcal{K})$ be a minimal normal extension of $S$ and $F$
be the semispectral measure of $S$. Then the following assertions hold:

  1. (i)

if $S$ is a contraction, then $\lim_{n\to\infty}\|S^{n}h\|^{2}=\langle
F({\mathbb{T}})h,h\rangle$ for every $h\in\mathcal{H}$,

  2. (ii)

the following conditions are equivalent:

    1. (ii-a)

$S$ is strongly stable, i.e., $\lim_{n\to\infty}S^{n}h=0$ for every
$h\in\mathcal{H}$,

    2. (ii-b)

$S$ is power bounded and $F({\mathbb{T}})=0$,

    3. (ii-c)

$S$ is a contraction and $F({\mathbb{T}})=0$,

  3. (iii)

$S$ is strongly stable if and only if $N$ is strongly stable.

###### Proof.

(i) Suppose $\|S\|\leqslant 1$. Then, by [8, Corollary II.2.17],
$\|N\|=\|S\|\leqslant 1$. It follows from the spectral theorem that

$\displaystyle\|S^{n}h\|^{2}$
$\displaystyle=\|N^{n}h\|^{2}\overset{\eqref{kompr}}{=}\int_{\bar{\mathbb{D}}}|z|^{2n}\langle
F(\mathrm{d}z)h,h\rangle,\quad n\in\mathbb{Z}_{+},\,h\in\mathcal{H},$

where $\bar{\mathbb{D}}=\\{z\in\mathbb{C}\colon|z|\leqslant 1\\}$. Since for
$z\in\bar{\mathbb{D}}$, the sequence $\\{|z|^{2n}\\}_{n=0}^{\infty}$ converges
to $\chi_{{\mathbb{T}}}(z)$ as $n\to\infty$, where $\chi_{{\mathbb{T}}}$ is
the characteristic function of ${\mathbb{T}}$, we deduce from Lebesgue’s
dominated convergence theorem that (i) holds.

(ii) Since subnormal operators are normaloid (see (4.3)), the assertion (ii)
follows from (i) and Proposition 4.1.

(iii) Recall that $\|S\|=\|N\|$, and also that $F({\mathbb{T}})=0$ if and only
if $E({\mathbb{T}})=0$, where $E$ is the spectral measure of $N$ (see (4.2)).
Combined with (ii) applied to both $S$ and $N$, this yields (iii). ∎

We now estimate the growth of norms of powers of an algebraic operator whose
spectral radius is less than or equal to $1$.

######  Proposition 4.3.

Suppose that $T\in\boldsymbol{B}(\mathcal{H})$ is an algebraic operator such
that $0<r(T)\leq 1$. Then there exists $\alpha\in(0,\infty)$ such that

$\displaystyle\|T^{n}\|\leqslant\alpha\,n^{\kappa}r(T)^{n},\quad n\geqslant
1,$ (4.4)

where $\kappa=(\deg p)-1$ and $p$ is the minimal polynomial of $T$.

###### Proof.

Since $r(T)>0$, in light of Lemma 2.3 and its proof it suffices to consider
the case when $(T-zI)^{i}=0$ for some $1\leqslant i\leqslant\deg p$ and for
some $z\in\mathbb{C}$ such that $0<|z|\leqslant 1$. Setting $N=T-zI$, it is
easily seen that

$\displaystyle T^{n}=(zI+N)^{n}=\sum_{j=0}^{i-1}\binom{n}{j}z^{n-j}N^{j},\quad
n\geqslant i-1,$

which together with $r(T)=|z|$ implies that

$\displaystyle\|T^{n}\|\leqslant\sum_{j=0}^{i-1}\binom{n}{j}\|N^{j}\||z|^{n-j}\leqslant\bigg{(}\sum_{j=0}^{i-1}\frac{\|N^{j}\|}{j!|z|^{j}}\bigg{)}n^{i-1}r(T)^{n},\quad
n\geqslant i-1.$

This completes the proof. ∎

######  Remark 4.4.

a) It is worth mentioning that if $T\in\boldsymbol{B}(\mathcal{H})$ is an
algebraic operator such that $r(T)=0$, then, in view of Lemma 2.3, the
estimate (4.4) still holds, but only for $n\geqslant\deg p$.

b) It follows from Gelfand’s formula for spectral radius that if
$T\in\boldsymbol{B}(\mathcal{H})$ is such that $r(T)<1$, then $T$ is uniformly
stable222A more detailed discussion of this issue can be found in [16,
Proposition 6.22]., i.e., $\lim_{n\to\infty}\|T^{n}\|=0$, and hence the
sequence $\\{T^{n}h\\}_{n=0}^{\infty}$ is convergent for every
$h\in\mathcal{H}$. The latter statement ceases to be true if the spectrum of
$T$ has a nonempty intersection with ${\mathbb{T}}\setminus\\{1\\}$, even if
$T$ is algebraic. It could be even worse, namely, if $T=zI$, where
$z=\operatorname{e}^{2\pi{\mathrm{i}\hskip 0.3014pt}\theta}$ and $\theta$ is
an irrational number, then by Jacobi’s theorem (see [10, Theorem I.3.13]) the
closure of the set $\\{T^{n}h\colon n\in\mathbb{Z}_{+}\\}$ is equal to
$\\{zh\colon z\in{\mathbb{T}}\\}$ for every $h\in\mathcal{H}$.

c) Note that if $T\in\boldsymbol{B}(\mathcal{H})$ is algebraic, then the
sequence $\\{\|T^{n}h\|^{1/n}\\}_{n=1}^{\infty}$ is convergent in
$\mathbb{R}_{+}$ for all $h\in\mathcal{H}$ (see [6, Proposition 6.2]). Without
the assumption that $T$ is algebraic, the sequence
$\\{\|T^{n}h\|^{1/n}\\}_{n=1}^{\infty}$ may not converge. For more details on
this issue, we refer the reader to [9]. $\diamondsuit$

We conclude this section by providing an example illustrating Theorem 1.1 and
Proposition 4.1.

######  Example 4.5.

Let $N\in\boldsymbol{B}(\mathcal{H})$ be a nonzero operator such that
$N^{2}=0$ and let $\alpha\in{\mathbb{T}}$. Then the operator
$T_{\alpha}:=\alpha I+N$ is algebraic because $p(T_{\alpha})=0$ with
$p(x)=(x-\alpha)^{2}$. Hence, by the spectral mapping theorem
$\sigma(T_{\alpha})=\\{\alpha\\}$ and so $T_{\alpha}$ is invertible in
$\boldsymbol{B}(\mathcal{H})$ and $r(T_{\alpha})=1$. However
$\|T_{\alpha}\|>1$ whenever $\|N\|>2$ (this can be achieved simply by
rescaling $N$). As a consequence, $T_{\alpha}$ is not normaloid, hence not
subnormal (see (4.3)). Observe that
$\mathcal{H}\setminus\mathscr{N}(N)\neq\emptyset$ and

$\lim_{n\to\infty}\|T_{\alpha}^{n}h\|=\infty$ if and only if
$h\in\mathcal{H}\setminus\mathscr{N}(N)$. (4.5)

Indeed, by Newton’s binomial formula (or simply by induction), we have

$\displaystyle(I+\bar{\alpha}N)^{n}=I+n\bar{\alpha}N,\quad
n\in\mathbb{Z}_{+},$

which implies that

$\displaystyle\|(\alpha
I+N)^{n}h\|^{2}=\|h\|^{2}+2n{\mathrm{R}e}(\alpha\langle
h,Nh\rangle)+n^{2}\|Nh\|^{2},\quad n\in\mathbb{Z}_{+},\,h\in\mathcal{H}.$

This yields (4.5). Since $\mathcal{H}\setminus\mathscr{N}(N)\neq\emptyset$, we
infer from (4.5) that $T_{\alpha}$ is not power bounded. A simple example of
this kind is $\mathcal{H}=\mathbb{C}^{2}$, $N=[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}]$ and $\alpha=1$. In this particular case $\|T_{1}\|>1$
without rescaling. $\diamondsuit$

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