SM

    Standard Model
CPV

    CP violating
CPC

    CP conserving
EH

    Euler-Heisenberg
BSM

    beyond the Standard Model
COM

    center of mass
QED

    Quantum Electrodynamics
EFT

    Effective Field Theory
ALP

    axion-like particle
EOM

    Equation of Motion
RHS

    right hand side
LHS

    left hand side
FP

    Fabry-Perot
EM

    Electromagnetism
SRF

    superconducting radio frequency
SNR

    signal to noise ratio
EP

    Equivalence Principle
CL

    confidence level
DM

    dark matter
ULDM

    ultra-light dark matter
VEV

    vacuum expectation value
EWSB

    Electroweak Symmetry Breaking

Parametric resonance in neutrino oscillations induced by ultra-light dark
matter and implications for KamLAND and JUNO

Marta Losada1, Yosef Nir2, Gilad Perez2, Inbar Savoray2 and Yogev Shpilman2

1New York University Abu Dhabi, PO Box 129188, Saadiyat Island, Abu Dhabi,
United Arab Emirates
2Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel

<EMAIL_ADDRESS>yosef.nir,gilad.perez,inbar.savoray,
<EMAIL_ADDRESS>
If ultra-light dark matter (ULDM) exists and couples to neutrinos, the
neutrino oscillation probability might be significantly altered by a
parametric resonance. This resonance can occur if the typical frequency of
neutrino flavor-oscillations $\Delta m^{2}/(2E)$, where $\Delta m^{2}$ is the
mass-squared difference of the neutrinos and $E$ is the neutrino energy,
matches the oscillation frequency of the ULDM field, determined by its mass,
$m_{\phi}$. The resonance could lead to observable effects even if the ULDM
coupling is very small, and even if its typical oscillation period, given by
$\tau_{\phi}=2\pi/m_{\phi}$, is much shorter than the experimental temporal
resolution. Defining a small parameter $\epsilon_{\phi}$ to be the ratio
between the contribution of the ULDM field to the neutrino mass and the vacuum
value of the neutrino mass, the impact of the resonance is particularly
significant if $\epsilon_{\phi}m_{\phi}L\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}4$, where $L$ is the
distance between the neutrino source and the detector. Such parametric
resonance can improve the fit to the KamLAND experiment measurements by about
$3.5\sigma$ compared to standard oscillations. This scenario will be tested by
the JUNO experiment.

##  1 Introduction

Light scalar fields may constitute ULDM candidates [1, 2, 3, 4]. If these
fields are sufficiently light, they oscillate in time, and can lead to
temporal variations in various constants of Nature. We are interested in the
possibility that the ULDM field couples to neutrinos, thus affecting their
masses and mixing [5, 6, 7, 8, 9, 10, 11, 12].

A scalar with mass $m_{\phi}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}{\rm eV}$ can be treated as a
classical bosonic field that oscillates with time,

$\displaystyle\phi=\phi_{0}\sin(m_{\phi}t+\varphi)\,,$ (1)

with initial phase $\varphi$. If it is to account for the dark matter (DM),
then

$\displaystyle\phi_{0}=\frac{\sqrt{2\rho_{\phi}^{\oplus}}}{m_{\phi}}\sim 2\
{\rm GeV}\ \left(\frac{10^{-12}\ {\rm eV}}{m_{\phi}}\right),$ (2)

where $\rho_{\phi}^{\oplus}$ corresponds to the ULDM density at the surface of
Earth, which we assume to match the DM density at the solar position given in
[13]. The oscillation period is given by

$\displaystyle\tau_{\phi}=\frac{2\pi}{m_{\phi}}\approx 4\times 10^{-3}\ {\rm
sec}\ \left(\frac{10^{-12}\ {\rm eV}}{m_{\phi}}\right)\,.$ (3)

In a previous work [10], we considered the range of $m_{\phi}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{-12}\ {\rm eV}$. For this mass range, $\tau_{\phi}$ is
larger than $\tau_{d}(=L)$, the source-to-detector distance, in which case the
effective neutrino masses and mixing do not change along the propagation from
the source to the detector. In this work, we consider the range

$\displaystyle m_{\phi}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{-12}\ {\rm eV},$ (4)

leading to $\tau_{\phi}<\tau_{d}$. This case opens the door to the possibility
of parametric resonance, which is the main focus of this paper.

Consider the effective mass and $\phi$-Yukawa terms for the neutrinos, in the
neutrino mass basis, arising from dimension-five and dimension-six terms in
the Lagrangian, respectively:

$\displaystyle{\cal
L}_{m_{\nu}}=m_{i}\nu_{i}^{T}\nu_{i}+\hat{y}_{ij}\phi\nu_{i}^{T}\nu_{j}\,.$
(5)

Treating $\phi$ as a classical field, it modifies the neutrino mass matrix as

$\displaystyle(\hat{m}_{\nu})_{ij}=m_{i}\delta_{ij}+\hat{y}_{ij}\phi\,.$ (6)

We are mainly interested in the off-diagonal $\hat{y}$ entries. We assume that
the $\phi$-dependent modifications are small, and define small parameters
$\epsilon_{ij}$ given by

$\displaystyle\epsilon_{ij}\equiv\frac{2\hat{y}_{ij}\phi_{0}}{m_{i}-m_{j}}.$
(7)

For the two generation case discussed below, we denote the small parameter by
$\epsilon_{\phi}$. In what follows, we show that a parametric resonance, that
is an ${\cal O}(1)$ modification of the neutrino oscillation probability, can
occur even for very small $\epsilon_{ij}$, as their effects are enhanced in
the $m_{\phi}L\gg 1$ limit. Such large contributions are generated when the
oscillation frequency of the neutrinos,

$\displaystyle\omega_{E}\equiv\Delta m^{2}_{ij}/(4E),$ (8)

(where $\Delta m^{2}_{ij}\equiv m_{i}^{2}-m_{j}^{2}$), matches half the
oscillation frequency of the scalar field,

$\displaystyle\omega_{R}\equiv m_{\phi}/2.$ (9)

The plan of this paper is as follows. In Section 2, we present our theoretical
framework, develop our formalism, and explain the conditions for and features
of the parametric resonance. In Section 3, we study the implications for
neutrino experiments. We argue that a feature in the spectrum measured by
KamLAND, if not a statistical fluctuation, can be accounted for by a
parametric resonance due to a ULDM field, and moreover that the JUNO
experiment will test this scenario. We summarize our conclusions in Section 4.

##  2 The theoretical framework

###  2.1 Two neutrino model

To understand the basic features of the ULDM-induced parametric resonance in
neutrino oscillations, we first study a two neutrino generations model. Since
we are interested in a resonance effect, the periodic perturbation induced by
the ULDM field should couple the two states of the system. We thus consider
the following Yukawa matrix in the neutrino mass basis:

$\displaystyle\hat{y}=y\begin{pmatrix}0&1\cr 1&0\end{pmatrix},$ (10)

with $y$ real. Two comments are in order:

  * •

Diagonal entries in $\hat{y}$ cause an effect of ${\cal
O}(\epsilon_{\phi}^{2})$ in the time-averaged neutrino oscillation
probability. (By “time-averaged” we mean an average of the oscillation
probability over the initial phase $\varphi$ of the scalar field in Eq. (1).)
Since this is a minuscule effect compared to the potential ${\cal O}(1)$
effect of the resonance, we set the diagonal $\hat{y}$ terms to zero.

  * •

If $y$ were complex, we could absorb its phase in the phase $\varphi$ (see
Appendix A). It will then have no effect on the time averaged transition
probability.

Considering the above, and the fact that $\hat{y}$ is symmetric, the matrix in
Eq. (10) is the most general matrix needed for this study.

The equations of motion for the neutrinos are given by

$\displaystyle
i\partial_{t}\begin{pmatrix}\nu_{\alpha}\cr\nu_{\beta}\end{pmatrix}={\cal
H}\begin{pmatrix}\nu_{\alpha}\cr\nu_{\beta}\end{pmatrix},$ (11)

where the effective Hamiltonian in the unperturbed mass basis is given by

$\displaystyle{\cal
H}=\frac{1}{2E}(m_{\nu}+\hat{y}\phi)^{\dagger}(m_{\nu}+\hat{y}\phi).$ (12)

For our choice of $\hat{y}$, and omitting terms proportional to the unit
matrix, which do not affect neutrino oscillations, we obtain

$\displaystyle{\cal
H}=\omega_{E}[\sigma_{z}+\epsilon_{\phi}\sin(m_{\phi}t+\varphi)\sigma_{x}],$
(13)

where $\sigma$ are the Pauli matrices.

We label the unperturbed neutrino mass eigenstates as $\nu^{0}_{1,2}$. Their
time evolution is given by

$\displaystyle\nu^{0}_{1,2}(t)=e^{\mp i\omega_{E}t}\nu^{0}_{1,2}(t=0).$ (14)

In the presence of the perturbation, the time evolution is given by

$\displaystyle\nu_{1,2}(t)=c_{1,2}(t)\nu^{0}_{1,2}(t),$ (15)

where $c_{1,2}$ obey the following equations:

$\displaystyle
i\partial_{t}c_{1,2}=\frac{\omega_{E}}{2i}\epsilon_{\phi}\left[e^{i(m_{\phi}\pm
2\omega_{E})t+i\varphi}-e^{-i(m_{\phi}\mp
2\omega_{E})t-i\varphi}\right]c_{2,1}.$ (16)

We further define a detuning parameter,

$\displaystyle\delta_{E}\equiv\frac{m_{\phi}-2\omega_{E}}{m_{\phi}}=\frac{\omega_{R}-\omega_{E}}{\omega_{R}}\,,$
(17)

and rewrite Eq. (16) in terms of $\delta_{E}$:

$\displaystyle
i\partial_{t}c_{1,2}=\frac{\omega_{E}}{2i}\epsilon_{\phi}\left\\{e^{im_{\phi}[1\pm(1-\delta_{E})]t+i\varphi}-e^{-im_{\phi}[1\mp(1-\delta_{E})]t-i\varphi}\right\\}c_{2,1}.$
(18)

For $\delta_{E}\ll 1$, we find a term that oscillates with frequency
$\delta_{E}m_{\phi}$ and a term that oscillates with frequency $2m_{\phi}$. If
the typical time scale for neutrino propagation satisfies

$\displaystyle\tau_{d}\gg\tau_{\phi},$ (19)

then the rapidly oscillating term can be neglected. This is known as the
“rotating wave approximation”, and it leads to a simplified equation of motion

$\displaystyle
i\partial_{t}c_{1,2}=\mp\frac{\omega_{E}}{2i}\epsilon_{\phi}e^{\mp
i\left(m_{\phi}\delta_{E}t+\varphi\right)}c_{2,1}\,,$ (20)

that can be solved analytically. Then, in the vicinity of the resonance, we
obtain the probability for an interaction eigenstate $\nu_{\beta}$ produced at
time $t$ to be measured as the interaction eigenstate $\nu_{\alpha}$ after
propagating a distance $L$,

$\displaystyle P_{\alpha\beta}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\left\\{\sin\left(\frac{\omega_{E}L}{2}\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\right)\left[\epsilon_{\phi}c_{2\theta}\sin\left(\varphi+2\omega_{R}t+\omega_{R}L\right)-2\delta_{E}s_{2\theta}\cos(\omega_{R}L)\right]\right.$
(21) $\displaystyle+$
$\displaystyle\left.\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}s_{2\theta}\sin(\omega_{R}L)\cos\left(\frac{\omega_{E}L}{2}\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\right)\right\\}^{2}$
$\displaystyle+$
$\displaystyle\frac{\epsilon_{\phi}^{2}}{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\cos^{2}\left(\varphi+2\omega_{R}t+\omega_{R}L\right)\sin^{2}\left(\frac{\omega_{E}L}{2}\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\right),$

where $s_{2\theta}\equiv\sin 2\theta$, $c_{2\theta}\equiv\cos 2\theta$, and
$\theta$ is the rotation angle from the interaction basis (that is also the
charged lepton mass basis) to the unperturbed neutrino mass basis. When
deriving the transition probability measured by experiments, where temporal
dependence is not considered, we have to average the above expression over the
initial ULDM phase $\varphi$. This would also be the case if the events are
time-stamped, but the ULDM oscillation time or coherence time are shorter than
the experimental resolution.

The transition probability exhibits a resonance at $\omega_{E}=\omega_{R}$,
which could lead to $\mathcal{O}\left(1\right)$ modifications to the standard
transition probability when $m_{\phi}L\epsilon_{\phi}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}4$.
This resonance is analogous to the parametric resonance in classical
mechanics, which is obtained when the frequency of the perturbation is twice
the oscillator frequency. Note that our system has two small parameters:
$\epsilon_{\phi}$ and $(m_{\phi}L)^{-1}$, and the transition probability
depends on the ratio $\epsilon_{\phi}/(m_{\phi}L)^{-1}$, which could be much
larger than $\epsilon_{\phi}$.

The resonance corresponds to a situation where each interaction of the
neutrinos with the scalar results in an exchange of the neutrino mass
eigenstate, with corresponding absorption or emission of energy quanta
$m_{\phi}$. The $\nu_{1}\leftrightarrow\nu_{2}$ transition would be most
efficient when energy conservation is satisfied. This is possible for any
$N=2n+1$ interactions with the scalar, leading to a resonance at
$\omega_{E}=(N/2)m_{\phi}$. However, a large $N$ resonance comes at the cost
of suppressed effective coupling $\epsilon_{N}$ [14]:

$\displaystyle\frac{\epsilon_{N}}{\epsilon_{\phi}}\approx\frac{(\epsilon_{\phi}N)^{N-1}}{2^{2N-2}(\frac{N-1}{2}!)^{2}}.$
(22)

####  2.1.1 $\theta=0$

Various features of the resonance are simplest to obtain in a two generation
model with zero mixing, $\theta=0$. With the initial condition

$\displaystyle\begin{pmatrix}\nu_{\alpha}\cr\nu_{\beta}\end{pmatrix}(t=0)=\begin{pmatrix}0\cr
1\end{pmatrix},$ (23)

the transition probability in vacuum is zero, as the neutrinos do not mix. The
interaction with the ULDM field, however, does allow transitions. For
$\theta=0$, we obtain the well-known Rabi oscillations probability111Rabi
oscillations would occur also for oscillatory matter profiles, see Ref. [15]
and references therein.:

$\displaystyle\langle
P_{\alpha\beta}\rangle_{\varphi}=\frac{\epsilon_{\phi}^{2}(1-\delta_{E})^{2}}{\epsilon_{\phi}^{2}(1-\delta_{E})^{2}+4\delta_{E}^{2}}\sin^{2}\left(\frac{m_{\phi}L}{4}\sqrt{\epsilon_{\phi}^{2}(1-\delta_{E})^{2}+4\delta_{E}^{2}}\right).$
(24)

First, we examine the height of the resonance. At the resonance energy
$E_{R}=\Delta m^{2}/(2m_{\phi})$, corresponding to $\omega_{E}=m_{\phi}/2$ and
$\delta_{E}=0$, we have for the phase-averaged transition probability,

$\displaystyle
P_{\alpha\beta}(E_{R})=\sin^{2}\left(\frac{\epsilon_{\phi}m_{\phi}L}{4}\right).$
(25)

(Here and below, we often omit the $\langle\ldots\rangle_{\phi}$ notation for
$P_{\alpha\beta}$.) We can distinguish two possible regions:

  * •

For $m_{\phi}L\epsilon_{\phi}/4\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1$, an ${\cal O}(1)$ transition
probability might be induced, in contrast to the zero probability in vacuum.

  * •

For $m_{\phi}L\epsilon_{\phi}/4\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1$, we have
$P_{\alpha\beta}(E_{R})\approx(\epsilon_{\phi}m_{\phi}L/4)^{2}$, so that even
for very small $\epsilon_{\phi}$, the effect can be enhanced by $m_{\phi}L\gg
1$.

Next, we examine the shape of the probability around the resonance. Note that
moving away from the resonance to higher energies, $E>E_{R}$, implies
$0\leq\delta_{E}\leq 1$, while moving away to lower energies, $E<E_{R}$,
implies $-\infty\leq\delta_{E}<0$. We distinguish two limits:

$\displaystyle
P_{\alpha\beta}\approx\left\\{\begin{matrix}\frac{\epsilon_{\phi}^{2}(1-\delta_{E})^{2}m_{\phi}^{2}L^{2}}{16}&\frac{m_{\phi}L}{4}\sqrt{4\delta_{E}^{2}+\epsilon_{\phi}^{2}(1-\delta_{E})^{2}}\ll
1\cr\frac{\epsilon_{\phi}^{2}}{\epsilon_{\phi}^{2}+(\frac{2\delta_{E}}{1-\delta_{E}})^{2}}&\frac{m_{\phi}L}{4}\sqrt{4\delta_{E}^{2}+\epsilon_{\phi}^{2}(1-\delta_{E})^{2}}\gg
1\end{matrix}\right..$ (26)

We obtain the width of the resonance:

$\displaystyle\frac{|\delta
E|}{E_{R}}=\frac{|\delta\omega_{E}|}{|\omega_{R}|}\sim\left\\{\begin{matrix}\epsilon_{\phi}&\epsilon_{\phi}m_{\phi}L\gg
1\cr 1/(m_{\phi}L)&\epsilon_{\phi}m_{\phi}L\ll 1\end{matrix}\right..$ (27)

The features of the resonance are demonstrated in Fig. 1.

Figure 1: $P_{\alpha\beta}(\delta_{E})$: (Left) $\epsilon_{\phi}m_{\phi}L\gg
1$: The height of the resonance is ${\cal O}(1)$, and the width is
$\epsilon_{\phi}$. (Right) $\epsilon_{\phi}m_{\phi}L\ll 1$: The height of the
resonance is $(\epsilon_{\phi}m_{\phi}L)^{2}/16$, and the width is
$1/(m_{\phi}L)$.

####  2.1.2 $\theta\neq 0$

Upon averaging over the initial phase $\varphi$, we obtain

$\displaystyle\langle P_{\alpha\beta}\rangle_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{s_{2\theta}^{2}}{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\left[-2\delta_{E}\sin\left(\frac{\omega_{E}L}{2}\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\right)\cos(\omega_{R}L)+\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\sin(\omega_{R}L)\cos\left(\frac{\omega_{E}L}{2}\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\right)\right]^{2}$
(28) $\displaystyle+$
$\displaystyle\frac{\epsilon_{\phi}^{2}}{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\frac{1+c_{2\theta}^{2}}{2}\sin^{2}\left(\frac{\omega_{E}L}{2}\sqrt{\epsilon_{\phi}^{2}+4\delta_{E}^{2}}\right).$

Given that the vacuum transition probability at $E=E_{R}$ is given by

$\displaystyle
P^{0}_{\alpha\beta}(E_{R})=s_{2\theta}^{2}\sin^{2}(\omega_{R}L),$ (29)

we obtain:

$\displaystyle
P_{\alpha\beta}(E_{R})=P^{0}_{\alpha\beta}(E_{R})+\sin^{2}\left(\frac{\epsilon_{\phi}m_{\phi}L}{4}\right)\left[1-\frac{1}{2}s_{2\theta}^{2}-P^{0}_{\alpha\beta}(E_{R})\right].$
(30)

Thus, the transition probability at the resonance is enhanced compared to the
vacuum transition probability if
$\sin^{2}(m_{\phi}L/2)<(1/s_{2\theta}^{2})-(1/2)$ and reduced if
$\sin^{2}(m_{\phi}L/2)>(1/s_{2\theta}^{2})-(1/2)$.

###  2.2 Three neutrino model

We consider the three neutrinos case, where the $\phi$-field couples off-
diagonally to only two mass eigenstates. Explicitly, we take $\hat{y}$ in the
neutrino vacuum mass basis of the following form:

$\displaystyle\hat{y}=y\begin{pmatrix}0&0&1\cr 0&0&0\cr 1&0&0\end{pmatrix}\,.$
(31)

In this model, $\nu_{2}$ evolves independently of $\nu_{1}$ and $\nu_{3}$.
Thus, the former approximately follows vacuum evolution, while for the latter
two we can apply our two generations analysis. The final result for the
$\varphi$-averaged survival probability is given by

$\displaystyle\langle P_{\alpha\alpha}\rangle_{\varphi}$
$\displaystyle=\left\\{2(|U_{\alpha 1}|^{2}+|U_{\alpha
3}|^{2})\left[s^{2}_{\epsilon}\cos\left(\frac{m_{\phi}(1-\eta_{E})}{2}L\right)+c^{2}_{\epsilon}\cos\left(\frac{m_{\phi}(1+\eta_{E})}{2}L\right)\right]+|U_{\alpha
2}|^{2}\cos\left(\frac{(\Delta m^{2}_{21}+\Delta
m^{2}_{23})L}{4E}\right)\right\\}^{2}$ $\displaystyle+\left\\{2(|U_{\alpha
1}|^{2}-|U_{\alpha
3}|^{2})\left[c^{2}_{\epsilon}\cos\left(\frac{m_{\phi}(1-\eta_{E})}{2}L\right)-s^{2}_{\epsilon}\cos\left(\frac{m_{\phi}(1+\eta_{E})}{2}L\right)\right]+|U_{\alpha
2}|^{2}\sin\left(\frac{(\Delta m^{2}_{21}+\Delta
m^{2}_{23})L}{4E}\right)\right\\}^{2}$ $\displaystyle+8|U_{\alpha
1}|^{2}|U_{\alpha
3}|^{2}s^{2}_{\epsilon}c^{2}_{\epsilon}\sin^{2}\left(\frac{m_{\phi}\eta_{E}}{2}L\right),$
(32)

where $s_{\epsilon}\equiv\sin\theta_{\epsilon}$,
$c_{\epsilon}\equiv\cos\theta_{\epsilon}$,

$\displaystyle\tan 2\theta_{\epsilon}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon_{31}(1-\delta_{E})}{2\delta_{E}},$
$\displaystyle\eta_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\epsilon_{31}^{2}(1-\delta_{E})^{2}+4\delta_{E}^{2}}\,,$
(33)

and $U_{\alpha i}$ are the entries of the PMNS matrix, which is the transition
matrix from the unperturbed neutrino mass basis to the interaction basis.

At the resonance neutrino energy $E_{R}=\Delta m^{2}_{31}/(2m_{\phi})$,
implying $\sin 2\theta_{\epsilon}=-1$ and $\eta_{E}=\epsilon_{31}/2$, the
survival probability is given by

$\displaystyle P_{\alpha\alpha}(E_{R})$ $\displaystyle=$
$\displaystyle\left[2(|U_{\alpha 1}|^{2}+|U_{\alpha
3}|^{2})\cos\frac{\epsilon_{31}m_{\phi}L}{4}\cos\frac{m_{\phi}L}{2}+|U_{\alpha
2}|^{2}\cos\frac{(\Delta m^{2}_{21}+\Delta m^{2}_{23})L}{4E}\right]^{2}$ (34)
$\displaystyle+$ $\displaystyle\left[2(|U_{\alpha 1}|^{2}-|U_{\alpha
3}|^{2})\sin\frac{\epsilon_{31}m_{\phi}L}{4}\sin\frac{m_{\phi}L}{2}+|U_{\alpha
2}|^{2}\sin\frac{(\Delta m^{2}_{21}+\Delta m^{2}_{23})L}{4E}\right]^{2}$
$\displaystyle+$ $\displaystyle 2|U_{\alpha 1}|^{2}|U_{\alpha
3}|^{2}\sin^{2}\frac{\epsilon_{31}m_{\phi}L}{4}.$

Our main interest concerns $P_{ee}$. Making the approximation $|U_{e3}|=0$, we
find, at the resonance,

$\displaystyle P_{ee}(E_{R})$ $\displaystyle\approx$ $\displaystyle
1-\sin^{2}2\theta_{12}\sin^{2}\left(\frac{\Delta m^{2}_{21}t}{4E}\right)$ (35)
$\displaystyle-$
$\displaystyle\left[\cos^{4}\theta_{12}\sin^{2}\left(\frac{\epsilon_{31}\Delta
m^{2}_{31}t}{8E}\right)+\sin^{2}2\theta_{12}\cos\left(\frac{\Delta
m^{2}_{21}t}{2E}\right)\sin^{2}\left(\frac{\epsilon_{31}\Delta
m^{2}_{31}t}{16E}\right)\right].$

##  3 Implications for experiments

In order to obtain good sensitivity to the neutrino oscillation parameters,
experiments where both the source and the detector are terrestrial are
designed with energy spectrum and baseline length such that $\Delta
m^{2}_{ij}L/(4E)={\cal O}(1)$ for some $\Delta m^{2}_{ij}$. A ULDM field
coupling these $\nu_{i}$ and $\nu_{j}$, and which is of mass $m_{\phi}=\Delta
m^{2}_{ij}/(2E_{R})$, where $E_{R}$ falls within the spectrum of the
experiment, will not undergo many oscillations during the neutrino
propagation. In this case, an $\mathcal{O}(1)$ modification of the oscillation
probability would require $\epsilon_{ij}\sim 1/(m_{\phi}L)\geq\mathcal{O}(1)$.
Furthermore, the width of the resonance, $\Delta E/E_{R}\gtrsim 1/(m_{\phi}L)$
will be at least of ${\cal O}(1)$, and thus spread over many energy bins
(assuming energy resolution of a few percent). Given measurements of a
neutrino oscillation experiment, one can derive bounds on the parameter space
in such scenario, which we leave for future work.

Nature has provided us, however, with a hierarchy of neutrino mass-squared
differences, $\Delta m^{2}_{21}\ll\Delta m^{2}_{31}\sim\Delta m^{2}_{32}$.
Consequently, a ULDM field of mass $m_{\phi}=\Delta m^{2}_{31}/(2E_{R})$ would
oscillate many times during the neutrino propagation in an experiment where
$\Delta m^{2}_{21}/(4E_{R})={\cal O}(1)$, such that a significant resonance
effect can be generated with $\epsilon_{13}\ll 1$. Furthermore, the width of
the resonance would be rather narrow,

$\displaystyle\frac{\Delta E}{E_{R}}\sim\frac{2E_{R}}{\Delta
m^{2}_{31}L}\sim\frac{\Delta m^{2}_{21}}{\Delta m^{2}_{31}}\approx 0.03,$ (36)

thus affecting a single, or very few adjacent energy bins.

We are thus led to consider, on the experimental side, the KamLAND experiment
and, on the theoretical side, the three neutrino model of Eq. (31).
($\hat{y}_{23}\neq 0$ gives a weaker improvement in the fit to KamLAND
compared to $\hat{y}_{13}\neq 0$ and, furthermore, might have additional
implications.)

###  3.1 KamLAND

KamLAND is an experiment designed to be sensitive to $\Delta m^{2}_{21}$ and
$\sin\theta_{12}$. The sources of the neutrinos are 53 nuclear reactors, with
an average flux-weighted distance of 180 km to the detector. We study the
sensitivity of KamLAND to the resonance effects by taking the following steps:

  1. 1.

Calculate the expected number of unoscillated $e$-events using the fluxes of
the nearest 21 reactors.

  2. 2.

Calculate the energy dependent survival probability for each reactor for a set
of values of $(m_{\phi},\hat{y}_{13})$.

  3. 3.

Calculate the expected number of oscillated $e$-events and obtaining the
overall survival probability $P_{\rm calc}(E_{i},m_{\phi},\hat{y}_{13})$.

  4. 4.

Perform a $\chi^{2}$ test,

$\displaystyle\chi^{2}(m_{\phi},\hat{y}_{13})=\sum_{i={\rm
bins}}\left[\frac{P_{\rm calc}(E_{i},m_{\phi},\hat{y}_{13})-P_{\rm
meas}(E_{i})}{\sigma(E_{i})}\right]^{2},$ (37)

where $P_{\rm meas}(E_{i})$ and $\sigma(E_{i})$ are the experimental values
provided by KamLAND [16]. In our calculation, we assumed the unperturbed
vacuum neutrino parameters to be
$\theta_{12}=0.606\,,\theta_{23}=0.83\,,\theta_{13}=0.1\,,\Delta
m^{2}_{21}=7.54\times 10^{-5}~{}\rm{eV}^{2}$ and $\Delta m^{2}_{32}=2.5\times
10^{-3}~{}\rm{eV}^{2}$. The standard matter propagation effects were taken
into account by using the matter-modified mixing angle $\theta_{12M}$ and mass
splitting $\Delta m^{2}_{21M}$ given in Eq. (3) and Eq. (4) of [16].

Our results are presented in Fig. 2. The blue curve corresponds to standard
neutrino oscillations in matter, produced using the fluxes and distances of
the nearest 21 reactors (see [10]), and is consistent with KamLAND’s result.
Including the ULDM, the best fit values of the parameters are

$\displaystyle m_{\phi}$ $\displaystyle=$ $\displaystyle 3.6\times 10^{-10}\
{\rm eV},$ $\displaystyle\hat{y}_{13}$ $\displaystyle=$ $\displaystyle 8\times
10^{-11}\ \frac{\sqrt{\Delta m^{2}_{31}}}{m_{3}+m_{1}}\sqrt{\frac{\rho_{\rm
DM}}{\rho^{\oplus}_{\rm DM}}}\,,$ (38)

where $\rho_{\rm DM}$ is the measured dark matter density from [8]. The most
prominent effect is the suppression of $P_{ee}$ in the bin around $E_{R}\sim
3.6\ {\rm MeV}$. This feature corresponds to the deviation near 50 km/MeV in
Fig. 5 of the KamLAND paper [16].222In the journal version of Ref. [16], the
following statement is made concerning this bin: “We have inspected the
apparent deviation near 50 km/MeV for systematic effects, and find none; it is
statistical in nature, and disappears with a different choice of binning”.
Indeed, the parametric resonance in the fit with the ULDM field is narrow,
$|\delta E|/E_{R}\sim 0.02$, and the consequent deficit would not be apparent
with a choice of larger bins. The fit using the above values for the
parameters is approximately $3.5\sigma$ better than standard matter
oscillations. The significance will, however, decrease when the look elsewhere
effect [17] is taken into account.

Figure 2: The KamLAND experiment: (Left) $P_{ee}(L/E)$: The experimental
results (black), the best fit of standard oscillations in matter (blue), and
the best fit for a ULDM field of $m_{\phi}=3.6\times 10^{-10}$ eV and
$\hat{y}_{13}=8\times 10^{-11}$ (red). Shown are both the binned (solid) and
the unbinned (dashed) probabilities. (Right) $\sqrt{\Delta\chi^{2}}$ for a
ULDM field of mass $m_{\phi}$, and coupling $\hat{y}_{13}$ , compared to the
minimum, which corresponds to $m_{\phi}=3.6\times 10^{-10}$ eV and
$\hat{y}_{13}=8\times 10^{-11}$.

Taking $m_{1}=0$ and $\rho^{\oplus}_{\rm DM}=\rho_{\rm DM}$, the best fit
values correspond to

$\displaystyle\epsilon_{31}=0.019,\ \ \ \epsilon_{31}m_{\phi}L/4=1.54.$ (39)

###  3.2 JUNO

JUNO is an experiment designed to be sensitive to both $\Delta m^{2}_{21}$ and
$\Delta m^{2}_{31}$, in order to determine the neutrino mass hierarchy [18].
The sources of the neutrinos are two reactors located at a similar distance of
53 km. We now study the effect that a ULDM field with the parameters of Eq.
(3.1), corresponding to the best fit to the KamLAND measurements, would have
on the survival probability measured by JUNO. The modification to the vacuum
oscillation probability is shown in Fig. 3. As expected, $P_{ee}$ is modified
around $E_{R}\sim 3.6\ {\rm MeV}$.

The high statistics and good energy resolution of JUNO will enable the
experiment to split the spectrum in the range of $2-8$ MeV into 200 energy
bins. We performed a $\chi^{2}$ analysis to estimate the sensitivity of JUNO
to the resonance effect, using

$\displaystyle\Delta\chi^{2}(m_{\phi},\hat{y}_{13})$ $\displaystyle=$
$\displaystyle\sum_{i={\rm bins}}\left[\frac{P_{\rm
uldm}(E_{i},m_{\phi},\hat{y}_{13})-P_{\rm
vac}(E_{i})}{\sigma(E_{i})}\right]^{2},$ $\displaystyle\sigma(E_{i})$
$\displaystyle=$ $\displaystyle\sqrt{\frac{P(E_{i})[1-P(E_{i})]}{N(E_{i})}},$
(40)

where $N(E_{i})$ is the expected number of unoscillated events in the $i$’th
bin.

Our results are presented in Fig. 3. If the best fit values for KamLAND, Eq.
(3.1), are realized in nature, the deviation from the standard matter
oscillation prediction will be signaled with a $3.5\sigma$ significance after
$60$ days of collecting data. It is also important to note that JUNO could
constrain other values of the UDLM parameters.

Figure 3:  The JUNO experiment: (Left) $P_{ee}(E)$: The expected result from
standard oscillations (blue), and the expected result for a ULDM field of
$m_{\phi}=3.6\times 10^{-10}$ eV and $\hat{y}_{13}=8\times 10^{-11}$ (red).
(Right): Expected $\sqrt{\Delta\chi^{2}}$ between vacuum oscillations and
oscillations in a ULDM field, assuming 60 days of data taking. KamLAND best
fit parameters are marked by a black dot.

###  3.3 Other experiments

We studied whether a ULDM field with the parameters of Eq. (3.1) will modify
the measurements of other neutrino experiments. Our conclusion, that currently
only JUNO can test this scenario, is based on the following considerations:

  * •

Solar neutrino experiments are unaffected because of our choice of
$\hat{y}_{i2}=\hat{y}_{2i}=0$, meaning that the ULDM field does not affect the
propagation of $\nu_{2}$.

  * •

Atmospheric neutrino experiments are unaffected because $E^{\rm
atm}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}25E_{R}^{\rm kam}$ and $L^{\rm atm}=(0.03-250)L^{\rm kam}$. A
resonant effect in both KamLAND and atmospheric neutrino experiments can only
occur if the one at KamLAND arises from a high harmonic, $N\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}25$.
Eq. (22) implies then a highly suppressed $\epsilon_{N}$ at KamLAND, with
baseline distance that is not large enough to generate a significant effect.

  * •

Accelerator neutrino experiments are unaffected because $E^{\rm
acc}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{3}E_{R}^{\rm kam}$, and the same argument as for the
atmospheric neutrinos holds.

  * •

The Daya Bay experiment, which has an energy spectrum similar to KamLAND, will
not be sensitive to the $\phi$-field because of its much shorter baseline,
$L^{\rm DB}\sim 1\ {\rm km}$.

Of course, it could be that the deficit around $E\sim 3.6\ {\rm MeV}$ in
KamLAND would turn out to be a statistical fluctuation, and a ULDM field
exists with parameters different from Eq. (3.1). We leave a study of the
sensitivity of neutrino experiments to other ranges of $\phi$-field parameters
to future work.

##  4 Conclusions

An ULDM field $\phi$ with a mass $m_{\phi}$ greater than $\sim 10^{-12}$ eV
would oscillate with a period $\tau_{\phi}$ shorter than a few milliseconds.
If it couples to neutrinos, this time scale is relevant to terrestrial
neutrino propagation, which is observed over baselines $L$ from tens to
thousands of km. We have shown that if the ULDM oscillates many times during
neutrino propagation i.e. $m_{\phi}L\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1$, a parametric
resonance effect could be generated if a neutrino energy value of
$E_{R}=\Delta m^{2}_{ij}/(2m_{\phi})$ falls within the window of sensitivity
for a given experiment. We have further shown that the contribution of the
ULDM field to the neutrino oscillation probability scales with
$\sin^{2}\left(\epsilon_{ij}m_{\phi}L/4\right)$, where $\epsilon_{ij}$ denotes
the ratio between the contribution of the ULDM field to the neutrino mass to
the vacuum value of the neutrino mass. This means that a resonant enhancement
of the ULDM effects would occur even for very small $\epsilon_{ij}$ values.

Given that $\Delta m^{2}_{31}\sim 10^{-5}\ {\rm eV}^{2}$ and assuming
$m_{\phi}\sim 10^{-10}\ {\rm eV}$, the effect would be most relevant for
$E\sim$ a few MeV. Indeed, we find that the long baseline reactor experiment
KamLAND is sensitive to this type of effects. In fact, KamLAND has measured a
significant deficit of events in one of its energy bins, around 3.6 MeV. Such
a deficit, if not a statistical fluctuation, can be accounted for by a ULDM
field of mass $m_{\phi}\sim 3.6\times 10^{-10}\ {\rm eV}$, and with effective
coupling to $\nu_{1}\nu_{3}$ of order $\hat{y}_{13}\sim 10^{-10}$. This
scenario improves the fit compared to standard matter oscillations with a
local significance $\sim 3.5\sigma$.

A future test of this scenario is expected from the medium baseline reactor
experiment JUNO. A ULDM field with parameters fitted to the KamLAND
measurement is expected to give a $3.5\sigma$ deficit of events, compared to
standard matter oscillations, in the energy bin around 3.6 MeV for 60 days of
collected data.

We expect other neutrino experiments to be sensitive to other regions in the
ULDM parameter space ($m_{\phi},\hat{y}_{ij}$). We leave a detailed study of
these constraints to future work.

### Acknowledgements

We thank M.C. Gonzalez Garcia for fruitful discussions regarding the KamLAND
results. YN is the Amos de-Shalit chair of theoretical physics, and is
supported by grants from the Israel Science Foundation (grant number 1124/20),
the United States-Israel Binational Science Foundation (BSF), Jerusalem,
Israel (grant number 2018257), by the Minerva Foundation (with funding from
the Federal Ministry for Education and Research), and by the Yeda-Sela (YeS)
Center for Basic Research. The work of GP is supported by grants from BSF-NSF,
Friedrich Wilhelm Bessel research award, GIF, ISF, Minerva, SABRA - Yeda-Sela
- WRC Program, the Estate of Emile Mimran, and The Maurice and Vivienne Wohl
Endowment. IS is supported by a fellowship from the Ariane de Rothschild Women
Doctoral Program.

##  Appendix A Resonance Equations of Motion for a Complex Coupling

We consider the following Yukawa matrix in the neutrino mass basis in the two-
generations picture:

$\displaystyle\hat{y}=y\begin{pmatrix}0&1\cr 1&0\end{pmatrix},$ (41)

where $y$ is complex. Omitting terms proportional to the unit matrix, which do
not affect neutrino oscillations, we obtain

$\displaystyle{\cal
H}=2\omega_{E}[\sigma_{z}+\left(\epsilon^{R}_{\phi}\sigma_{x}+\epsilon^{I}_{\phi}\sigma_{y}\right)\sin(m_{\phi}t+\varphi)],$
(42)

where $\sigma$ are the Pauli matrices and

$\displaystyle\epsilon^{R}_{\phi}\equiv\frac{2\phi_{0}\text{Re}\left(y\right)}{m_{1}-m_{2}}\qquad$
$\displaystyle\,,\qquad\epsilon^{I}_{\phi}\equiv\frac{2\phi_{0}\text{Im}\left(y\right)}{m_{1}+m_{2}}\,.$
(43)

Following the same prescription from the main text, Eqs. (14)-(16), we find
the following equations for $c_{1,2}$,

$\displaystyle
i\partial_{t}c_{1,2}=\frac{\omega_{E}}{2i}\left(\epsilon^{R}_{\phi}\pm
i\epsilon^{I}_{\phi}\right)\left[e^{i(m_{\phi}\pm
2\omega_{E})t+i\varphi}-e^{-i(m_{\phi}\mp
2\omega_{E})t-i\varphi}\right]c_{2,1}\,,$ (44)

which in the rotating wave approximation, keeping only the slowly oscillating
term, yield

$\displaystyle i\partial_{t}c_{1,2}=\pm
i\frac{\omega_{E}}{2}r_{\epsilon}e^{\mp
i\left(m_{\phi}\delta_{E}t+\varphi-\varphi_{\epsilon}\right)}c_{2,1}\,,$ (45)

where
$r_{\epsilon}\equiv\sqrt{{\epsilon^{R}_{\phi}}^{2}+{\epsilon^{I}_{\phi}}^{2}}$
and
$\varphi_{\epsilon}\equiv\arctan\left(\epsilon^{I}_{\phi}/\epsilon^{R}_{\phi}\right)$.
Therefore, in the proximity of the resonance, the phase $\varphi_{\epsilon}$
can be absorbed into the ULDM phase $\varphi$, and will thus have no
observable effect in the time-averaged case.

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  *[ULDM]: ultra-light dark matter
  *[DM]: dark matter


We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $PGL(3,\mathbb{F}_q[t])$ quotient of $\widetilde{A}_2$-type building. We prove that the set of non-trivial approximate eigenvalues $(\lambda^+,\lambda^-)$ of the weighted adjacency operators $A_w^\pm$ on the quotient induced from the colored adjacency operators $A^\pm$ on the building for $PGL_3$ contains the simultaneous spectrum of $A^\pm$ and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))/PGL(3,\mathbb{F}_q[\![t^{-1}]\!])$ is not a Ramanujan complex, from a combinatorial aspect.

Soonki Hong
Soonki Hong
Department of Mathematical Education
Catholic Kwandong University
Gangneung 25601
Republic of Korea

Sanghoon Kwon
Sanghoon Kwon*
Room 506 Department of Mathematical Education
Catholic Kwandong University
Gangneung 25601
Republic of Korea

2020 Mathematics Subject Classification. Primary 20E42, 20G25; Secondary 47A25

§ INTRODUCTION

A finite $k$-regular graph $X$ is called a Ramanujan graph if for every eigenvalue $\lambda$ of the adjacency matrix $A_X$ of $X$ satisfies either $\lambda=\pm k$ or $|\lambda|\le 2\sqrt{k-1}$. An eigenvalue $\lambda$ is called trivial if $\lambda=\pm k$. Since the interval $[-2\sqrt{k-1},2\sqrt{k-1}]$ is equal to the spectrum $\mathcal{S}^2$ of the adjacency operator of $k$-regular tree $\mathcal{T}_k$, we note that a finite $k$-regular graph $X$ is Ramanujan if and only if every non-trivial spectrum of $A_X$ is contained in the spectrum $\mathcal{S}^2$ of the adjacency operator $A$ on $L^2(\mathcal{T}_{k})$.

Such graphs can be constructed as quotients of the Bruhat-Tits tree associated to $PGL(2,\mathbb{Q}_p)$ by congruence subgroups of uniform lattices of $PGL(2,\mathbb{Q}_p)$ [8], using Ramanujan conjecture for classical modular forms. More examples were given by Morgenstern [11], replacing $\mathbb{Q}_p$ by $\mathbb{F}_q(\!(t^{-1})\!)$. One significant difference between $PGL(2,\mathbb{Q}_p)$ and $PGL_2(\mathbb{F}_q(\!(t^{-1})\!))$ is that $PGL(2,\mathbb{F}_q(\!(t^{-1})\!))$ has a non-uniform lattice $\Gamma=PGL(2,\mathbb{F}_q[t])$. For congruence subgroups $\Lambda$ of $\Gamma$, the quotient graphs are infinite but the edges and vertices come with suitable weights $w$ so that the total volume associated to the weight is finite. Under these weights on vertices and edges, the adjacency operator $A$ on $\mathcal{T}_{q+1}$ induces the weighted operator $A_{X}$ on the quotient $X=\Lambda\backslash\mathcal{T}_{q+1}$.

In [12], the author defined Ramanujan diagrams as such weighted objects satisfying the similar bound for non-trivial spectrum of $A_X$.
In this case, every non-trivial spectrum of $A_X$ on $L^2_w(X)$ is contained in the interval $[-2\sqrt{q},2\sqrt{q}]$ and hence it is a Ramanujan diagram. For example, the adjacency operator on $L^2_w(PGL(2,\mathbb{F}_q[t])\backslash\mathcal{T}_{q+1})$ has discrete spectrum $\pm(q+1)$ and continuous spectrum $[-2\sqrt{q},2\sqrt{q}]$ (see Figure <ref>).

[dashed,->] (-5,0) – (5,0);

[line width=1.2pt,black] (2.5,0) – (-2.5,0);

[fill] (4,0) circle (0.05);
[fill] (-4,0) circle (0.05);

at (4,-0.4) $q+1$;
at (-4.2,-0.4) $-q-1$;

at (2.5,-0.4) $2\sqrt{q}$;
at (-2.6,-0.4) $-2\sqrt{q}$;

Spectrum of $A_X$ on $PGL(2,\mathbb{F}_q[t])\backslash \mathcal{T}_{q+1}$

For graphs or diagram coming from a quotient of $G=PGL(2,\mathbb{F}_q(\!(t^{-1})\!))$ by an arithmetic lattice $\Gamma$, being a Ramanujan graph or Ramanujan diagram can be understood via representation-theoretic reformulation. Namely, $\Gamma\backslash \mathcal{T}_{q+1}$ is Ramanujan if and only if all the infinite-dimensional spherical irreducible $G$-representations which are weakly contained in $L^2(\Gamma\backslash G)$ are not from the complementary series. See [7] for the detail.

The authors in [2] suggested a generalization of the notion of Ramanujan graphs to the simplicial complexes obtained as finite quotients of the Bruhat-Tits building for $PGL(d,F)$ for a non-Archimedean local field $F$. Let $\mathcal{B}$ be the building associated to $PGL(d,F)$. The colored adjacency operator $A_j:L^2(\mathcal{B})\rightarrow L^2(\mathcal{B})$ is defined for $f\in L^2(\mathcal{B})$ by

$$ A_j f(x)=\sum_{\substack{ y\sim x\\ \tau(y)=\tau(x)+j}}f(y),$$

where $y\sim x$ implies that there is an edge between $y$ and $x$ in $\mathcal{B}$ and $\tau\colon \mathcal{B}^0\to\mathbb{Z}/d\mathbb{Z}$ is a color function (see Section <ref> for the precise definition). Let $\mathcal{S}^d$ be the simultaneous spectrum of colored adjacency operators $(A_1,\ldots,A_{d-1})$ on $L^2(\mathcal{B}^0)$, which may be computed explicitly as a subset of $\mathbb{C}^{d-1}$ (see Theorem 2.11 of [9] and also Proposition 4.5 of [1] for $d=3$). In fact, $\mathcal{S}^d$ is equal to the set $\sigma(S)$ for

$$S=\{(z_1,\ldots,z_d)\colon |z_1|=\cdots=|z_d|=1\textrm{ and }z_1z_2\cdots z_d=1\}$$
and $\sigma\colon S\to\mathbb{C}^{d-1}$ be the map given by $(z_1,\ldots,z_d)\mapsto (\lambda_1,\ldots,\lambda_{d-1})$ where

A finite complex $X$ arising as a quotient of $\mathcal{B}(G)$ is called Ramanujan if every non-trivial automorphic spectrum $(\lambda_1,\ldots,\lambda_{d-1})$ of $A_{X,j}$ acting on $L^2(X)$ is contained in the simultaneous spectrum $\mathcal{S}^d$ of $A_j$ on $\mathcal{B}$.
In [9], [10], [6] and [15], the authors constructed higher dimensional Ramanujan complexes arising as finite quotients of $PGL(d,F)$.

In [14], the author investigated non-uniform Ramanujan quotients of the Bruhat-Tits building $\mathcal{B}_d$ of $PGL(d,\mathbb{F}_q(\!(t^{-1})\!))$, generalizing the finite Ramanujan complexes constructed in [9], [10], [6] and [15]. She proved using the representation-theoretic arugment that if $d>2$, then for $G=PGL(d,\mathbb{F}_q(\!(t^{-1})\!))$, $\Gamma=PGL(d,\mathbb{F}_q[t])$ and $\mathcal{B}_d$ the Bruhat-Tits building of $G$, the quotient $\Lambda\backslash \mathcal{B}_d$ is not Ramanujan for any finite index subgroup $\Lambda$ of $\Gamma$. This mainly comes from the following fact: the Ramanujan conjecture in positive characteristic for $PGL_d$ for $d>2$, achieved by Lafforgue, gives bounds on the cuspidal spectrum, but the other parts of the spectrum do not satisfy the same bounds as the cuspidal spectrum.

While there are significant differences from the point of view of representation theory, the combinatorial distinction between the cuspidal spectrum and the other parts of the spectrum was not clear, as mentioned in [14]. In this paper, we explore the combinatorial characterization of the automorphic spectrum of the natural weighted adjacency operator on the non-uniform simplicial complex $PGL(3,\mathbb{F}_q[t])\backslash\mathcal{B}_3$.

Let $\Gamma=PGL(3,\mathbb{F}_q[t])$ and $G=PGL(3,\mathbb{F}_q(\!(t^{-1})\!))$.

Since $\Gamma$ acts with torsion on $\mathcal{B}(G)$ the degree of vertices in $\Gamma\backslash \mathcal{B}(G)$ is not constant, the colored adjacency operators $A^+=A_1$ and $A^-=A_2$ on $\mathcal{B}(G)$ induces the weighted adjacency operators $A^+_w$ and $A_w^-$ on $L^2_w(\Gamma\backslash \mathcal{B}(G))$ for a suitable weight function $w$.

More precisely, the weighted adjacency operators $A_w^+$ and $A_w^-$ on $L_w^2(\Gamma\backslash\mathcal{B}(G))$ is defined for any $f\in L^2_w(\Gamma\backslash\mathcal{B}(G))$ by

$$A_w^\pm f(u):=\sum_{\substack{(u,v)\in E\\ \tau(v)=\tau(u)\pm1}}\frac{w(u,v)}{w(v)}f(v).$$

See Section <ref> for the exact calculation of $w$. These operators $A_w^\pm$ satisfies that $(A_w^+)^*=A_w^-$ (see [14], Remark 4.1). Now we state our main theorem.

Let $A_w^+$ be weighted adjacency operator on $L^2_w(\Gamma\backslash \mathcal{B}(G))$ induced from the colored adjacency operator $A^+$ on $\mathcal{B}(G)$. The spectrum of the operator $A_w^+$ contains
$\Sigma_0\cup\Sigma_1\cup\Sigma_2$ where

$$\Sigma_0=\{q^2+q+1,(q^2+q+1)e^{\frac{2\pi i}{3}},(q^2+q+1)e^{\frac{4\pi i}{3}}\}$$
is a set of three distinct points,

$$\Sigma_1=\{q^{\frac{3}{2}}e^{i\theta}+qe^{-2i\theta}+q^{\frac{1}{2}}e^{i\theta}\colon \theta\in\mathbb{R}\}$$
is a hypocycloid with three cusps $(q^{\frac{3}{2}}+q+q^{\frac{1}{2}})e^{\frac{2k\pi i}{3}}$ for $k=0,1,2$ and

$$\Sigma_2=\{q(s_1+s_2+s_3)\in\mathbb{C}\colon s_1s_2s_3=1\textrm{ and }|s_1|=|s_2|=|s_3|=1\}$$
is a hypocycloid with three cusps $3qe^{\frac{2k\pi i}{3}}$ for $k=0,1,2$ and its interior (See Figure <ref>).

[dashed,->] (-5,0) – (5,0);
[dashed,->] (0,-5) – (0,5);

0.5 1̱.̱5̱

[fill,line width=1pt,black] plot[samples=100,domain=0:360,smooth,variable=] ((-̱)*cos()+*cos((-̱)*/,(-̱)*sin()-*sin((-̱)*/);

1.2 3̱.̱6̱
[line width=1pt,black] plot[samples=100,domain=0:360,smooth,variable=] ((-̱)*cos()+*cos((-̱)*/,(-̱)*sin()-*sin((-̱)*/);

[fill] (4.5,0) circle (0.05);
[fill] (120:4.5) circle (0.05);
[fill] (240:4.5) circle (0.05);

at (1.5,1.5) $3q$;
[->] (1.5,1.3) – (1.5,0.1);
at (3.6,1.2) $\sqrt{q}(q+\sqrt{q}+1)$;
[->] (3.6,1) – (3.6,0.1);
at (4.5,-1) $q^2+q+1$;
[->] (4.5,-0.8) – (4.5,-0.1);

$\Sigma_0\cup\Sigma_1\cup\Sigma_2$ in complex plane

Since $A_w^+$ and $A_w^-$ are normal and commute with each other, if $\lambda^+$ is a spectrum of $A^+_w$, then $\lambda^-=\overline{\lambda^+}$ is also a spectrum of $A^-_w$ and vice versa. Thus, the automorphic spectrum of $A^\pm_w$ is defined as the vectors $(\lambda^+,\lambda^-)$ in $\mathbb{C}^2$ for which there exists a sequence of unit vectors $f_n\in L^2_w(\Gamma\backslash \mathcal{B}(G))$ such that

$$\lim_{n\to\infty} (A_w^\pm f_n-\lambda^\pm f_n)=0.$$
Since $\mathcal{S}^3=\Sigma_2$ and the points in $\Sigma_1$ are also in the automorphic spectra of $A^\pm_w$, we obtain the following corollary.

$PGL(3,\mathbb{F}_q[t])\backslash\mathcal{B}(G)$ is not a Ramanujan complex.

If $\Gamma$ acts properly and cocompactly on an $\widetilde{A}_2$ building $\mathcal{B}$, then in general by [2] and [3] the $L^2$-spectrum of the adjacency operator $A^+$ consists of three points $\Sigma_0$ together with a subset of the region bounded by $\Sigma_1$. By definition, the quotient complex $\Gamma\backslash\mathcal{B}$ is Ramanujan if every non-trivial $L^2$-spectrum is contained in $\Sigma_2(=\mathcal{S}^3)$.

This article is organized as follows. In Section <ref>, we review the definition of the Bruhat-Tits building associated to the group $PGL_d$ over a non-Archimedean local field and compact Ramanujan complexes in their combinatorial and representation-theoretic forms. In Section <ref>, we present the structure of the non-uniform quotient $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))$. Colored adjacency operator with weights, the natural operator on the quotient space, is discussed in Section <ref>. In Section <ref> and <ref>, we explore the simultaneous eigenfunctions and the spectrum of the weighted adjacency operators.

§ BUILDING AND COMPACT RAMANUJAN COMPLEX

Let $F$ be a non-Archimedean local field with a discrete valuation $\nu$ and $\mathcal{O}$ be the valuation ring of $F$. Let $\pi$ be the uniformizer of $\mathcal{O}$ for which $\pi\mathcal{O}$ is the unique maximal ideal of $\mathcal{O}$. Let $G$ be the projective general linear group
$$PGL(d,F)=GL(d,F)/\{\lambda I\colon\lambda\in F\}$$
and let $W$ be the image of the map from $GL(d,\mathcal{O})$ to $PGL(d,F)$ defined by

$$g\rightarrow g\{\lambda I\colon\lambda\in F\}.$$
In this section, we review the affine building of type $\widetilde{A}_{d-1}$, colored adjacency operators and compact Ramanujan complexes arising as a quotient of the building for $PGL(d,F)$. For details, we refer to [9].

The Bruhat-Tits building $\mathcal{B}(G)$ associated with $G$ is the $(d-1)$-dimensinal contractible simplicial complex defined as follows. We say two $\mathcal{O}$-lattices $L$ and $L'$ of rank $d$ are in the same equivalence class if $L=sL'$ for some $s\in F^\times$. The set $\mathcal{B}(G)^0$ of vertices of $\mathcal{B}(G)$ is the set of the equivalence classes $[L]$. For given $k$-vertices $[L_1],[L_2],\cdots,[L_k]$, they form a $k$-dimensional simplex in $\mathcal{B}(G)$ if

\begin{equation}\label{eq:1.1}
\pi L_1'\subset L_k'\subset L_{k-1}'\subset \cdots L_2'\subset L_1'
\end{equation}

for some $L_i'\in [L_i].$ In general, we denote by $\mathcal{B}(G)^k$ the $k$-skeleton of $\mathcal{B}(G)$.

Let $\mathcal{O}^d$ be the standard $\mathcal{O}$-lattice. The action of a matrix $M$ in $G$ on the set of $\mathcal{O}$-lattices transfers the standard one $\mathcal{O}^d$ to the $\mathcal{O}$-lattice of which basis consists of the column vectors $M$ and every scalar matrix $\lambda I$ preserves every equivalence class. Thus, the group $G$ acts transitively on $\mathcal{B}(G)^0.$ Since the action of $G$ on $\mathcal{B}(G)^0$ defined by left multiplication satisfies the relation (<ref>), it follows that $G$ acts isometrically on the quotient space. Since the group $W$ is the stabilizer of the vertex $[\mathcal{O}^d]$, the set of vertices of Bruhat-Tits building associated to $G$ is identified with the quotient space $G/W.$

The color $\tau:\mathcal{B}(G)^0\rightarrow \mathbb{Z}/d\mathbb{Z}$ is defined by

$$\tau([L]):=\log_q [\mathcal{O}^d:\pi^i L],$$

for a sufficiently large positive integer $i$ with $\pi^i L\subset \mathcal{O}^d.$
Since $[\pi^iL:\pi^{i+1}L]=d$, the color $\tau([L])$ is independent of the choice of the lattice in $[L]$ and hence is well-defined.

$L^2(\mathcal{B}(G))$ be the space of functions $f\colon\mathcal{B}(G)^0\rightarrow \mathbb{C}$ satisfying
$$\sum_{x\in \mathcal{B}(G)^0}|f(x)|^2<\infty.$$

The colored adjacency operator $A_i:L^2(\mathcal{B}(G))\rightarrow L^2(\mathcal{B}(G))$ is defined for $f\in L^2(\mathcal{B}(G))$ by

\begin{equation}\label{eq:1.2}
A_if(x)=\sum_{\substack{ y\sim x\\ \tau(y)=\tau(x)+i}}f(y),
\end{equation}

where $y\sim x$ implies that there is an edge between $y$ and $x$ in $\mathcal{B}(G)$. These operators are bounded and commutative. Moreover, since $A_i^*=A_{k-i}$, the operators are normal. Let $\mathcal{S}^d\subset\mathbb{C}^{d-1}$ be the simultaneous spectrum of $(A_1,A_2,\ldots,A_{d-1})$ acting on $L^2(\mathcal{B}(G))$. More precisely, the spectrum $\mathcal{S}^d$ is the subset of $d$-tuples $(\lambda_1,\cdots,\lambda_{d-1})$ in $\mathbb{C}^{d-1}$ such that there exists a sequence of functions $f_n\in L^2(\mathcal{B}(G))$ with $\|f_n\|_2=1$ satisfying, for any  $i=1,\ldots, d-1,$

Although the non-uniform lattice $PGL(3,\mathbb{F}_q[t])$ of $PGL(3,\mathbb{F}_q(\!(t^{-1})\!))$ is the main topic of this paper, in order to motivate the reader, we review the definition of compact Ramanujan complexes arising as a quotient of $PGL(d,F)$. Let $\Gamma$ be a torsion-free cocompact discrete subgroup of $G$. Then $\Gamma$ acts on $\mathcal{B}(G)^0=G/W$ by left translation, and $\Gamma\backslash \mathcal{B}(G)$ is a finite complex. The color function defined on $\mathcal{B}(G)^0$ may not be preserved by $\Gamma$. However, the colors defined on the set $\mathcal{B}(G)^1$ of edges by

$$\tau(x,y)=\tau(x)-\tau(y)\quad(\textrm{mod }d)$$
are preserved by $\Gamma$.

Let $L^2(\Gamma\backslash \mathcal{B}(G))$ be the space of functions $f$ on $\Gamma\backslash\mathcal{B}(G)^0$ with $\|f\|_{2}<+\infty$. Here, $\|f\|_2=\sum_{x\in\Gamma\backslash\mathcal{B}(G)^0}|f(x)|^2$. Using the equation (<ref>), it follows that operators $A_i$ induce colored adjacency operators

$$A_{X,i}:L^2(\Gamma\backslash \mathcal{B}(G))\rightarrow L^2(\Gamma\backslash \mathcal{B}(G))$$

on $\Gamma\backslash \mathcal{B}(G)$.
The eigenfunction $f$ of $A_{X,i}$ on $L^2(\Gamma \backslash \mathcal{B}(G))$ is called trivial if the function $f$ is of the form

for some $d$-th root $\xi$ of unity, i.e., $\xi^d=1.$
The quotient complex $\Gamma \backslash \mathcal{B}(G)$ is called Ramanujan if for every non-trivial simultaneous eigenfunction $f$ of the colored adjacency operators $A_{X,i}$ acting on $L^2(\Gamma\backslash\mathcal{B}(G))$, the simultaneous eigenvalue $(\lambda_1,\ldots,\lambda_{d-1})$ for $f$ is contained in $\mathcal{S}^d$.

We remark that the authors in [9] proved that if $d$ is a prime and $\Gamma$ is an arithmetic uniform lattice of inner type, then $\Gamma \backslash \mathcal{B}(G)$ is Ramanujan.

§ NON-UNIFORM ARITHMETIC QUOTIENT $PGL(3,\MATHBB{F}_Q[T])\BACKSLASH PGL(3,\MATHBB{F}_Q(\!(T^{-1})\!))$

Let $\mathbb{F}_q$ be the finite field of order $q$ and let $\mathbb{F}_q(t)$ be the field of rational functions over $\mathbb{F}_q$. The absolute value $\|\cdot\|$ of $\mathbb{F}_q(t)$ is defined for any $f\in \mathbb{F}_q(t)$, by

$$\|{f}\|:=q^{\deg (g)-\deg (h)},$$

where $g,h$ are polynomial over $\mathbb{F}_q$ satisfying $f=\frac{g}{h}$.
The completion of $\mathbb{F}_q(t)$ with respect to $\|\cdot\|$, the field of formal Laurent series in $t^{-1}$, is denoted by $\mathbb{F}_q(\!(t^{-1})\!)$, i.e.,

$$\mathbb{F}_q(\!(t^{-1})\!):=\left\{\sum_{n=-N}^\infty a_nt^{-n}:N\in \mathbb{Z}, a_n\in \mathbb{F}_q\right\}.$$

The valuation ring $\mathcal{O}$ is the subring of power series

$$\mathbb{F}_q\mathbb{[\![}t^{-1}]\!]:=\left\{\sum_{n=0}^\infty a_nt^{-n}: a_n\in \mathbb{F}_q\right\}.$$

Let $G$ be the group $PGL(3,\mathbb{F}_q(\!(t^{-1})\!))$, $\Gamma$ be its non-uniform lattice $PGL(3,\mathbb{F}_q[t])$ and $W=PGL(3,\mathcal{O})$. In this section and throughout, we focus on the building $\mathcal{B}(G)$ for $G$ and the quotient $\Gamma\backslash \mathcal{B}(G)$ for $\Gamma=PGL(3,\mathbb{F}_q[t])$. In general, given a simple and simply-connected Chevalley group scheme $H$ defined over $\mathbb{Z}$, the action of $H(\mathbb{F}_q[t])$ on $\mathcal{B}(H)$ is described in [16].

We recall that the set of vertices $\mathcal{B}(G)^0$ is identified with the coset space $G/W$. Since the right multiplication by a matrix in $W$ is considered as the elementary column $\mathcal{O}$-operation, we may find a representative $[A]=AW$ of vertex in $\mathcal{B}(G)$ with the following conditions:

* The matrix $A$ is upper diagonal.

* The diagonal entries of $A$ are of the form $t^{n}$ where $n\in \mathbb{Z}$.

* The $(i,j)$-entry $a_{ij}$ of $A$ is contained in $t^{n+1}\mathbb{F}_q[t]$ whenever $a_{ii}=t^n$.

* If $a_{ii}=a_{jj}$, then $a_{ij}=a_{ji}=0$.

Similarly, the left action of $\Gamma$ is considered as the elementary row operation. The following lemmata describe the fundamental domain for $\Gamma$-action on $\mathcal{B}(G)^0$. These actually follow from the general theorem in [16], but to be self-contained we give the elementary proof of the statements.

Given every $g\in PGL(2,\mathbb{F}_q(\!(t^{-1})\!))$, there exists a unique non-negative integer $n$ such that

$$g=\gamma\begin{pmatrix} t^n & 0 \\ 0 & 1 \end{pmatrix}w$$

for some $\gamma\in PGL(2,\mathbb{F}_q[t])$ and $w\in PGL(2,\mathcal{O})$.

Let $\Gamma_2=PGL(2,\mathbb{F}_q[t])$, $W_2=PGL(2,\mathcal{O})$ and $[\alpha]$ be the polynomial part of $\alpha$. Let $T\colon \mathbb{F}_q(\!(t^{-1})\!)\to\mathbb{F}_q(\!(t^{-1})\!)$ be the mapping given by $T(\alpha)=\frac{1}{\alpha-[\alpha]}$. From the above observation, every $g\in G$ has a representative

$$\begin{pmatrix} t^m & \alpha \\ 0 & 1 \end{pmatrix}$$

for $m\in\mathbb{Z}$ and $\alpha\in t^{m+1}\mathbb{F}_q[t]$. If $m\ge 0$, then $\alpha\in\mathbb{F}_q[t]$ and hence

\begin{align*}
\Gamma_2\begin{pmatrix} t^m & \alpha \\ 0 & 1 \end{pmatrix}W_2=\Gamma_2\begin{pmatrix} t^m & 0 \\ 0 & 1 \end{pmatrix}W_2
\end{align*}

$$\begin{pmatrix} t^m & \alpha \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}\begin{pmatrix} t^m & 0 \\ 0 & 1 \end{pmatrix}.$$

If $m<0$, then we may assume that $\alpha=a_{-m+1}t^{-m+1}+\cdots+a_{-1}t^{-1}$ and we have

\begin{align*}
&\Gamma_2\begin{pmatrix} t^m & \alpha \\ 0 & 1 \end{pmatrix}W_2=\Gamma_2\begin{pmatrix} t^m & \alpha-[\alpha] \\ 0 & 1 \end{pmatrix}W_2=\Gamma_2\begin{pmatrix} 0 & \alpha-[\alpha] \\ -\frac{t^m}{\alpha-[\alpha]} & 1 \end{pmatrix}W_2\\
=\,&\Gamma_2\begin{pmatrix} -t^mT(\alpha) & 1 \\ 0 & \alpha-[\alpha] \end{pmatrix}W_2=\Gamma_2\begin{pmatrix} -t^mT(\alpha)^2 & T(\alpha) \\ 0 & 1 \end{pmatrix}W_2.
\end{align*}

Since $\alpha$ is rational, we have $T^{k}(\alpha)=0$ for large enough $k$ which implies that the above reduction eventually stops. After exchanging row and column if necessary, we get

$$\Gamma\begin{pmatrix} t^m & \alpha \\ 0 & 1 \end{pmatrix}W=\Gamma\begin{pmatrix}t^n & 0 \\ 0 & 1 \end{pmatrix}W$$

for some $n\ge 0$.

Given every $g\in G$, there exists a unique pair of non-negative integers $(m,n)$ with $0\le n\le m$ such that

$$g\in\Gamma\begin{pmatrix}t^m & 0 & 0 \\ 0 & t^n & 0 \\ 0 & 0 & 1\end{pmatrix} W.$$

From the above observation, every $g\in G$ may be written by

$$\begin{pmatrix}t^{m_0} & a_{12} & a_{13} \\ 0 & t^{n_0} & a_{23} \\ 0 & 0 & t^{\ell_0}\end{pmatrix}w$$

for some $a_{ij}\in\mathbb{F}_q(\!(t^{-1})\!)$, matrix $w\in W$ and non-negative integers $m_0, n_0$ and $\ell_0$. If $m_0< n_0$, then multiplying suitable $\gamma$ on the left and $w$ on the right so that we may assume $a_{12}$ is zero. Similary, if $n_0< \ell_0$, then we may assume $a_{23}$ is zero. If either $a_{12}$ or $a_{23}$ is zero, then it reduces to the case where it is not so it suffices to consider the case $m_0\ge n_0\ge \ell_0$. In this case, applying the Lemma <ref> to the upper-left $2\times 2$ block of the matrix, we may find $\gamma_1\in \Gamma$ such that

$$\begin{pmatrix}t^{m_0} & a_{12} & a_{13} \\ 0 & t^{n_0} & a_{23} \\ 0 & 0 & t^{\ell_0}\end{pmatrix}=\gamma_1\begin{pmatrix}t^{m_1} & 0 & \star \\ 0 & t^{k_1} & \star \\ 0 & 0 & t^{\ell_0}\end{pmatrix}w_1$$

with $m_0\ge m_1\ge k_1\ge n_0$.
Now, applying the Lemma <ref> (and the proof) to the lower-right $2\times 2$ block, we may find $\gamma_1'\in \Gamma$ and $w_1'\in W$ such that

$$\begin{pmatrix}t^{m_1} & 0 & \star \\ 0 & t^{k_1} & \star \\ 0 & 0 & t^{\ell_0}\end{pmatrix}=\gamma_1'\begin{pmatrix}t^{m_1} & \star & \star \\ 0 & t^{n_1} & 0 \\ 0 & 0 & t^{\ell_1}\end{pmatrix}w_1'$$

with $k_1\ge n_1\ge \ell_1\ge \ell _0$. While the reducing process
$$(m_i,n_i,\ell_i)\rightarrow (m_{i+1},n_{i+1},\ell_{i+1})$$
goes on, both $(1,2)$-entry and $(2,3)$-entry become eventually zero since $m_{i}> m_{i+1}\ge \ell_{i+1}>\ell_{i}$ unless both entries are zero. Continuing this process until it eventually stops, we may obtain $\gamma,\gamma'\in \Gamma$ and $w,w'\in W$ for which

$$\begin{pmatrix}t^{m_0} & 0 & \star \\ 0 & t^{n_0} & \star \\ 0 & 0 & t^{\ell_0}\end{pmatrix}=\gamma'\begin{pmatrix}t^{m_t} & 0 & \star \\ 0 & t^{n_t} & 0 \\ 0 & 0 & t^{\ell_t}\end{pmatrix}w'=\gamma\begin{pmatrix}t^{m} & 0 & 0 \\ 0 & t^{n} & 0 \\ 0 & 0 & 1\end{pmatrix}w$$

with $m\ge n\ge 0$.

For $m,n$ with $m\ge n\ge 0$, let $v_{m,n}$ be the vertex of the quotient complex $\Gamma\backslash \mathcal{B}(G)$ corresponds to

$$\Gamma \begin{pmatrix} t^m & 0 & 0 \\ 0 & t^n & 0 \\ 0 & 0 & 1\end{pmatrix}W.$$

The definition of the building $\mathcal{B}(G)$ implies that there is an edge between two vertices $v_{m,n}$ and $v_{m',n'}$ if and only if the following hold:

(m',n')∈{(m±1,n),(m,n±1),(m±1,n±1)}     if m>n>0

(m',n')∈{(m±1,n),(m,n+1),(m+1,n+1)}     if m>n=0

(m',n')∈{(m+1,n),(m,n-1),(m±1,n±1)}     if m=n>0

(m',n')∈{(1,0),(1,1)}    if m=n=0.

Combining above facts, the quotient complex $\Gamma\backslash\mathcal{B}(G)$ is described as Figure <ref>.

[thick](-5,0) – (5.5,0);   [thick](-0.5,+sqrt(60.75)) – (-5,0);[thick](-4,+sqrt(3)) – (-3,0);[thick] (-3,+sqrt(12)) – (-1,0);[thick](-2,sqrt(27)) – (1,0);[thick] (-1,sqrt(48))–(3,0);[thick] (-4,+sqrt(3)–(5.5,+sqrt(3));[thick](-3,+sqrt(12))–(5.5,+sqrt(12));[thick](-2,+sqrt(27))–(5.5,+sqrt(27));[thick](-1,+sqrt(48))–(-1,+sqrt(48));[thick](1.5,+sqrt(60.75))–(-3,0);[thick](3.5,+sqrt(60.75))–(-1,0);[thick](5.5,+sqrt(60.75))–(1,0);[thick](5.5,+sqrt(75)/2)–(3,0);[thick](5,0)–(0.5,+sqrt(60.75));[thick] (-1,+sqrt(48))–(5.5,+sqrt(48));[thick](5.5,+sqrt(27)/2)–(2.5,+sqrt(60.75));[thick] (4.5,+sqrt(60.75))–(5.5,+sqrt(147)/2);[thick] (5,0)–(5.5,+sqrt(3)/2);
at (-5,-0.3) $v_{0,0}$;at (-3,-0.3) $v_{1,0}$;at (-1,-0.3) $v_{2,0}$;at (1,-0.3) $v_{3,0}$;at(3,-0.3) $v_{4,0}$;at (5,-0.3) $v_{5,0}$;at (6.5,3) $\cdots$;at (-4.6,1.9) $v_{1,1}$;at (-2.6,1.9) $v_{2,1}$;at (-0.6,1.9) $v_{3,1}$;at (1.4,1.9) $v_{4,1}$;at (3.4,1.9) $v_{5,1}$;at (-3.6,3.65) $v_{2,2}$;at (-1.6,3.65) $v_{3,2}$;at (0.4,3.65) $v_{4,2}$;at(2.4,3.65) $v_{5,2}$;at (4.4,3.65) $v_{6,2}$;at (-2.6,5.35) $v_{3,3}$;at (-0.6,5.35) $v_{4,3}$;at (1.4,5.35) $v_{5,3}$;at (3.4, 5.35) $v_{6,3}$;at (-1.6, 7.1) $v_{4,4}$;at (0.4, 7.1) $v_{5,4}$;at (2.4, 7.1) $v_{6,4}$;at (4.4, 7.1) $v_{7,4}$;
(-5,0)    circle (3pt); (-3,0)    circle (3pt); (-1,-0)    circle (3pt);(1,-0)    circle (3pt); (3,0) circle (3pt);(5,0) circle (3pt);(-4,+sqrt(3))    circle (3pt); (-3,+sqrt(12))    circle (3pt); (-2,sqrt(27))    circle (3pt);(-1,+sqrt(48))    circle (3pt); (-2,+sqrt(3)) circle (3pt);(-0,+sqrt(3)) circle (3pt);(0,+sqrt(3)) circle (3pt);(2,sqrt(3)) circle (3pt);(4,+sqrt(3)) circle (3pt);(-3,sqrt(12)) circle (3pt); (-1,sqrt(12)) circle (3pt);(1,sqrt(12)) circle (3pt);(3,sqrt(12)) circle (3pt);(3,sqrt(12)) circle (3pt);(5,sqrt(12)) circle (3pt);(-2,sqrt(27)) circle (3pt);(0,sqrt(27)) circle (3pt);(2,sqrt(27)) circle (3pt); (4,sqrt(27)) circle (3pt);(1,sqrt(48)) circle (3pt); (3,sqrt(48)) circle (3pt);(5,sqrt(48)) circle (3pt);

The fundamental domain for $\Gamma\backslash\mathcal{B}(G)$

§ COLORED ADJACENCY OPERATOR WITH WEIGHTS

Generalizing the idea of [12], the colored adjacency operators with weights are introduced in [14]. In this section, we study the combinatorics of the natural weighted adjacency operators $A_w^+$ and $A_w^-$. Since $\Gamma$ acts on $\mathcal{B}(G)$ with torsions, the induced operators on the quotient from the colored adjacency operators on $L^2(\mathcal{B}(G))$ come with weights envolving the cardinality of the stabilizer of the vertices and edges.

Given a simplicial complex $X$, let $V=V(X)$ and $E=E(X)$ be the set of vertices and edges of $X$, respectively. We denote by $(u,v)$ the edge with vertices $u$ and $v$.

The weight function $w:V\cup E\rightarrow (0,1]$ is a function such that $w(x)\leq w(e)$ holds for every $v\in V$ and $e\in E$ with $v\in e$. The function $\theta(u,v)=\frac{w(e)}{w(u)}$ is called the entering degree of $e$ to $u$, where $e=(u,v)$. The in-degree of a vertex $u$ is $\text{in-degree}(u):=\underset{(u,v)\in E}\sum \theta(u,v).$ A simplicial complex $X$ with weight function $w$ is $k$-regular if $\text{in-degree}(u)=k$ for any $u\in V$.
The weight function $w$ allows to define a measure $\mu$ on $V$, for any $S\subset V$, by $\mu(S)=\underset{v\in S}\sum w(v).$ Using the measure $\mu$, we define the $L^2$-norm of a function $f$ by

$$\|f\|_2:=\biggl(\sum_{u\in V}|f(u)|^2w(u)\biggr)^{1/2}$$

and let $L^2_w(X)$ be the space of functions $f\colon V(X)\to\mathbb{C}$ with $\|f\|_2<\infty$.

The weighted adjacency operators $A^+_w$ and $A^-_w$ on $L^2_w(\Gamma\backslash\mathcal{B}(G))$ is defined for any $f\in L^2_w(\Gamma\backslash\mathcal{B}(G))$ by

$$A_w^\pm f(u):=\sum_{\substack{(u,v)\in E\\ \tau(v)=\tau(u)\pm1}}\frac{w(u,v)}{w(v)}f(v).$$

The operators $A_w^\pm$ satisfies that $(A_w^+)^*=A_w^-$ (see [14], Remark 4.1).

Let $\Gamma_{m,n}$ be the stabilizer of the vertex
$$x_{m,n}W=\begin{pmatrix} t^m & 0 & 0 \\ 0 & t^n & 0 \\ 0 & 0 & 1\end{pmatrix}W$$
in $\mathcal{B}(G)$. Let $w$ be the weight function defined by

w(v_m,n):=q^3(q+1)(q-1)^2/ |Γ_m,n|  and  w(e):=q^3(q+1)(q-1)^2/|Γ_v_m,n∩Γ_v_m',n'|,

where $e=(v_{m,n},v_{m',n'})$ and $|S|$ denotes the cardinality of the set $S$. With respect to these weights $w$, the colored adjacency operators on $\mathcal{B}(G)$ induce the weighted adjacency operators $A_w^+$ and $A_w^-$ on the quotient $L^2_w(\Gamma\backslash \mathcal{B}(G))$.
The rest part of this section is devoted to compute the values of the weight function $w$.

Let $N_{m,n}=|\Gamma_{m,n}|$. We have

\begin{equation}\label{eq:3.1}
\begin{cases}
q^3(q+1)(q^2+q+1)(q-1)^2 &\text{ if }\, m=n=0\\
q^{2m+3}(q+1)(q-1)^2 &\text{ if }\,m>n=0\\
q^{2m+3}(q-1)^2 &\text{ if }\, m>n>0\\
q^{2m+3}(q+1)(q-1)^2&\text{ if }\, m=n>0.
\end{cases}
\end{equation}

Since $\gamma x_{m,n}W=x_{m,n}W$ if and only if $\gamma\in\Gamma_{m,n},$ we have

$$\Gamma_{m,n}=\{\gamma\in \Gamma:x_{m,n}^{-1}\gamma x_{m,n}\in W\}.$$

Thus, $\Gamma_{0,0}=\Gamma\cap PGL(3,\mathcal{O})=PGL(3,\mathbb{F}_q)$ and

\begin{equation}\label{eq:3.2}
\begin{split}
\Gamma_{m,0}&=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&a_{32}&a_{33}\end{pmatrix}\in GL(3,\mathbb{F}_q(\!(t^{-1})\!)):\substack{a_{11},a_{22},a_{23},a_{32},a_{33}\in \mathbb{F}_q, \\a_{12},a_{13}\in P^m(t)}\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\}\\
\Gamma_{m,n}&=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\end{pmatrix}\in GL(3,\mathbb{F}_q(\!(t^{-1})\!)):\substack{a_{11},a_{22},a_{33}\in \mathbb{F}_q, a_{12}\in P^{m-n}(t),\\a_{13}\in P^m(t),a_{23}\in P^n(t)}\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\}\\
\Gamma_{m,m}&=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\0&0&a_{33}\end{pmatrix}\in GL(3,\mathbb{F}_q(\!(t^{-1})\!)):\substack{a_{11},a_{21},a_{22},a_{23},a_{33}\in \mathbb{F}_q, \\a_{13},a_{23}\in P^m(t)}\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\},
\end{split}
\end{equation}

for $m>n>0$, where $P^n(t)$ is the space of polynomials of degree less than or equal to $n$.

it follows that $N_{0,0}=|PGL(3,\mathbb{F}_q)|=q^3(q+1)(q^2+q+1)(q-1)^2.$

For any $\gamma\in \Gamma_{m,0}$, $a_{11}\neq 0$, two vectors $(a_{22},a_{32})$ and $(a_{32},a_{33})$ are linearly independent. This shows that

Simliarly, we also have $N_{m,m}=q^{2m+3}(q+1)(q-1)^2.$

Since $a_{ii}\neq 0$ for any $\gamma\in \Gamma_{m,n}$ with $m>n>0$, it follows that

which completes the proof of the proposition.

Proposition <ref> implies that

\begin{equation}\label{eq:3.3}
\begin{cases}\vspace{0.5em}
\dfrac{1}{q^2+q+1} &\text{ if }\, m=n=0\\\vspace{1em}
\dfrac{1}{q^{2m}}&\text{ if }\,m>n=0\\\vspace{0.5em}
\dfrac{q+1}{q^{2m}} &\text{ if }\, m>n>0\\
\dfrac{1}{q^{2m}}&\text{ if }\, m=n>0.
\end{cases}
\end{equation}

It follows from (<ref>) that for any $m,n$ with $m>n>0,$

\begin{equation}\label{eq:3.4}
\begin{split}
&\Gamma_{m,m}\cap\Gamma_{m+1,m+1}=\Gamma_{m,m} \text{ and }\\
\end{split}
\end{equation}

The last equation of (<ref>) holds when $n=0$.
The following proposition gives the value of the weight function $w$ on edges.
Under the above notation, we have

\begin{equation}\label{eq:3.5}
\begin{split}
&|\Gamma_{0,0}\cap \Gamma_{1,1}|=q^3(q+1)(q-1)^2\\
&|\Gamma_{1,0}\cap \Gamma_{1,1}|=q^4(q-1)^2\\
\end{split}
\end{equation}

Applying (<ref>),

\begin{equation}\label{3.5}
\nonumber\begin{split}
&\Gamma_{0,0}\cap\Gamma_{1,0}=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&a_{32}&a_{33}\end{pmatrix}:a_{11},a_{12},a_{13},a_{22},a_{23},a_{32}a_{33}\in \mathbb{F}_q\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\}\\
&\Gamma_{0,0}\cap\Gamma_{1,0}=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\0&0&a_{33}\end{pmatrix}:a_{11},a_{12},a_{13},a_{21}a_{22},a_{23},a_{33}\in \mathbb{F}_q\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\}\\
&\Gamma_{1,0}\cap\Gamma_{1,1}=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\end{pmatrix}:\substack{a_{11},a_{12},a_{22},a_{23},a_{33}\in \mathbb{F}_q\\ a_{13}\in P^1(t)}\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\}
\end{split}
\end{equation}

\begin{equation}\nonumber
\Gamma_{m,n-1}\cap\Gamma_{m,n}=\biggl\{\begin{pmatrix} a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\end{pmatrix}:\substack{a_{11},a_{22},a_{33}\in \mathbb{F}_q,a_{12}\in P^{m-n}(t)\\a_{13}\in P^m(t),a_{23}\in P^{n-1}(t)}\biggr\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\}
\end{equation}

Simliar to the proof of Proposition <ref>, we get (<ref>).

§ EIGENFUNCTIONS OF THE WEIGHTED ADJACENCY OPERATORS

Let $S=\{(s_1,s_2,s_3)\in\mathbb{C}^3\colon s_1s_2s_3=1\text{ and }\overline{s_1+s_2+s_3}=s_1^{-1}+s_2^{-1}+s_3^{-1}\}.$
Since $(A_w^+)^*=A_w^-$, a pair $(\lambda^+,\lambda^-)\in\mathbb{C}^2-\{(0,0)\}$ of simultaneous eigenvalues of $A_w^+$ and $A_w^-$ satisfies $\overline{\lambda^+}=\lambda^-$. We note that every $(\lambda^+,\lambda^-)$ can be described by

\begin{equation}\label{eq:4.1}
\lambda^+:={q}(s_1+s_2+s_3)\text{ and }\lambda^-:={q}\left(\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}\right).\\
\end{equation}

for some point  $(s_1,s_2,s_3)$ in $S$.
In this section, we investigate all eigenfunctions of $A_w^\pm$ in $C(\Gamma\backslash\mathcal{B}(G))$, using the parametrization (<ref>) and a family of recursive relations of $A_w^\pm$. Although not every eigenfunction involves automorphic spectrum, this enables us in Section <ref> to find approximate eigenvalues of $A_w^\pm$ on $L^2_w(\Gamma\backslash\mathcal{B}(G))$.

For any $(s_1,s_2,s_3)\in S\backslash\{(s_1,s_2,s_3)\in S:|s_1|=|s_2|=|s_3|=1\}$, there exists a permutation $\sigma$ on $\{1,2,3\}$ such that

\begin{equation}\label{a}
\overline{s_{\sigma(1)}}=s_{\sigma(3)}^{-1}\text{ and }|s_{\sigma(2)}|=1.
\end{equation}

Without loss of generality, we may assume that $|s_1|>1$ and $|s_2|<1$. Put
$$s_1=ae^{i\theta_1},\,\,s_2=be^{i\theta_2}\,\,\text{ and }\,\,s_3=\frac{1}{ab}e^{-i(\theta_1+\theta_2)},$$

where $a,b>0.$
Since $|\overline{s_1}-s_1^{-1}+\overline{s_2}-s_2^{-1}|^2=|s_3^{-1}-\overline{s_3}|^2,$ we have

\begin{equation}
\begin{split}
\end{split}
\end{equation}

Using the above equation, we have $2\cos (\theta_1-\theta_2)=ab+\frac{1}{ab}$. This holds only if $ab=1$ and $\theta_1=\theta_2$. Thus it is possible to choose $\sigma$ satisfying (<ref>).

Now we will solve the equations for eigenfunctions of $A_w^+$ and $A_w^-$. In Chapter 3 of [1] where the authors investigate the spherical functions on $\mathcal{B}(G)$, similar recurrence formulas appear with different coefficients.

Let $f$ be an eigenfunction of $A_w^+$ and $A_w^-$ with eigenvalues $\lambda^+$ and $\lambda^-$. Suppose that $s_i\neq s_j$ for any $i\neq j$. Then, $f$ is given by

$$f(v_{m,n})=\sum_{\substack{(i,j)\in \{1,2,3\}^2\\i\neq j}}B_{i,j}q^ms_i^ms_{j}^n$$

Let $f$ be an eigenfunction of $A_w^+$ and $A_w^-$ with eigenvalues $\lambda^+$ and $\lambda^-$. For convenience, suppose that $f(v_{0,0})=1$. The definition of $A^\pm_w$ together with the equations (<ref>), (<ref>) and (<ref>) show that the function $f$ satisfies

\begin{eqnarray}
\label{2}A_w^+f(v_{0,0})&=&{(q^2+q+1)f(v_{1,0})}=\lambda^+\\
\label{3}A_w^-f(v_{0,0})&=&(q^2+q+1)f(v_{1,1})=\lambda^-\\
\label{4}A_w^+f(v_{1,0})&=&{f(v_{2,0})+(q^2+q)f(v_{1,1})}=\lambda^+f(v_{1,0})=(\lambda^+)^2\\
\label{5}A_w^-f(v_{1,0})&=&{q^2f(v_{0,0})+(q+1)f(v_{2,1})}=\lambda^-f(v_{1,0})=\lambda^+\lambda^-\\
\label{6}A_w^+f(v_{1,1})&=&{q^2f(v_{0,0})+(q+1)f(v_{2,1})}=\lambda^+f(v_{1,1})=\lambda^+\lambda^-\\
\label{7}A_w^-f(v_{1,1})&=&{(q^2+q)f(v_{1,0})+f(v_{2,2})}=\lambda^-f(v_{1,1})=(\lambda^-)^2.
\end{eqnarray}

For $m\geq 1$, the function $f$ have the following properties:

\begin{eqnarray}
\label{8}A_w^+f(v_{m,0})&=&{f(v_{m+1,0})+(q^2+q)f(v_{m,1})}=\lambda^+f(v_{m,0})\\
\label{9}A_w^-f(v_{m,0})&=&{q^2f(v_{m-1,0})+(q+1)f(v_{m+1,1})}=\lambda^-f(v_{m,0})\\
\label{10}A_w^+f(v_{m,m})&=&{q^2f(v_{m-1,m-1})+(q+1)f(v_{m+1,m})}=\lambda^+f(v_{m,m})\\
\label{11}A_w^-f(v_{m,m})&=&{(q^2+q)f(v_{m,m-1})+f(v_{m+1,m+1})}=\lambda^-f(v_{m,m}).
\end{eqnarray}

Let us denote $\alpha=\frac{\lambda^+}{q}$, $\beta=\frac{\lambda^-}{q}$ and $a_{m,n}=\frac{f(v_{m,n})}{q^{m+n}}$. Using (<ref>) and (<ref>), for any $m\geq 2,$ we have

\begin{eqnarray}
\label{12}&&a_{m+1,0}+q(q+1)a_{m,1}=\alpha a_{m,0}\\
\label{13}&&a_{m-2,0}+q(q+1)a_{m,1}=\beta a_{m-1,0}
\end{eqnarray}

It follows from (<ref>) and (<ref>) that for any $m\geq 2,$

\begin{equation}
a_{m+1,0}-\alpha a_{m,0}+\beta a_{m-1,0}+a_{m-2,0}=0.
\end{equation}

Since the point $(s_1,s_2,s_3)$ is the solution of the equation $X^3-\alpha X^2+\beta X-1$, $a_{m,0}$ is of the form

The numbers $A_{1,0}$, $A_{2,0}$ and $A_{3,0}$ are the solution of the following system of equations

\begin{eqnarray}
\nonumber &&a_{0,0}=A_{1,0}+A_{2,0}+A_{3,0}=1\\
\nonumber &&a_{1,0}=A_{1,0}s_1+A_{2,0}s_2+A_{2,0}s_3=\frac{s_1+s_2+s_3}{q^2+q+1}\\
\nonumber &&a_{2,0}=A_{1,0}s_1^2+A_{2,0}s_2^2+A_{3,0}s_3^2=\frac{(s_1+s_2+s_3)^2-(q+1)(s_1s_2+s_2s_3+s_3s_1)}{q^2+q+1}.
\end{eqnarray}

Thus for any $i\in \{1,2,3\},$ we have

where $j\neq k\in\{1,2,3\}\backslash \{i\}. $
The equation (<ref>) shows that for any $m\geq 1,$

For $m>n\geq 1$, we have

\begin{eqnarray}
\end{eqnarray}

This implies that for any $m>n\geq 1,$

\begin{eqnarray}
\label{17}&&\alpha a_{m,n}=\frac{1}{q}a_{m-1,n-1}+qa_{m,n+1}+a_{m+1,n}\\
\label{18}&&\beta  a_{m,n}=a_{m-1,n}+\frac{1}{q}a_{m,n-1}+qa_{m+1,n+1}.
\end{eqnarray}

Denote $a_{m,n}=A_{n,1}s_1^m+A_{n,2}s_2^m+A_{n,3}s_3^m.$

It follows from (<ref>) that for any $n\in \mathbb{N}$, $i\in\{1,2,3\}$ and distinct $j, k\in \{1,2,3\}\backslash\{i\},$

Since $\frac{s_{j}}{q}$ and $\frac{s_{k}}{q}$ are the solution of the equation of $q^2X^2-q(s_{j}+s_{k})X+s_{j}s_{k},$ we have

Since we know $A_{i,0}$ and $A_{i,1},$ for any distinct $j,k\in\{1,2,3\}\backslash\{i\},$

Thus the eigenfunction $f$ is defined by

$$f(v_{m,n})=q^{m+n}a_{m,n}=\sum_{\substack{(i,j)\in \{1,2,3\}^2\\i\neq j}}B_{i,j}q^ms_i^ms_{j}^n.$$

This completes the proof.

The following proposition gives the formula of simultaneous eigenfunctions in the singular cases, that is, when the numbers $s_j$ are not distinct. It is possible to solve the recurrence relations by appropriate modifications of the above methods. Rather than this, we obtain the desired formula by taking limits of the nonsingular case.

Let $f$ be an eigenfunction of $A_w^+$ and $A_w^-$ with eigenvalues $\lambda^+$ and $\lambda^-$. If $s_1\ne s_2=s_3$, then

\begin{align*}f(v_{m,n})=\frac{q^m}{(s_1-s_2)^2(q+1)(q^2+q+1)}&\biggl[\{(s_1-qs_2)^2\{(1-q)n+(q+1)\}s_1^ms_2^n\\
\end{align*}

If $s_1=s_2=s_3$, then

\begin{align*}
\end{align*}

Assume first that $s_1\ne s_2=s_3$. Let us consider the following three limits:

\begin{equation}
\nonumber\lim_{s_3\rightarrow s_2} B_{1,2}s_2^n+B_{1,3}s_3^n=\frac{(s_1-qs_2)^2\{(1-q)n+(q+1)\}}{(s_1-s_2)^2(q+1)(q^2+q+1)}s_2^n,
\end{equation}

\begin{equation}
\nonumber\lim_{s_3\rightarrow s_2} B_{2,3}s_2^ms_3^n+B_{3,2}s_2^ns_3^m=\frac{(s_2-qs_1)^2\{(1-q)(m+n)+(q+1)\}}{(s_1-s_2)^2(q+1)(q^2+q+1)}s_2^{m+n},
\end{equation}

\begin{equation}
\begin{split}
\nonumber\lim_{s_3\rightarrow s_2} B_{2,1}s_2^m+B_{3,1}s_3^m=&m\frac{(q-1)(s_1-qs_2)(s_2-qs_1)}{(s_1-s_2)^2(q+1)(q^2+q+1)}s_2^{m},\\
\end{split}
\end{equation}

The above equations and L'Hôpital's law show that

\begin{equation}\label{eq:4.19}
\begin{split}
\end{split}
\end{equation}

Now assume that $s_1=s_2=s_3$.
The right hand side of  (<ref>) and its first derivative are zero when $s_1=s_2$. Using L'Hôpital's law twice, we have

\begin{equation}\label{eq:4.20}
\begin{split}
\end{split}
\end{equation}

which completes the proof of the proposition.

§ AUTOMORPHIC SPECTRA OF THE WEIGHTED ADJACENCY OPERATORS

In this section, we investigate the automorphic spectra of the weighted adjacency operators $A^\pm_w$ on $L^2_w(\Gamma\backslash\mathcal{B}(G))$. We will prove that there are no discrete spectrum except trivial eigenvalues. Furthermore, we will verify that there are continuous spectra of $A_w^\pm$ which are outside of the spectrum of $A^\pm$ on $L^2(\mathcal{B}(G)).$

Fix $\mathbf{s}=(s_1,s_2,s_3)\in S$. Let us denote by $f_{\mathbf{s}}$ the simultaneous eigenfunction of $A_w^{\pm}$ with eigenvalues defined in (<ref>).

The trivial eigenfunctions are the only eigenfunctions $f_\mathbf{s}$ of $A_w^\pm$ which belongs to $L^2_w(\Gamma\backslash\mathcal{B}(G))$.

By definition, we have

\begin{equation}\label{eq:5.1}
\begin{split}
\|f_\mathbf{s}\|^2_2&=\sum_{m=0}^\infty\sum_{n=0}^m |f(v_{m,n})|^2w(v_{m,n}).
\end{split}
\end{equation}

Suppose that $s_1=s_2=s_3$. By assumption, we must have $|s_1|=1$. Since the sequence $|f_\mathbf{s}(v_{m,0})|^2w(v_{m,0})$ is a polynomial in $m$ by (<ref>) and (<ref>), it follows that $\|f_\mathbf{s}\|^2_2=\infty$.

Let us consider the case when $s_1\neq s_2=s_3$. By Lemma <ref>, $|s_1|=|s_2|=1.$ Since

for some nonzero constants $A,B$ and $C$, $|f_\mathbf{s}(v_{m,0})|^2w(v_{m,0})$ diverges and $\|f_\mathbf{s}\|^2_2=\infty$.

In the case $s_1\neq s_2 \neq s_3$ and $|s_1|=|s_2|=|s_3|=1$,

Since every $A_{i,0}$ is nonzero, $|f_\mathbf{s}(v_{m,0})|^2w(v_{m,0})$ does not converge to zero as $m$ goes to infinity and $\|f_\mathbf{s}\|=\infty.$
Suppose that $|s_1|>|s_2|>|s_3|$. Since $|s_1|>1$, $|s_2|=1$ and $|s_3|=|s_1|^{-1}$ by Lemma <ref>, if $B_{1,2}$ or $B_{2,1}$ is nonzero, $|f_\mathbf{s}(v_{m,m})|^2w(v_{m,m})$ diverges as $m$ goes to infinity and $\|f_\mathbf{s}\|_2=\infty$. The 3-tuple $(s_1,s_2,s_3)$ satisfies $s_1=qs_2$ and $s_2=qs_3$  when $B_{1,2}=B_{2,1}=0$. The 3-tuple $(s_1,s_2,s_3)$ is of the form $e^{\frac{2k\pi i}{3}}(q,1,1/q)$ for any $k\in \mathbb{Z}$. Hence the eigenfunctions are described by

\begin{equation}
\nonumber f_\mathbf{s}(v_{m,n})=\omega^{m+n}\quad (\omega=e^{\frac{2k\pi i}{3}}).
\end{equation}

The remaining part is to show that the above function is $L^2$. In fact,

\begin{equation}
\begin{split}
\|f_{\mathbf{s}}\|^2_2=&\sum_{m=0}^\infty\sum_{n=0}^m |f_\mathbf{s}(v_{m,n})|^2w(v_{m,n})\\
=&\frac{B_{3,2}^2}{q^2+q+1}+\sum_{m=1} |f_\mathbf{s}(v_{m,0})|^2w(v_{m,0})+\sum_{m=1}^\infty\sum_{n=1}^{m-1} |f_\mathbf{s}(v_{m,n})|^2w(v_{m,n})\\
=&\frac{1}{q^2+q+1}+2\sum_{m=1} q^{-2m} +\sum_{m=1}^\infty\sum_{n=1}^{m-1} (q+1)q^{-2m}\\
=&\frac{1}{q^2+q+1}+2\sum_{m=1} q^{-2m} +\sum_{m=1}^\infty(m-1)(q+1)q^{-2m}<\infty.
\end{split}
\end{equation}

This yields the proof of the proposition.

From the proof of the above statement, we obtain the following two corollaries.
Every nontrivial eigenfunction $f_{\mathbf{s}}$ of $A_w^\pm$ indexed by $\mathbf{s}\in S$ is not in $L^2(\Gamma\backslash\mathcal{B}(G)).$

The operators $A_w^\pm$ are bounded operator on $L^2_w(\Gamma\backslash\mathcal{B}(G))$ with operator norm $q^2+q+1.$

If $\mathbf{s}=(q,1,\frac{1}{q})$, the eigenfunction $f_{\mathbf{s}}\equiv 1$ is positive and $A_w^\pm f_{\mathbf{s}}=(q^2+q+1) f_{\mathbf{s}}$. By Schur's test (see [13], page 102), the operator norm is $q^2+q+1.$

Given $(s_1,s_2,s_3)\in S$ and $\epsilon>0$, let us define the function $f_{\mathbf{s}}^\epsilon$ by

Let $\mathbf{s}$ be a point in $S$ such that $s_1$, $s_2$ and $s_3$ are distinct and $|s_i|=1$ for any $ i\in\{1,2,3\}$. For any $\epsilon\in (0,1/2)$, $f_\mathbf{s}^\epsilon$ is $L^2$-function and

\begin{equation}\label{eq:5.3}
\lim_{\epsilon \rightarrow 0} \frac{\|A_w^\pm f_\mathbf{s}^\epsilon-\lambda^\pm f_\mathbf{s}^\epsilon\|_2}{\|f_\mathbf{s}^\epsilon\|_2}=0,
\end{equation}

where $\lambda^\pm=q(s_1^{\pm1}+s_2^{\pm1}+s_3^{\pm1}).$ The norm $\|f_\mathbf{s}^\epsilon\|_2$ goes infinity as $\epsilon$ goes to zero.

\begin{equation}\label{eq:5.4}
|f_{\mathbf{s}}(v_{m,n})|\leq q^m\sum_{\substack{i,j\\i\neq j}}|B_{i,j}|
\end{equation}

hold for any $m\ge n\ge 0$, it follows that
$f_{\mathbf{s}}^\epsilon$ is in $L^2_w(\Gamma\backslash\mathcal{B}(G))$. By Proposition <ref>, $\|f_\mathbf{s}^\epsilon\|_2$ goes to infinity as $\epsilon$ goes to zero.

It remains to show that (<ref>). Since $\lambda^+f_\mathbf{s}^\epsilon(v_{m,n})=(1-\epsilon)^mA_w^+f_{\mathbf{s}}(v_{m,n}),$ we have

\begin{align*}\label{eq:5.5}
A_w^+f_\mathbf{s}^\epsilon(v_{0,0})-\lambda^+f_\mathbf{s}^\epsilon(v_{0,0})=& A_w^+ f_\mathbf{s}^\epsilon(v_{0,0})-A_w^+f_\mathbf{s}(v_{0,0})\\
=&-\epsilon(q^2+q+1) f(v_{1,0})\\
A_w^+f_\mathbf{s}^\epsilon(v_{m,0})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,0})=&A_w^+f_\mathbf{s}^\epsilon(v_{m,0})-(1-\epsilon)^m A_w^+f_\mathbf{s}(v_{m,0})\\
A_w^+f_\mathbf{s}^\epsilon(v_{m,m})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,m})=&A_w^+f_\mathbf{s}^\epsilon(v_{m,m})-(1-\epsilon)^m A_w^+f_\mathbf{s}(v_{m,m})\\ \displaybreak[0]
&-(1-\epsilon)^m q^2f(v_{m-1,m-1})-(1-\epsilon)^{m}(q+1)f(v_{m+1,m})\\
A_w^+ f_\mathbf{s}^\epsilon(v_{m,n})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,n})=&A_w^+f_\mathbf{s}^\epsilon(v_{m,n})-(1-\epsilon)^m A_w^+f_\mathbf{s}(v_{m,n})\\
=& (1-\epsilon)^{m-1}q^2f(v_{m-1,n})+(1-\epsilon)^mqf(v_{m,n-1})\\
\end{align*}

Denote $B:=\sum_{\substack{i,j\\i\neq j}}|B_{i,j}|.$
By (<ref>) and the above formulas together with the triangle inequality, we have

\begin{equation}\label{eq:5.6}
\begin{split}
&|A_w^+f_\mathbf{s}^\epsilon(v_{0,0})-\lambda^+f_\mathbf{s}^\epsilon(v_{0,0})|\leq \epsilon q(q^2+q+1)B\\
&|A_w^+f_\mathbf{s}^\epsilon(v_{m,0})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,0})|\leq\epsilon(1-\epsilon)^{m}q^{m+1} B\\
&|A_w^+ f_\mathbf{s}^\epsilon(v_{m,n})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,n})|\leq2\epsilon(1-\epsilon)^{m-1}(q^2+1)q^{m+1}B.
\end{split}
\end{equation}

Using (<ref>), we have

\begin{align*}
&\,\,\|A_w^\pm f_\mathbf{s}^\epsilon-\lambda^\pm f_\mathbf{s}^\epsilon\|_2^2=\sum_{m=0}^\infty\sum_{n=0}^m|A_w^+f_\mathbf{s}^\epsilon(v_{m,n})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,n})|^2w(v_{m,n})\\
=&\,\,|A_w^+f_\mathbf{s}^\epsilon(v_{0,0})-\lambda^+f_\mathbf{s}^\epsilon(v_{0,0})|^2w(v_{0,0})+\sum_{m=1}^\infty|A_w^+f_\mathbf{s}^\epsilon(v_{m,0})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,0})|^2w(v_{m,0})\\ \displaybreak[0]
&+\sum_{m=1}^\infty|A_w^+f_\mathbf{s}^\epsilon(v_{m,m})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,m})|^2w(v_{m,m}) \\ \displaybreak[0]
&+\sum_{m=1}^\infty\sum_{n=1}^{m-1}|A_w^+ f_\mathbf{s}^\epsilon(v_{m,n})-\lambda^+f_\mathbf{s}^\epsilon(v_{m,n})|^2w(v_{m,n})\\ \displaybreak[0]
\leq&\,\,\epsilon^2(q^2+q+1)q^2B^2+\epsilon^2q^2B^2\sum_{m=1}^\infty(1-\epsilon)^{2m}+4\epsilon^2q^2(q^2+q+1)^2B^2\sum_{m=1}^\infty(1-\epsilon)^{2m-2}\\ \displaybreak[0]
&+4\epsilon^2q^2(q+1)(q^2+1)^2B^2\sum_{m=1}^\infty (m-1)(1-\epsilon)^{2m-2}\\
=&\,\,\epsilon^2q^2(q^2+q+1)B^2+\frac{\epsilon^2q(1-\epsilon)^2B^2}{2\epsilon-\epsilon^2}+\frac{4\epsilon ^2q^2(q^2+q+1)^2B^2}{2\epsilon-\epsilon^2}\\ \displaybreak[0]
\leq&\,\,\epsilon^2q(q^2+q+1)B^2+{\epsilon qB^2}+{4\epsilon q(q^2+q+1)^2B^2}+{4q(q+1)(q^2+1)^2B^2}.
\end{align*}

Since $\|f_\mathbf{s}^\epsilon\|_2$ goes to infinity as $\epsilon$ goes to zero, from the above inequality we have (<ref>).

Let $\mathbf{s}$ be a point in $S$ such that $s_1>s_2>s_3$. For any $\epsilon\in (0,1/2)$, $f_\mathbf{s}^\epsilon$ is $L^2$-function and satisfies (<ref>) only if $\mathbf{s}=(\sqrt{q}e^{i\theta},e^{-2i\theta},\frac{e^{i\theta}}{\sqrt{q}})$ for any $\theta\in \mathbb{R}$ or $e^{\frac{2\pi k}{3}i}(q,1,1/q)$.

As in the proof of Proposition <ref>, for any $\mathbf{s}\in S$ with $|s_1|>|s_2|>|s_3|$, the function $f_\mathbf{s}^\epsilon$ is not in $L^2_w(\Gamma\backslash\mathcal{B}(G))$ for some $\epsilon \in (0,1/2)$ unless $B_{1,2}=B_{1,3}=B_{2,1}=0$. This holds when $s_1=qs_2$ and $s_2=qs_3$, or $s_1=qs_3$. By Lemma <ref>, $\mathbf{s}=(q\omega,\omega,\frac{\omega}{q})$ for $\omega^3=1$ or $\mathbf{s}=(\sqrt{q}e^{i\theta},e^{-2i\theta},\frac{e^{i\theta}}{\sqrt{q}})$ for $\theta\in \mathbb{R}$. Since Proposition <ref> deals with the first case, it is enough to consider the second case. In this case, we also have (<ref>). Following the proof of Lemma <ref>, we obtain Corollary <ref>.

This gives the proof of Theorem <ref>.

By Lemma <ref>, the discrete spectrum of $A_w^+$ on $L^2_w(\Gamma\backslash\mathcal{B}(G))$ is $\Sigma_0$. Lemma <ref> implies that $\Sigma_2$ is in the automorphic spectra of $A_w^+$. Corollary <ref> shows that $\Sigma_1$ is also contained in the automorphic spectra of $A_w^+$ on $L^2_w(\Gamma\backslash\mathcal{B}(G))$.

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