# WEIGHT ERGODIC THEOREMS FOR PROBABILITY MEASURES ON LOCALLY COMPACT GROUPS

H. S. MUSTAFAYEV  Van Yuzuncu Yil University, Faculty of Science, Department
of Mathematics, VAN-TURKEY<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a locally compact group with the left Haar measure $m_{G}$. A
probability measure $\mu$ on $G$ is said to be strictly aperiodic if the
support of $\mu$ is not contained in a proper closed left coset of $G.$ Assume
that $\mu$ is a strictly aperiodic measure on $G$ and
$\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded good weight for the
mean ergodic theorem [8, Definition 21.1]. We show that if $G$ is compact,
then

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}=\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\right)\overline{m}_{H},$

where $H$ is the closed subgroup of $G$ generated by
$\mathop{\mathrm{s}upp}\mu$ and $\overline{m}_{H}$ is the measure on $G$
defined by $\overline{m}_{H}\left(E\right)=m_{H}\left(E\cap H\right)$ for
every Borel subset $E$ of $G.$ If $H$ is not compact, then

$w^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}=0.$

Some related problems are also discussed.

###### Key words and phrases:

Mean ergodic theorem, locally compact group, probability measure, convergence.

###### 2010 Mathematics Subject Classification:

28A33, 43A10, 43A30, 43A77.

##  1\. Introduction

Let $G$ be a locally compact group with the left Haar measure $m_{G}$ (in the
case when $G$ is compact, $m_{G}$ will denote normalized Haar measure on $G$)
and let $M\left(G\right)$ be the convolution measure algebra of $G.$ As usual,
$C_{0}\left(G\right)$ will denote the space of all complex valued continuous
functions on $G$ vanishing at infinity. Since
$C_{0}\left(G\right)^{\ast}=M\left(G\right),$ the space $M\left(G\right)$
carries the weak∗ topology
$\sigma\left(M\left(G\right),C_{0}\left(G\right)\right).$ In the following,
the w∗-topology on $M\left(G\right)$ always means this topology. For a subset
$S$ of $G,$ by $\left[S\right]$ we will denote the closed subgroup of $G$
generated by $S.$ A probability measure $\mu$ on $G$ is said to be adapted if
$\left[\mathop{\mathrm{s}upp}\mu\right]=G.$ Also, a probability measure $\mu$
on $G$ is said to be strictly aperiodic if the support of $\mu$ is not
contained in a proper closed left coset of $G.$ For $n\in\mathbb{N},$ by
$\mu^{n}$ we will denote $n$-th convolution power of $\mu\in M\left(G\right)$,
where $\mu^{0}:=\delta_{e}$ is the Dirac measure concentrated at the unit
element $e$ of $G.$

A classical Kawada-Itô theorem [14, Theorem 7] asserts that if $\mu$ is an
adapted measure on a compact metrizable group $G,$ then the sequence of
probability measures
$\left\\{\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\right\\}_{n\in\mathbb{N}}$ weak∗
converges to the Haar measure on $G$ (see also [11, Theorem 3.2.4]). If $\mu$
is an adapted and strictly aperiodic measure on a compact metrizable group
$G,$ then w∗-$\lim_{n\rightarrow\infty}\mu^{n}=m_{G}$ [14, Theorem 8]. If
$\mu$ is an adapted measure on a second countable non-compact group $G$, then
w∗-$\lim_{n\rightarrow\infty}\mu^{n}=0$ [18, Theorem 2]. In [3, Théorème 8],
it was proved that if $\mu$ is a strictly aperiodic measure on a non-compact
locally compact group $G,$ then w∗-$\lim_{n\rightarrow\infty}\mu^{n}=0$. For
related results see also, [1], [2], [11].

In this paper, we present some results concerning convergence of probability
measures on locally compact groups.

##  2\. Weak∗ convergence

Let $X$ be a complex Banach space and let $B\left(X\right)$ be the algebra of
all bounded linear operators on $X$. An operator $T\in B\left(X\right)$ is
said to be mean ergodic if the limit

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}T^{i}x\text{ \ exists in
norm }\forall x\in X.$

If $T$ is mean ergodic, then

$P_{T}x:=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}T^{i}x\text{ \
}\left(x\in X\right)$

is the projection onto $\ker\left(T-I\right)$. The projection $P_{T}$ will be
called mean ergodic projection associated with $T.$

An operator $T\in B\left(X\right)$ is said to be power bounded if

$C_{T}:=\sup_{n\geq 0}\left\|T^{n}\right\|<\infty.$

A power bounded operator $T$ on a Banach space $X$ is mean ergodic if and only
if

(2.1) $X=\overline{\text{ran}\left(T-I\right)}\oplus\ker\left(T-I\right).$

It is well known that a power bounded operator on a reflexive Banach space is
mean ergodic (for instance see, [15, Chapter 2]).

The following result is an immediate consequence of the identity (2.1).

######  Proposition 2.1.

Let $T$ be a power bounded operator on a Banach space $X$ and assume that
$\lim_{n\rightarrow\infty}\left\|T^{n+1}x-T^{n}x\right\|=0$ for all $x\in X.$
If $T$ is mean ergodic $($so if $X$ is reflexive$)$, then
$\lim_{n\rightarrow\infty}\left\|T^{n}x-P_{T}x\right\|=0$ for all $x\in X,$
where $P_{T}$ is the mean ergodic projection associated with $T$.

The open unit disc and the unit circle in the complex plane will be denoted by
$\mathbb{D}$ and $\mathbb{T}$, respectively. If $T\in B\left(X\right)$ is
power bounded then clearly,
$\sigma\left(T\right)\subseteq\overline{\mathbb{D}}$, where
$\sigma\left(T\right)$ is the spectrum of $T.$ The classical Katznelson-
Tzafriri theorem [13] states that if $T\in B\left(X\right)$ is power bounded,
then $\lim_{n\rightarrow\infty}\left\|T^{n+1}-T^{n}\right\|=0$ if and only if
$\sigma\left(T\right)\cap\mathbb{T}\subseteq\left\\{1\right\\}.$

If $T\in B\left(X\right)$ is mean ergodic, then $T$ is Cesàro bounded, that
is,

$\sup_{n\in\mathbb{N}}\left\|\frac{1}{n}\sum_{i=1}^{n}T^{i}\right\|<\infty.$

It follows from the spectral mapping theorem that $r\left(T\right)\leq 1,$
where $r\left(T\right)$ is the spectral radius of $T.$

Let $N$ be a normal operator on a complex Hilbert space $\mathcal{H}$ with the
spectral measure $E$ $($recall that
$\sigma\left(N\right)=\mathop{\mathrm{s}upp}E$). If $N$ is mean ergodic, then
$\left\|N\right\|=r\left(N\right)\leq 1$ ( notice also that a normal operator
is power bounded if and only it is a contraction). It easily follows from the
spectral theorem that if $N$ is a normal contraction operator on a Hilbert
space $\mathcal{H}$, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=i}^{n}N^{i}x=E\left\\{1\right\\}x\text{\
\ in norm }\forall x\in\mathcal{H}.$

On the other hand, a few lines of computation show that

$\lim_{n\rightarrow\infty}\left\|N^{n+1}x-N^{n}x\right\|^{2}=\mathop{\displaystyle\int}\limits_{\sigma\left(N\right)\cap\mathbb{T\diagdown}\left\\{1\right\\}}\left|z-1\right|^{2}d\langle
E\left(z\right)x,x\rangle\text{, \ }\forall x\in\mathcal{H}\text{.}$

From this identity and from Proposition 2.1 it follows that the sequence
$\left\\{N^{n}x\right\\}_{n\in\mathbb{N}}$ converges in norm for every
$x\in\mathcal{H}$ if and only if $\left\|N\right\|\leq 1$ and
$E\left[\sigma\left(N\right)\cap\mathbb{T\diagdown}\left\\{1\right\\}\right]=0.$
Under these conditions we have

$\lim_{n\rightarrow\infty}\left\|N^{n}x-E\left\\{1\right\\}x\right\|=0\text{,
\ }\forall x\in\mathcal{H}.$

Recall from [8, Definition 21.1] that a sequence
$\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ in $\mathbb{C}$ is called good
weight for the mean ergodic theorem (briefly good weight) if for every Hilbert
space $\mathcal{H}$ and every contraction $T$ on $\mathcal{H}$ the limit

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}T^{i}x\text{ \ exists
}\forall x\in\mathcal{H}\text{.}$

By Theorem 21.2 of [8], a bounded sequence
$\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a good weight if and only if the
limit

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\xi^{i}=:a\left(\xi\right)\text{
\ exists }\forall\xi\in\mathbb{T}\text{.}$

Let $\left(\Omega,\Sigma,m\right)$ be a probability space and let
$\varphi:\Omega\rightarrow\Omega$ be a measure-preserving transformation. It
follows from the Wiener-Wintner theorem [8, Corollary 21.6] that the sequence
$\left(f\left(\varphi^{n}\left(\omega\right)\right)\right)_{n\in\mathbb{N}}$
is a bounded good weight for any $f\in L^{\infty}\left(\Omega\right)$ and for
all almost every $\omega\in\Omega$.

If $\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded good weight, then
for every contraction $T$ on a Hilbert space $\mathcal{H}$ one has

(2.2)
$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}T^{i}x=\sum_{\xi\in\sigma_{p}\left(T\right)\cap\mathbb{T}}a\left(\xi\right)P_{\xi}x\text{\
\ in norm }\forall x\in\mathcal{H},$

where $P_{\xi}$ are the orthogonal projections onto the mutually orthogonal
eigenspaces $\ker\left(T-\xi I\right)$ for
$\xi\in\sigma_{p}\left(T\right)\cap\mathbb{T}$, where
$\sigma_{p}\left(T\right)$ is the point spectrum of $T$ [8, Theorem 21.2] (it
follows that $a\left(\xi\right)\neq 0$ for at most countably many
$\xi\in\mathbb{T}$).

Let $N$ be a normal contraction operator on a Hilbert space $\mathcal{H}$ with
the spectral measure $E.$ Assume that $E\left\\{\xi\right\\}=0$ for all
$\xi\in\mathbb{T}\diagdown\left\\{1\right\\}.$ It follows from the identity
(2.2) that if $\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded good
weight, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}N^{i}x=\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\right)E\left\\{1\right\\}x\text{\
\ in norm }\forall x\in\mathcal{H}\text{.}$

Let $G$ be a locally compact group with the left Haar measure $m_{G}$. If $H$
is a closed subgroup of $G$, then the measure $m_{H}$ may be regarded as a
measure on $G$ by putting $\overline{m}_{H}\left(E\right)=m_{H}\left(E\cap
H\right)$ for every Borel subset $E$ of $G.$ Note that
$\mathop{\mathrm{s}upp}\overline{m}_{H}=H.$

One of the main results of the paper is the following.

######  Theorem 2.2.

Let $\mu$ be a strictly aperiodic measure on a locally compact group $G$ and
let $\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ be a bounded good weight. The
following assertions hold:

$\left(a\right)$ If $G$ is compact, then

$w^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}=\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\right)\overline{m}_{\left[\mathop{\mathrm{s}upp}\mu\right]}.$

$\left(b\right)$ If $\left[\mathop{\mathrm{s}upp}\mu\right]$ is not compact,
then

$w^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}=0.$

Below, we collect some preliminary results that will needed later on.

Let $G$ be a locally compact group. A representation $\pi$ of $G$ on a Banach
space $X_{\pi}$ (the representation space of $\pi)$ is a homomorphism from $G$
to the group of invertible isometries on $X_{\pi}.$ We assume that $\pi$ is
strongly continuous. Then, for an arbitrary $\mu\in M\left(G\right)$ we can
define a bounded linear operator $\widehat{\mu}\left(\pi\right)$ on $X_{\pi}$
by

$\widehat{\mu}\left(\pi\right)x=\int_{G}\pi\left(g\right)xd\mu\left(g\right)\text{,
\ }x\in X_{\pi}.$

The map $\mu\rightarrow\widehat{\mu}\left(\pi\right)$ is linear,
multiplicative, and contractive;
$\left\|\widehat{\mu}\left(\pi\right)\right\|\leq\left\|\mu\right\|_{1},$
where $\left\|\mu\right\|_{1}$ is the total variation norm of $\mu\in
M\left(G\right).$

By $\widehat{G}$ we will denote unitary dual of $G$, the set of all
equivalence classes of irreducible continuous unitary representations of $G$
with the Fell topology. Recall that $\pi_{0}\in\widehat{G}$ is a limit point
of $M\subset\widehat{G}$ in the Fell topology, if the matrix function
$g\rightarrow\langle\pi_{0}\left(g\right)x_{0},x_{0}\rangle$
$\left(x_{0}\in\mathcal{H}_{\pi_{0}}\right)$ can be uniformly approximated on
every compact subset of $G$ by the matrix functions
$g\rightarrow\langle\pi\left(g\right)x,x\rangle$ $\left(\pi\in M,\text{
}x\in\mathcal{H}_{\pi}\right)$ (in the case when $G$ is abelian, Fell topology
coincides with the usual topology of $\widehat{G},$ the dual group of $G$).
The function $\pi\rightarrow\widehat{\mu}\left(\pi\right)$
$\left(\pi\in\widehat{G}\right)$ is called Fourier-Stieltjes transform of
$\mu\in M\left(G\right)$ (if $G$ is abelian, the Fourier-Stieltjes transform
of $\mu\in M\left(G\right)$ will be denoted simply by $\widehat{\mu}$). If
$\widehat{\mu}\left(\pi\right)=0$ for all $\pi\in\widehat{G},$ then $\mu=0$
(for instance see, [5, §18]). It is well known that if $G$ is compact, then
every $\pi\in\widehat{G}$ is finite dimensional. Also, we know that if $G$ is
compact (resp. compact and metrizable) then $\widehat{G}$ is discrete (resp.
countable). These facts are consequences of the Peter-Weyl theory [17, Chapter
4].

By $B_{X}$ and $S_{X}$ respectively, we denote the closed unit ball and the
unit sphere of a Banach space $X.$ Notice that ext$B_{X}\subseteq S_{X},$
where ext$B_{X}$ is the set of all extreme points of $B_{X}.$ For the sake of
convenience, $X$ will be called $A$-space if ext$B_{X}=S_{X}$. For example,
uniformly convex Banach spaces, in particular, Hilbert spaces and $L^{p}$
$\left(1<p<\infty\right)$ spaces are $A$-spaces.

The proof of the following result is similar to the proof of [4, Proposition
2.1].

######  Lemma 2.3.

Let $\mu$ be a probability measure on a locally compact group $G$ and let
$\pi$ be a continuous Banach representation of $G$. If the representation
space $X_{\pi}$ is a Banach $A$-space, then for an arbitrary
$\xi\in\mathbb{T}$ we have

$\ker\left[\widehat{\mu}\left(\pi\right)-\xi I_{\pi}\right]=\left\\{x\in
X_{\pi}:\pi\left(g\right)x=\xi x,\text{ }\forall
g\in\mathop{\mathrm{s}upp}\mu\right\\}.$

###### Proof.

Assume that $\widehat{\mu}\left(\pi\right)x=\xi x$ for some $x\in X_{\pi}$
with $\left\|x\right\|=1$ and $\xi\in\mathbb{T}.$ Let
$s\in$$\mathop{\mathrm{s}upp}\mu$ and let $\left\\{U_{i}\right\\}_{i\in I}$ be
a directed (by reverse inclusion) basis neighborhood system of $s.$ Then
$\mu\left(U_{i}\right)>0$ for all $i\in I.$ There is no restriction to assume
that $\mu\left(U_{i}\right)<1$ for all $i\in I.$ If

$x_{i}:=\frac{1}{\mu\left(U_{i}\right)}\int_{U_{i}}\pi\left(g\right)xd\mu\left(g\right)\text{
\ and \ \ }y_{i}:=\frac{1}{1-\mu\left(U_{i}\right)}\int_{G\diagdown
U_{i}}\pi\left(g\right)xd\mu\left(g\right),$

then $\left\|x_{i}\right\|\leq 1$ and $\left\|y_{i}\right\|\leq 1$. The
identity

$\xi x=\mu\left(U_{i}\right)x_{i}+\left[1-\mu\left(U_{i}\right)\right]y_{i}$

shows that $\xi x$ is a convex combination of $x_{i}$ and $y_{i}.$ Since
$\left\|\xi x\right\|=1,$ we have $\xi x=x_{i}$ for all $i\in I$ and
therefore,

$\frac{1}{\mu\left(U_{i}\right)}\int_{U_{i}}\left[\pi\left(g\right)x-\xi
x\right]d\mu\left(g\right)=0\text{, \ }\forall i\in I.$

As $U_{i}\rightarrow\left\\{s\right\\}$ we obtain $\pi\left(s\right)x=\xi x.$

The following result was proved in [20, Lemma 2.3].

######  Lemma 2.4.

Let $\mu$ be a strictly aperiodic measure on a locally compact group $G$ and
let $\pi$ be a continuous unitary representation of $G$. Then, the operator
$\widehat{\mu}\left(\pi\right)$ cannot have unitary eigenvalues except
$\xi=1.$

Now, let $\mu$ be a strictly aperiodic measure on a locally compact group $G$
and let $\pi$ be a continuous unitary representation of $G$. By Lemma 2.4, the
operator $\widehat{\mu}\left(\pi\right)$ cannot have unitary eigenvalues
except $\xi=1.$ In view of the identity (2.2) and Lemma 2.3, we have the
following.

######  Corollary 2.5.

Let $\mu$ be a strictly aperiodic measure on a locally compact group $G$ and
let $\pi$ be a continuous unitary representation of $G$ on a Hilbert space
$\mathcal{H}_{\pi}$. If $\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded
good weight, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\widehat{\mu}\left(\pi\right)^{i}x=\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\right)P_{\mu}^{\pi}x\text{
\ in norm }\forall x\in\mathcal{H}_{\pi}\text{,}$

where $P_{\mu}^{\pi}$ is the orthogonal projection onto

$\left\\{x\in\mathcal{H}_{\pi}:\pi\left(g\right)x=x:\forall
g\in\left[\mathop{\mathrm{s}upp}\mu\right]\right\\}.$

In addition if $\mu$ is adapted and $\pi\in\widehat{G}\diagdown id,$ then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\widehat{\mu}\left(\pi\right)^{i}x=0\text{
\ in norm }\forall x\in\mathcal{H}_{\pi}.$

Let $G$ be a locally compact group. For an arbitrary $f\in
L^{p}\left(G\right)$ $\left(1<p<\infty\right),$ we put
$f^{\vee}\left(g\right):=f\left(g^{-1}\right)$ and
$\widetilde{f}:=\overline{f\left(g^{-1}\right)}$. Notice that for every
$u,\upsilon\in L^{2}\left(G\right),$ the function $u\ast\widetilde{\upsilon}$
is in $C_{0}\left(G\right)$ and

$\langle\mu,u\ast\widetilde{\upsilon}\rangle=\langle\mu\ast\overline{\upsilon},\overline{u}\rangle\text{,
\ }\forall\mu\in M\left(G\right).$

It follows that $\left\\{u\ast\widetilde{\upsilon}:u,\upsilon\in
L^{2}\left(G\right)\right\\}$ is linearly dense in $C_{0}\left(G\right).$
Notice also that if $f\in L^{p}\left(G\right)$ $\left(p\neq 2\right)$ and
$h\in L^{q}\left(G\right)$ ($1/p+1/q=1$), then $h\ast f^{\vee}\in
C_{0}\left(G\right)$ and

$\langle\mu,h\ast f^{\vee}\rangle=\langle\mu\ast f,h\rangle,\text{ \
}\forall\mu\in M\left(G\right).$

It follows that $\left\\{h\ast f^{\vee}:h\in L^{q}\left(G\right),\text{ }f\in
L^{p}\left(G\right)\right\\}$ is linearly dense in $C_{0}\left(G\right).$

Let $\pi$ be the left regular representation of $G$ on $L^{p}\left(G\right)$
$\left(1\leq p<\infty\right)$, where

$\pi\left(g\right)f\left(s\right)=f\left(g^{-1}s\right):=f_{g}\left(s\right).$

Then, $\pi$ is continuous and $\widehat{\mu}\left(\pi\right)f=\mu\ast f$ for
every $\mu\in M\left(G\right).$ We will denote this operator by
$\lambda_{p}\left(\mu\right).$ It is well known that the left convolution
operator $\lambda_{p}\left(\mu\right)f:=\mu\ast f$ is a bounded linear
operator on $L^{p}\left(G\right),$ that is,

$\left\|\lambda_{p}\left(\mu\right)f\right\|_{p}\leq\left\|\mu\right\|_{1}\left\|f\right\|_{p}\text{
\ and \
}\left\|\lambda_{1}\left(\mu\right)\right\|_{1}=\left\|\mu\right\|_{1}.$

From the identity (2.2) and Lemmas 2.3 and 2.4, we obtain the following.

######  Corollary 2.6.

Let $\mu$ be a strictly aperiodic measure on a locally compact group $G$. If
$\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded good weight, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}\ast
f=\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\right)P_{\mu}f\text{
\ in }L^{2}\text{-norm }\forall f\in L^{2}\left(G\right),$

where $P_{\mu}$ is the orthogonal projection onto

$\left\\{f\in L^{2}\left(G\right):f_{g}=f:\forall
g\in\left[\mathop{\mathrm{s}upp}\mu\right]\right\\}.$

If $\left[\mathop{\mathrm{s}upp}\mu\right]$ is not compact, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}\ast f=0\text{
\ in }L^{2}\text{-norm for every }f\in L^{2}\left(G\right).$

It follows from Lemma 2.3 that if $\mu\in M\left(G\right)$ is a probability
measure and $1<p<\infty,$ then for every $\xi\in\mathbb{T}$,

$\ker\left[\lambda_{p}\left(\mu\right)-\xi I\right]=\left\\{f\in
L^{p}\left(G\right):f_{g}=\xi f,\text{ }\forall
g\in\mathop{\mathrm{s}upp}\mu\right\\}.$

From this identity we can deduce that if the support of the probability
measure $\mu\in M\left(G\right)$ is not compact, then the operator
$\lambda_{p}\left(\mu\right)$ $\left(1<p<\infty\right)$ has no unitary
eigenvalues. On the other hand, we know [7, Chapter II, Theorem 4.1] that if
$T$ is a power bounded operator on a reflexive Banach space $X,$ then $T$ has
no unitary eigenvalues if and only if

$\lim_{n\rightarrow\infty}\frac{1}{n}\mathop{\displaystyle\sum}\limits_{i=1}^{n}\left|\langle
T^{n}x,\varphi\rangle\right|=0\text{ \ for all }x\in X\text{ and }\varphi\in
X^{\ast}.$

Hence we have the following.

######  Corollary 2.7.

If $\mu\in M\left(G\right)$ is a probability measure with non-compact support,
then for every $f\in L^{p}\left(G\right)$ $\left(1<p<\infty\right)$ and $h\in
L^{q}\left(G\right)$ $\left(1/p+1/q=1\right),$ we have

$\lim_{n\rightarrow\infty}\frac{1}{n}\mathop{\displaystyle\sum}\limits_{i=1}^{n}\left|\langle\mu^{n}\ast
f,h\rangle\right|=0.$

We now in a position to prove Theorem 2.2.

###### Proof of theorem 2.2.

Let $H:=\left[\mathop{\mathrm{s}upp}\mu\right]$. If

$m_{n}:=\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i},$

then

$\sup_{n\in\mathbb{N}}\left\|m_{n}\right\|_{1}\leq\sup_{n\in\mathbb{N}}\left\\{\left|a_{n}\right|\right\\}<\infty.$

(a) Let $\pi\in\widehat{G}$ and let
$\left\\{e_{\pi}^{\left(1\right)},...,e_{\pi}^{\left(n_{\pi}\right)}\right\\}$
be the basic vectors of the representation space $\mathcal{H}_{\pi}$
($\dim\mathcal{H}_{\pi}=n_{\pi}$). Since

$\widehat{m_{n}}\left(\pi\right)=\frac{1}{n}\sum_{i=1}^{n}a_{i}\widehat{\mu}\left(\pi\right)^{i},$

by Corollary 2.6,

$\lim_{n\rightarrow\infty}\widehat{m_{n}}\left(\pi\right)=aP_{\mu}^{\pi}\text{
\ in the strong operator topology,}$

where

$a:=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}$

and $P_{\mu}^{\pi}$ is the orthogonal projection onto
$\ker\left[\widehat{\mu}\left(\pi\right)-I\right].$ Let $f_{i,j}^{\pi}$ be the
matrix functions of $\pi$, where

$f_{i,j}^{\pi}\left(g\right)=\langle\pi\left(g\right)e_{\pi}^{\left(i\right)},e_{\pi}^{\left(j\right)}\rangle\text{
\ }\left(i,j=1,...,n_{\pi}\right).$

Then we can write

$\lim_{n\rightarrow\infty}\langle
m_{n},f_{i,j}^{\pi}\rangle=\lim_{n\rightarrow\infty}\langle\widehat{m_{n}}\left(\pi\right)e_{\pi}^{\left(i\right)},e_{\pi}^{\left(j\right)}\rangle=\langle
aP_{\mu}^{\pi}e_{\pi}^{\left(i\right)},e_{\pi}^{\left(j\right)}\rangle.$

By the Peter-Weyl $C$-Theorem [17, Chapter 4], the system of matrix functions

$\left\\{f_{i,j}^{\pi}:\pi\in\widehat{G},\text{ }i,j=1,...,n_{\pi}\right\\}$

is linearly dense in $C\left(G\right),$ the space of all complex valued
continuous functions on $G.$ Since the sequence
$\left\\{m_{n}\right\\}_{n\in\mathbb{N}}$ is norm bounded, the limit
$\lim_{n\rightarrow\infty}\langle m_{n},f\rangle$ exists for all $f\in
C\left(G\right).$ If $a=0,$ then there is nothing to prove. Because in this
case, $\lim_{n\rightarrow\infty}\langle m_{n},f\rangle=0$ for all $f\in
C\left(G\right)$ and therefore, w∗-$\lim m_{n}=0.$ Hence, we may assume that
$a\neq 0.$

Further, since

$f\mapsto\lim_{n\rightarrow\infty}\langle m_{n},f\rangle$

is a bounded linear functional on $C\left(G\right)$, there exists
$\theta_{\mu}\in M\left(G\right)$ such that

$\lim_{n\rightarrow\infty}\langle
m_{n},f\rangle=\langle\theta_{\mu},f\rangle\text{, \ }\forall f\in
C\left(G\right).$

So we have

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}m_{n}=\theta_{\mu}.$

It follows that

$\widehat{m_{n}}\left(\pi\right)\rightarrow\widehat{\theta_{\mu}}\left(\pi\right)\text{
\ in the weak operator topology for all }\pi\in\widehat{G}.$

Consequently we have $\widehat{\theta_{\mu}}\left(\pi\right)=aP_{\mu}^{\pi}$
for all $\pi\in\widehat{G}$. Now, we must show that
$\theta_{\mu}=a\overline{m}_{H}.$ To see this, let $\pi\in\widehat{G}$ be
given. Since $\overline{m}_{H}$ is an idempotent measure,
$\widehat{\overline{m}_{H}}\left(\pi\right)$ is an orthogonal projection.
Since $\mathop{\mathrm{s}upp}\overline{m}_{H}=H,$ by Lemma 2.3,

$\displaystyle\widehat{\overline{m}_{H}}\left(\pi\right)\mathcal{H}_{\pi}$
$\displaystyle=$
$\displaystyle\ker\left[\widehat{\overline{m}_{H}}\left(\pi\right)-I_{\pi}\right]$
$\displaystyle=$
$\displaystyle\left\\{x\in\mathcal{H}_{\pi}:\pi\left(g\right)x=x,\text{
}\forall g\in H\right\\}.$

For the same reason,

$\displaystyle\frac{1}{a}\widehat{\theta_{\mu}}\left(\pi\right)\mathcal{H}_{\pi}$
$\displaystyle=$ $\displaystyle
P_{\mu}^{\pi}\mathcal{H}_{\pi}=\ker\left[\widehat{\mu}\left(\pi\right)-I_{\pi}\right]$
$\displaystyle=$
$\displaystyle\left\\{x\in\mathcal{H}_{\pi}:\pi\left(g\right)x=x,\text{
}\forall g\in H\right\\}.$

So we have
$\widehat{\theta_{\mu}}\left(\pi\right)=a\widehat{\overline{m}_{H}}\left(\pi\right)$
for all $\pi\in\widehat{G}.$ It follows that $\theta=a\overline{m}_{H}.$

(b) Assume that $H$ is not compact. As we have noted above, if $u,\upsilon\in
L^{2}\left(G\right),$ then $u\ast\widetilde{\upsilon}\in C_{0}\left(G\right)$
and the set $\left\\{u\ast\widetilde{\upsilon}:u,\upsilon\in
L^{2}\left(G\right)\right\\}$ is linearly dense in $C_{0}\left(G\right)$.
Since the sequence $\left\\{m_{n}\right\\}_{n\in\mathbb{N}}$ is uniformly
bounded, it suffices to show that

$\lim_{n\rightarrow\infty}\langle m_{n},u\ast\widetilde{\upsilon}\rangle=0.$

By Corollary 2.6,

$\lim_{n\rightarrow\infty}\left(m_{n}\ast\overline{\upsilon}\right)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}\ast\overline{\upsilon}=0\
\ \text{in }L^{2}\text{-norm.}$

So we have

$\langle
m_{n},u\ast\widetilde{\upsilon}\rangle=\lim_{n\rightarrow\infty}\langle
m_{n}\ast\overline{\upsilon},\overline{u}\rangle=0.$

Next, we have the following.

######  Proposition 2.8.

Let $G$ be a compact group and let $\left\\{\mu_{n}\right\\}_{n\in\mathbb{N}}$
be a bounded sequence in $M\left(G\right)$. The following conditions are
equivalent:

$\left(a\right)$ $w^{\ast}$-$\lim_{n\rightarrow\infty}\mu_{n}=\mu$ for some
$\mu\in M\left(G\right).$

$\left(b\right)$ $\lim_{n\rightarrow\infty}\left(\mu_{n}\ast f\right)=\mu\ast
f$ uniformly on $G$ for every $f\in C\left(G\right).$

###### Proof.

(a)$\Rightarrow$(b) Let $\pi\in\widehat{G}$ and let $\mathcal{H}_{\pi}$ be the
representation space of $\pi.$ If

$f_{x,y}^{\pi}\left(g\right):=\langle\pi\left(g\right)x,y\rangle\text{ \
}(x,y\in\mathcal{H}_{\pi}),$

then as

$\langle\theta,f_{x,y}^{\pi}\rangle=\langle\widehat{\theta}\left(\pi\right)x,y\rangle\text{,
\ }\forall\theta\in M\left(G\right),$

we have

$\langle\widehat{\mu_{n}}\left(\pi\right)x,y\rangle=\langle\mu_{n},f_{x,y}^{\pi}\rangle\rightarrow\langle\mu,f_{x,y}^{\pi}\rangle=\langle\widehat{\mu}\left(\pi\right)x,y\rangle\text{.}$

Consequently,
$\widehat{\mu_{n}}\left(\pi\right)\rightarrow\widehat{\mu}\left(\pi\right)$ in
the strong operator topology. A few lines of computation show that

(2.3) $\left(\theta\ast
f_{x,y}^{\pi}\right)\left(g\right)=\langle\pi\left(g\right)x,\widehat{\theta}\left(\pi\right)y\rangle\text{,
\ }\forall\theta\in M\left(G\right).$

Now let $f\in C\left(G\right)$ be given. Since the system of matrix functions
is linearly dense in $C\left(G\right),$ for an arbitrary $\varepsilon>0$ there
exist complex numbers $\lambda_{1},...,\lambda_{k}$ and
$\pi_{1},...,\pi_{k}\in\widehat{G}$ such that

$\left|f\left(g\right)-\lambda_{1}\langle\pi_{1}\left(g\right)x_{1},y_{1}\rangle-...-\lambda_{k}\langle\pi_{k}\left(g\right)x_{k},y_{k}\rangle\right|<\varepsilon\text{
\ }\left(\forall g\in G\right),$

where $x_{i},y_{i}\in\mathcal{H}_{\pi_{i}}$ $(i=1,...,k).$ By (2.3),

$\left|\left(\mu_{n}\ast
f\right)\left(g\right)-\lambda_{1}\langle\pi_{1}\left(g\right)x_{1},\widehat{\mu_{n}}\left(\pi_{1}\right)y_{1}\rangle-...-\lambda_{k}\langle\pi_{k}\left(g\right)x_{k},\widehat{\mu_{n}}\left(\pi_{k}\right)y_{k}\rangle\right|<\varepsilon
C$

and

$\left|\left(\mu\ast
f\right)\left(g\right)-\lambda_{1}\langle\pi_{1}\left(g\right)x_{1},\widehat{\mu}\left(\pi_{1}\right)y_{1}\rangle-...-\lambda_{k}\langle\pi_{k}\left(g\right)x_{k},\widehat{\mu}\left(\pi_{k}\right)y_{k}\rangle\right|<\varepsilon
C,$

where $C:=\sup_{n\in\mathbb{N}}\left\|\mu_{n}\right\|.$ It follows that

$\displaystyle\sup_{g\in G}\left|\left(\mu_{n}\ast
f\right)\left(g\right)-\left(\mu\ast f\right)\left(g\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\lambda_{1}\right|\left\|\widehat{\mu_{n}}\left(\pi_{1}\right)y_{1}-\widehat{\mu}\left(\pi_{1}\right)y_{1}\right\|\left\|x_{1}\right\|+...$
$\displaystyle+\left|\lambda_{k}\right|\left\|\widehat{\mu_{n}}\left(\pi_{k}\right)y_{k}-\widehat{\mu}\left(\pi_{k}\right)y_{k}\right\|\left\|x_{k}\right\|+2\varepsilon
C\text{.}$

Since
$\widehat{\mu_{n}}\left(\pi\right)x\rightarrow\widehat{\mu}\left(\pi\right)x$
in norm for all $\pi\in\widehat{G}$ and $x\in\mathcal{H}_{\pi},$ from the
preceding inequality we can deduce that $\mu_{n}\ast f\rightarrow\mu\ast f$
uniformly on $G.$

(b)$\Rightarrow$(a) For an arbitrary $f\in C\left(G\right),$

$\int_{G}fd\mu_{n}-\int_{G}fd\mu=\left(\mu_{n}\ast
f\right)\left(e\right)-\left(\mu\ast f\right)\left(e\right)\rightarrow 0.$

From Theorem 2.2 and Proposition 2.8 we have the following.

######  Corollary 2.9.

Let $\mu\in M\left(G\right)$ be an adapted and strictly aperiodic measure on a
compact group $G.$ If $\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded
good weight, then for every $f\in C\left(G\right),$

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}\ast
f=\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\right)\int_{G}fdm_{G}\text{
\ uniformly on }G.$

Recall that a measure $\mu\in M\left(G\right)$ is said to be power bounded if

$C_{\mu}:=\sup_{n\geq 0}\left\|\mu^{n}\right\|_{1}<\infty.$

If $\mu\in M\left(G\right)$ is power bounded, then so is the operator
$\lambda_{p}\left(\mu\right)$ $\left(1<p<\infty\right).$

The following result was proved in [10, Theorem 3.4]. The same result for
locally compact abelian groups was obtained earlier in [19, Proposition 2.5].
For completeness we give the proof of this result. Our proof is different.

######  Proposition 2.10.

If $\mu\in M\left(G\right)$ is power bounded, then there exists an idempotent
measure $\theta_{\mu}$ in $M\left(G\right)$ such that

$w^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu^{i}=\theta_{\mu}$

$(\theta_{\mu}$ will be called limit measure associated with $\mu).$

###### Proof.

Let $u,\upsilon\in L^{2}\left(G\right)$. Since the operator
$\lambda_{2}\left(\mu\right)$ is mean ergodic, there exists $w\in
L^{2}\left(G\right)$ such that

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\ast\overline{\upsilon}=w\text{
\ in }L^{2}\text{-norm.}$

Therefore we have

$\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}\mu^{i},u\ast\widetilde{\upsilon}\rangle=\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\ast\overline{\upsilon},\overline{u}\rangle=\langle
w,\overline{u}\rangle.$

Since the sequence
$\left\\{\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\right\\}_{n\in\mathbb{N}}$ is
bounded and the set $\left\\{u\ast\widetilde{\upsilon}:u,\upsilon\in
L^{2}\left(G\right)\right\\}$ is linearly dense in $C_{0}\left(G\right),$ the
limit $\lim_{n\rightarrow\infty}\langle m_{n},f\rangle\ \ $exists for all
$f\in C_{0}\left(G\right).$ Consequently, there exists $\theta_{\mu}\in
M\left(G\right)$ such that

$\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}\mu^{i},f\rangle=\langle\theta_{\mu},f\rangle\text{,
\ }\forall f\in C_{0}\left(G\right).$

So we have

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu^{i}=\theta_{\mu}.$

It is easy to check that $\theta_{\mu}$ is an idempotent measure.

######  Remark 2.11.

Let $G$ be a locally compact abelian group and let $\mu\in M\left(G\right)$ be
power bounded. If $\theta_{\mu}$ is the limit measure associated with $\mu,$
then $\widehat{\theta_{\mu}}=\mathbf{1}_{\text{int}\mathcal{F}_{\mu}},$ where
$\mathcal{F}_{\mu}:=\left\\{\chi\in\widehat{G}:\widehat{\mu}\left(\chi\right)=1\right\\}$
and $\mathbf{1}_{\text{int}\mathcal{F}_{\mu}}$ is the characteristic function
of int$\mathcal{F}_{\mu}$. Consequently, int$\mathcal{F}_{\mu}$ is a clopen
subset of $\widehat{G}$ [19, Proposition 2.5]. It is easy to check that if
$\mu$ is a probability measure, then $\mathcal{F}_{\mu}$ is a closed subgroup
of $\widehat{G},$ that is,
$\mathcal{F}_{\mu}=\left[\mathop{\mathrm{s}upp}\mu\right]^{\bot}.$

######  Corollary 2.12.

If $\mu\in M\left(G\right)$ is power bounded and $1<p<\infty,$ then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\ast
f=\theta_{\mu}\ast f\text{ \ in }L^{p}\text{-norm for every }f\in
L^{p}\left(G\right),$

where $\theta_{\mu}$ is the limit measure associated with $\mu$
$(f\rightarrow\theta_{\mu}\ast f$ is the mean ergodic projection associated
with $\lambda_{p}\left(\mu\right)).$

###### Proof.

Let $f\in L^{p}\left(G\right)$ and $h\in L^{q}\left(G\right)$ $(1/p+1/q=1).$
Since

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu^{i}=\theta_{\mu}\text{
\ and \ }h\ast f^{\vee}\in C_{0}\left(G\right),$

we can write

$\displaystyle\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\ast
f,h\rangle$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}\mu^{i},h\ast
f^{\vee}\rangle$ $\displaystyle=$ $\displaystyle\langle\theta_{\mu},h\ast
f^{\vee}\rangle=\langle\theta_{\mu}\ast f,h\rangle.$

This shows that

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\ast
f=\theta_{\mu}\ast f\ \ \text{weakly.}$

Since the operator $\lambda_{p}\left(\mu\right)$ is mean ergodic, we have

$\lim\frac{1}{n}\sum_{i=1}^{n}\mu^{i}\ast f=\theta_{\mu}\ast f\text{ \ in
}L^{p}\text{-norm.}$

Let $\mu\in M\left(G\right)$ be power bounded and $\xi\in\mathbb{T}$. Denote
by $\theta_{\mu}^{\xi}$ the limit measure associated with $\xi\mu.$ By
Proposition 2.10, $\theta_{\mu}^{\xi}$ is an idempotent measure and

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\xi^{i}\mu^{i}=\theta_{\mu}^{\xi}.$

######  Theorem 2.13.

Let $G$ be a second countable locally compact group and let $\mu\in
M\left(G\right)$ be a measure with $\left\|\mu\right\|_{1}\leq 1.$ The
following assertions hold:

$\left(a\right)$
$\sigma_{p}\left(\lambda_{2}\left(\mu\right)\right)\cap\mathbb{T}$ is at most
countable.

$\left(b\right)$ If $\left\\{a_{n}\right\\}_{n\in\mathbb{N}}$ is a bounded
good weight and
$\sigma_{p}\left(\lambda_{2}\left(\mu\right)\right)\cap\mathbb{T=}\left\\{\xi_{1},\xi_{2},...\right\\},$
then

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}=\sum_{i=1}^{\infty}a\left(\xi_{i}\right)\theta_{\mu}^{\xi_{i}},$

where $\theta_{\mu}^{\xi_{i}}$ is the limit measure associated with
$\xi_{i}\mu$ and

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}a_{k}\xi_{i}^{k}=a\left(\xi_{i}\right).$

###### Proof.

(a) Since $L^{2}\left(G\right)$ is separable and $\lambda_{2}\left(\mu\right)$
is a contraction, by the Jamison theorem [12],
$\sigma_{p}\left(\lambda_{2}\left(\mu\right)\right)\cap\mathbb{T}$ is at most
countable.

(b) Let $f\in L^{2}\left(G\right)$ and $\xi\in\mathbb{T}$ be given. By
Corollary 2.12,

$\frac{1}{n}\sum_{i=1}^{n}\xi^{i}\lambda_{2}\left(\mu\right)^{i}f\rightarrow\theta_{\mu}^{\xi}\ast
f\ \ \text{in }L^{2}\text{-norm.}$

Notice that $f\mapsto\theta_{\mu}^{\xi}\ast f$ is the orthogonal projection
onto $\ker\left[\lambda_{2}\left(\mu\right)-\xi I\right].$ Taking into account
the identity (2.2), for an arbitrary $u,\upsilon\in L^{2}\left(G\right)$ we
can write

$\displaystyle\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i},u\ast\widetilde{\upsilon}\rangle$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\langle\frac{1}{n}\sum_{i=1}^{n}a_{i}\lambda_{2}\left(\mu\right)^{i}\overline{u},\overline{v}\rangle$
$\displaystyle=$
$\displaystyle\langle\sum_{i=1}^{\infty}a\left(\xi_{i}\right)\theta_{\mu}^{\xi_{i}}\ast\overline{u},\overline{v}\rangle$
$\displaystyle=$
$\displaystyle\langle\sum_{i=1}^{\infty}a\left(\xi_{i}\right)\theta_{\mu}^{\xi_{i}},u\ast\widetilde{\upsilon}\rangle.$

Since

$\sup_{n\in\mathbb{N}}\left\|\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}\right\|_{1}\leq\sup_{n\in\mathbb{N}}\left|a_{n}\right|<\infty$

and the set $\left\\{u\ast\widetilde{\upsilon}:u,\upsilon\in
L^{2}\left(G\right)\right\\}$ is linearly dense in $C_{0}\left(G\right),$ we
have

$\text{w}^{\ast}-\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}\mu^{i}=\sum_{i=1}^{\infty}a\left(\xi_{i}\right)\theta_{\mu}^{\xi_{i}}.$

##  3\. The sequence $\left\\{\mu^{n}\right\\}_{n\in\mathbb{N}}$

As we have noted above,
$\left\|\lambda_{1}\left(\mu\right)\right\|=\left\|\mu\right\|_{1}$ for all
$\mu\in M\left(G\right).$ Moreover, we have
$\sigma\left(\lambda_{1}\left(\mu\right)\right)=\sigma_{M\left(G\right)}\left(\mu\right)$
for all $\mu\in M\left(G\right),$ where
$\sigma_{M\left(G\right)}\left(\mu\right)$ is the spectrum of $\mu$ with
respect to the algebra $M\left(G\right).$

######  Proposition 3.1.

If $\mu\in M\left(G\right)$ is power bounded with
$\sigma\left(\lambda_{1}\left(\mu\right)\right)\cap\mathbb{T\subseteq}\left\\{1\right\\},$
then there exists an idempotent measure $\theta\in M\left(G\right)$ such that

$w^{\ast}-\lim_{n\rightarrow\infty}\mu^{n}=\theta.$

###### Proof.

It suffices to show that the sequence
$\left\\{\mu^{n}\right\\}_{n\in\mathbb{N}}$ has only one weak∗ cluster point.
Since
$\sigma\left(\lambda_{1}\left(\mu\right)\right)\cap\mathbb{T\subseteq}\left\\{1\right\\}$,
by the Katznelson-Tzafriri theorem mentioned above,

$\lim_{n\rightarrow\infty}\left\|\mu^{n+1}-\mu^{n}\right\|_{1}=\lim_{n\rightarrow\infty}\left\|\lambda_{1}\left(\mu\right)^{n+1}-\lambda_{1}\left(\mu\right)^{n}\right\|=0.$

Assume that

$\theta_{1}=\text{w}^{\ast}\text{-}\lim_{\alpha}\mu^{n_{\alpha}}\text{ \ and \
}\theta_{2}=\text{w}^{\ast}\text{-}\lim_{\beta}\mu^{m_{\beta}},$

for two subnets $\left\\{\mu^{n_{\alpha}}\right\\}_{\alpha}$ and
$\left\\{\mu^{m_{\beta}}\right\\}_{\beta}$ of
$\left\\{\mu^{n}\right\\}_{n\in\mathbb{N}}.$ Since the multiplication on
$M\left(G\right)$ is separately w∗-continuous,

$\mu\ast\theta_{1}=\theta_{1}\ast\mu=\text{w}^{\ast}\text{-}\lim_{\alpha}\mu^{n_{\alpha}+1}.$

Consequently we have

$\left\|\mu\ast\theta_{1}-\theta_{1}\right\|_{1}\leq\underline{\lim}_{\alpha}\left\|\mu^{n_{\alpha}+1}-\mu^{n_{\alpha}}\right\|=0$

and

$\left\|\theta_{1}\ast\mu-\theta_{1}\right\|_{1}\leq\underline{\lim}_{\alpha}\left\|\mu^{n_{\alpha}+1}-\mu^{n_{\alpha}}\right\|=0.$

Hence $\mu\ast\theta_{1}=\theta_{1}\ast\mu=\theta_{1}.$ Now, passing to the
limit (in the w∗-topology) in the identities

$\mu^{m_{\beta}}\ast\theta_{1}=\theta_{1}\ast\mu^{m_{\beta}}=\theta_{1},$

we have $\theta_{2}\ast\theta_{1}=\theta_{1}\ast\theta_{2}=\theta_{1}.$
Similarly, $\theta_{2}\ast\theta_{1}=\theta_{1}\ast\theta_{2}=\theta_{2}.$ If
$\theta:=\theta_{1}=\theta_{2},$ then $\theta^{2}=\theta.$

Let $G$ be a locally compact abelian group. It is well known that
$M\left(G\right)$ is a commutative semisimple unital Banach algebra, but
$M\left(G\right)$ fails to be (Shilov) regular, in general. However, there
exists a largest closed regular subalgebra of $M\left(G\right)$ which we will
denote by $M_{\text{reg}}\left(G\right).$ Since the algebra
$L^{1}\left(G\right)$ and the discrete measure algebra $M_{d}\left(G\right)$
are subalgebras of $M_{\text{reg}}\left(G\right),$ we have
$L^{1}\left(G\right)+M_{d}\left(G\right)\subseteq
M_{\text{reg}}\left(G\right),$ but in general,
$L^{1}\left(G\right)+M_{d}\left(G\right)\neq M_{\text{reg}}\left(G\right)$
[16, Example 4.3.11]. This shows that the algebra
$M_{\text{reg}}\left(G\right)$ is remarkable large. Recall [16, Proposition
4.12.5] that if $\mu\in M_{\text{reg}}\left(G\right)$, then the operator
$\lambda_{1}\left(\mu\right)$ has natural spectrum, that is,
$\sigma\left(\lambda_{1}\left(\mu\right)\right)=\overline{\widehat{\mu}\left(\widehat{G}\right)}$.
It follows that if $\mu\in M_{\text{reg}}\left(G\right)$ is a probability
measure, then
$\sigma\left(\lambda_{1}\left(\mu\right)\right)\cap\mathbb{T\subseteq}\left\\{1\right\\}$
if and only if for any neighborhood $V$ of $1,$ $\sup_{\chi\notin
V}\left|\widehat{\mu}\left(\chi\right)\right|<1.$

######  Corollary 3.2.

If $\mu\in M\left(G\right)$ is power bounded, then there exists an idempotent
measure $\theta\in M\left(G\right)$ such that

$w^{\ast}-\lim_{n\rightarrow\infty}\left(\frac{\delta_{e}+\mu}{2}\right)^{n}=\theta.$

Moreover, for an arbitrary $f\in L^{p}\left(G\right)$
$\left(1<p<\infty\right)$ we have

$\lim_{n\rightarrow\infty}\left[\left(\frac{\delta_{e}+\mu}{2}\right)^{n}\ast
f\right]=\theta\ast f\text{ \ in }L^{p}\text{-norm.}$

###### Proof.

If $\nu:=\frac{\delta_{e}+\mu}{2}$, then $\nu$ is power bounded, that is,
$\sup_{n\in\mathbb{N}}\left\|\nu^{n}\right\|_{1}\leq C_{\mu}$. Therefore, the
operator $\lambda_{1}\left(\nu\right)=\frac{I+\lambda_{1}\left(\mu\right)}{2}$
is power bounded. Notice also that if $f\left(z\right):=\frac{1+z}{2}$
$\left(z\in\mathbb{C}\right),$ then $f\left(1\right)=1$ and
$\left|f\left(z\right)\right|<1$ for all
$z\in\overline{\mathbb{D}}\diagdown\left\\{1\right\\}.$ It follows from the
spectral mapping theorem that
$\sigma\left(\lambda_{1}\left(\nu\right)\right)\cap\mathbb{T\subseteq}\left\\{1\right\\}.$
By Proposition 3.1, there exists an idempotent measure $\theta\in
M\left(G\right)$ such that w∗-$\lim_{n\rightarrow\infty}\nu^{n}=\theta.$

As in the proof of Proposition 2.12 we have

$\lim_{n\rightarrow\infty}\left(\nu^{n}\ast f\right)=\theta\ast f\text{ \
weakly for every }f\in L^{p}\left(G\right).$

On the other hand, by the Katznelson-Tzafriri theorem,

$\lim_{n\rightarrow\infty}\left\|\lambda_{1}\left(\nu\right)^{n+1}-\lambda_{1}\left(\nu\right)^{n}\right\|=0.$

Now, it follows from Proposition 2.1 that

$\lim_{n\rightarrow\infty}\left(\nu^{n}\ast f\right)=\theta\ast f\text{\ \ in
}L^{p}\text{-norm.}$

As we have noted above, if $\mu$ is a strictly aperiodic measure on a non-
compact locally compact group $G,$ then
w∗-$\lim_{n\rightarrow\infty}\mu^{n}=0$ [3, Théorème 8]. In [18, Theorem 2] it
was proved that if $\mu$ is an adapted measure on a second countable locally
compact non-compact group $G$, then w∗-$\lim_{n\rightarrow\infty}\mu^{n}=0$.

As is well known, equipped with the involution given by
$d\widetilde{\mu}\left(g\right)=\overline{d\mu\left(g^{-1}\right)}$, the
algebra $M\left(G\right)$ becomes a Banach $\ast$-algebra. If $\mu$ is a
probability measure on a locally compact group $G,$ then as
$\mathop{\mathrm{s}upp}\widetilde{\mu}=\left(\mathop{\mathrm{s}upp}\mu\right)^{-1}$,
we have

$\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)=\overline{\left\\{\left(\mathop{\mathrm{s}upp}\mu\right)^{-1}\cdot\left(\mathop{\mathrm{s}upp}\mu\right)\right\\}}$

(for instance see, [11, Theorem 2.2.2]).

The following result is certainly known, but we have not been able to find a
precise reference.

######  Proposition 3.3.

A measure $\mu\in M\left(G\right)$ is strictly aperiodic if and only if the
measure $\widetilde{\mu}\ast\mu$ is adapted.

###### Proof.

Assume that the measure $\widetilde{\mu}\ast\mu$ is not adapted. This means
that

$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]:=H\neq
G.$

It follows that $g^{-1}\cdot\left(\mathop{\mathrm{s}upp}\mu\right)\subseteq H$
or $\mathop{\mathrm{s}upp}\mu\subseteq gH$ for all
$g\in\mathop{\mathrm{s}upp}\mu.$ This shows that $\mu$ is not strictly
aperiodic. Now, assume that $\mu$ is not strictly aperiodic. Then
$\mathop{\mathrm{s}upp}\mu\subseteq gH$ for some closed proper subgroup $H$ of
$G$ and $g\in G$. It follows that
$\left(\mathop{\mathrm{s}upp}\mu\right)^{-1}\subseteq Hg^{-1}$ and therefore
$\left(\mathop{\mathrm{s}upp}\mu\right)^{-1}\cdot\left(\mathop{\mathrm{s}upp}\mu\right)\subseteq
H.$ Thus we have

$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]\subseteq
H\neq G.$

This shows that the measure $\widetilde{\mu}\ast\mu$ is not adapted.

Next, we have the following.

######  Theorem 3.4.

Let $\mu$ be a probability measure on a locally compact group $G.$ If one of
the subgroups
$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$ and
$\left[\mathop{\mathrm{s}upp}\left(\mu\ast\widetilde{\mu}\right)\right]$ is
not compact, then

$w^{\ast}-\lim_{n\rightarrow\infty}\mu^{n}=0.$

###### Proof.

Recall that a contraction $T$ on a Hilbert space is said to be completely non-
unitary if it has no proper reducing subspace on which it acts as a unitary
operator. By the Nagy-Foiaş theorem [7, Chapter II, Theorem 3.9] if $T$ is a
completely non-unitary contraction, then $T^{n}\rightarrow 0$ in the weak
operator topology. Now, assume that
$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$ is
not compact. Let us show that $\lambda_{2}\left(\mu\right)$ is a completely
non-unitary contraction. Since
$\lambda_{2}\left(\mu\right)^{\ast}=\lambda_{2}\left(\widetilde{\mu}\right),$
it suffices to show that
$\lambda_{2}\left(\widetilde{\mu}\right)\lambda_{2}\left(\mu\right)f=f$, where
$f\in L^{2}\left(G\right)$ implies $f=0.$ By Lemma 2.3,

$\displaystyle\left\\{f\in
L^{2}\left(G\right):\lambda_{2}\left(\widetilde{\mu}\right)\lambda_{2}\left(\mu\right)f=f\right\\}$
$\displaystyle=$ $\displaystyle\left\\{f\in
L^{2}\left(G\right):\lambda_{2}\left(\widetilde{\mu}\ast\mu\right)f=f\right\\}$
$\displaystyle=$ $\displaystyle\left\\{f\in L^{2}\left(G\right):f_{g}=f\text{,
}\forall
g\in\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]\right\\}.$

Since $\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$
is not compact, from the identity $f_{g}=f$ for all
$g\in\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$,
we have $f=0$ (a.e.). Hence $\lambda_{2}\left(\mu\right)$ is a completely non-
unitary contraction. Consequently, $\lambda_{2}\left(\mu\right)^{n}\rightarrow
0$ in the weak operator topology. Then, for an arbitrary $u,\upsilon\in
L^{2}\left(G\right)$ we can write

$\langle\mu^{n},u\ast\widetilde{\upsilon}\rangle=\langle\mu^{n}\ast\overline{\upsilon},\overline{u}\rangle=\langle\lambda_{2}\left(\mu\right)^{n}\overline{\upsilon},\overline{u}\rangle\rightarrow
0.$

Since the set $\left\\{u\ast\widetilde{\upsilon}:u,\upsilon\in
L^{2}\left(G\right)\right\\}$ is linearly dense in $C_{0}\left(G\right),$ we
have $\langle\mu^{n},f\rangle\rightarrow 0$ for all $f\in
C_{0}\left(G\right).$ Hence w∗-$\lim_{n\rightarrow\infty}\mu^{n}=0.$

######  Corollary 3.5.

Let $\mu$ be a probability measure on a locally compact group $G.$ If one of
the subgroups
$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$ and
$\left[\mathop{\mathrm{s}upp}\left(\mu\ast\widetilde{\mu}\right)\right]$ is
not compact, then

$\mu^{n}\ast f\rightarrow 0\text{ \ weakly for all }f\in
L^{p}\left(G\right)\text{ }\left(1<p<\infty\right).$

###### Proof.

If $f\in L^{p}\left(G\right)$ and $h\in L^{q}\left(G\right)$ $(1/p+1/q=1),$
then $h\ast f^{\vee}\in C_{0}\left(G\right)$. By Theorem 3.4,
w∗-$\lim_{n\rightarrow\infty}\mu^{n}=0$ and therefore we have

$\lim_{n\rightarrow\infty}\langle\mu^{n}\ast
f,h\rangle=\lim_{n\rightarrow\infty}\langle\mu^{n},h\ast f^{\vee}\rangle=0.$

However, we have the following.

######  Proposition 3.6.

Let $\mu$ be a probability measure on a locally compact group $G$ and assume
that one of the subgroups
$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$ and
$\left[\mathop{\mathrm{s}upp}\left(\mu\ast\widetilde{\mu}\right)\right]$ is
not compact. If the Lebesgue measure of
$\sigma\left(\lambda_{2}\left(\mu\right)\right)\cap\mathbb{T}$ is zero, then

$\lim_{n\rightarrow\infty}\left\|\mu^{n}\ast f\right\|_{p}=0,\text{ \ }\forall
f\in L^{p}\left(G\right)\text{ }\left(1<p<\infty\right).$

###### Proof.

As in the proof of Theorem 3.4 we can see that if one of the subgroups
$\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]$ and
$\left[\mathop{\mathrm{s}upp}\left(\mu\ast\widetilde{\mu}\right)\right]$ is
not compact, then $\lambda_{2}\left(\mu\right)$ is a completely non-unitary
contraction. On the other hand, Nagy-Foias theorem [7, Chapter II, Proposition
6.7] states that if $T$ is a completely non-unitary contraction on a Hilbert
space $\mathcal{H}$ and if $\sigma\left(T\right)\cap\mathbb{T}$ is of the
Lebesgue measure zero, then $\lim_{n\rightarrow\infty}\left\|T^{n}x\right\|=0$
for all $x\in\mathcal{H}.$ Therefore we have
$\lim_{n\rightarrow\infty}\left\|\mu^{n}\ast f\right\|_{2}=0$ for all $f\in
L^{2}\left(G\right).$ Now, assume that $p\neq 2.$ Let $C_{c}\left(G\right)$ be
the space of all complex valued continuous functions on $G$ with compact
support and let $f\in C_{c}\left(G\right).$ Then, $\nu\ast f\in
L^{p}\left(G\right)$ for any $\nu\in M\left(G\right)$ and for all $1\leq
p\leq\infty.$ By the Riesz-Thorin Convexity Theorem [6, Chapter VI, Lemma
10.9], $\alpha\mapsto\log\left\|\nu\ast f\right\|_{\frac{1}{\alpha}}$ is a
convex function on $\left[0,1\right]$. Let $q$ be chosen such that $q>p$ for
$p>2$ and $1<q<p$ for $1<p<2.$ If $\lambda:=\frac{2q-2p}{pq-2p},$ then
$0<\lambda<1$ and $\frac{1}{p}=\frac{1-\lambda}{q}+\frac{\lambda}{2}.$
Consequently, we can write

$\left\|\nu\ast f\right\|_{p}\leq\left\|\nu\ast
f\right\|_{q}^{1-\lambda}\left\|\nu\ast f\right\|_{2}^{\lambda},\text{ \
}\forall\nu\in M\left(G\right).$

Hence we have

$\left\|\mu^{n}\ast f\right\|_{p}\leq\left\|\mu^{n}\ast
f\right\|_{q}^{1-\lambda}\left\|\mu^{n}\ast
f\right\|_{2}^{\lambda}\leq\left\|f\right\|_{q}^{1-\lambda}\left\|\mu^{n}\ast
f\right\|_{2}^{\lambda},\text{ \ }\forall n\in\mathbb{N}.$

It follows that $\lim_{n\rightarrow\infty}\left\|\mu^{n}\ast f\right\|_{p}=0$
for all $f\in C_{c}\left(G\right).$ Since $C_{c}\left(G\right)$ is dense in
$L^{p}\left(G\right),$ we have $\lim_{n\rightarrow\infty}\left\|\mu^{n}\ast
f\right\|_{p}=0$ for all $f\in L^{p}\left(G\right).$

######  Remark 3.7.

Let $G$ be a locally compact abelian group and $\mu\in M\left(G\right).$ The
Fourier-Plancherel transform estabilishes a unitary equivalence between the
convolution operator $\lambda_{2}\left(\mu\right)$ on $L^{2}\left(G\right)$
and the multiplication operator $M_{\widehat{\mu}}$ on
$L^{2}\left(\widehat{G}\right)$. Therefore, we have
$\sigma\left(\lambda_{2}\left(\mu\right)\right)=\overline{\widehat{\mu}\left(\widehat{G}\right)}.$
If
$\mathcal{E}_{\mu}:=\left\\{\chi\in\widehat{G}:\left|\widehat{\mu}\left(\chi\right)\right|=1\right\\}$,
then $\mathcal{E}_{\mu}=\mathcal{F}_{\widetilde{\mu}\ast\mu}$. Notice also
that
$\widehat{\mu}\left(\mathcal{E}_{\mu}\right)\subseteq\sigma\left(\lambda_{2}\left(\mu\right)\right)\cap\mathbb{T}$.
If $\mu$ is a probability measure, then $\mathcal{E}_{\mu}$ is a closed
subgroup of $\widehat{G},$ that is,
$\mathcal{E}_{\mu}=\left[\mathop{\mathrm{s}upp}\left(\widetilde{\mu}\ast\mu\right)\right]^{\bot}$
$($see, Remark 2.11$).$

## References

  * [1] P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc. New York, 1999\.
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  * [3] Y.Derriennic, Lois ”zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré Sect. B (N.S.) 12(1976), 111-129.
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# Flat and almost flat bands in the quasi-one-dimensional Josephson junction
array

Daryna Bukatova and Yaroslav Zolotaryuk  Bogolyubov Institute for Theoretical
Physics, National Academy of Sciences of Ukraine, Kyiv 03143, Ukraine
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>(corresponding author)

###### Abstract

The dispersion law for the linear waves in the quasi-one-dimensional array of
inductively coupled Josephson junctions (JJ) is derived. The array has a
multiladder structure that consists of the finite number of rows ($N\geq 2$)
in $Y$ direction and is infinite in $X$ direction. The spectrum of the linear
waves (Josephson plasmons) consists of $2N-1$ branches. Among these branches
there is a $N$-fold completely flat degenerate one that coincides with the
Josephson plasma frequency. The remaining $N-1$ branches have a standard
Josephson plasmon dispersion law typical for 1D JJ arrays. Application of the
uniform dc bias on the top of each vertical column of junctions lifts the
degeneracy and only one flat branch remains unchanged. The rest of the
previously flat branches become weakly dispersive. The parameter range where
the flatness of these branches is maximal has been discussed.

###### pacs:

74.81.Fa,74.50.+r,73.20.Mf

††: J. Phys. C: Solid State Phys.

Keywords: Weak superconductivity, Josephson junctions, Josephson junction
arrays, dispersion, plasmons, flat bands.

##  1 Introduction

The concept of flat bands (FB) appears in different fields of modern physics
[1]. A FB in a spectrum (either energy or phonon) which is a completely
dispersionless band $\mathbf{\nabla}_{\mathbf{q}}\,\omega(\mathbf{q})=0$ where
$\omega(\mathbf{q})$ is the dispersion law of the respective system. The
concept of FBs was introduced at first theoretically for electron states in
the two-dimensional dice lattice model [2] and, some years later, for the
itinerant electrons in the Hubbard model [3]. It was later extended to
magnetic systems [4], Josephson junctions (JJs) [5, 6] and Dirac materials [7,
8]. A remarkable consequence of the absence of dispersion is the existence of
localized states without breaking the translational invariance of the lattice.
Although the first FB models appeared in the late 1980s, only about 10-15
years ago it became possible to observe FBs in different systems and to
construct artificial FB systems. For example, several lattice structures with
a FB have been realized in photonic lattices (such as waveguide arrays) [9]
and cold atomic gases in optical lattices [10].

The Josephson transmission lines (JTL) or Josephson junction arrays (JJAs) are
examples of artificial superconducting systems that can support FBs. For
example, the frustrated JJA with FBs has been studied in [6]. It should be
mentioned that JJAs are widely studied for applications in quantum computing,
for example, possibility of application of JTLs for quantum states readout was
discussed theoretically in [11] and studied experimentally later in [12].
Since there has been active research conducted on propagation of current
pulses through JTLs [13], it is important to study their dispersion relations.

Ladders of JJs have been actively investigated due to observation of various
nonlinear phenomena such as vortex propagation [14, 15], meandering [16] and,
finally, prediction [17] and experimental discovery [18, 19, 20] of discrete
breathers [21]. It should be noted that while Josephson vortices can exist in
the standard 1D JJA, discrete breathers require at least a simple JJ ladder.
In [19, 22] it was shown that the Josephson plasmon spectrum for the
anisotropic Josephson junction ladder (JJL) with two horizontal rows has a
flat band. It is natural to generalize this ladder into a quasi-one-
dimensional ladder-like structure with an arbitrary number of horizontal rows
$N$. We will call it an $N$-row JJA or a quasi-one-dimensional Josephson
junction array (Q1D JJA).

The aim of this work is to find and analyse the plasmon spectrum and the the
corresponding wave amplitudes of the Q1D JJA with the arbitrary number of
rows, and, in particular, to establish whether this spectrum still has a FB
for any number of rows as it does for the 2-row case.

This paper is organized as follows. In the next section we present the
equations of motion for the Q1D JJA with $N$ rows. In the third section the
dispersion law for an arbitrary number of rows is obtained and its main
properties are discussed. Outlines and conclusions are given in the last
section.

##  2 Equations of motion

We consider a ladder-like array of JJs that consists of a finite number ($N$)
of rows in Y direction as shown in figure 1. The length of each of the rows in
the X direction is supposed to be much greater than $N$. Since we are
interested in the plane waves of the array we may assume it to be infinite.

$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\phi^{(v)}_{n-1,N-1}$$\phi^{(v)}_{n,N-1}$$\phi^{(v)}_{n+1,N-1}$$\phi^{(h)}_{n-1,N}$$\phi^{(h)}_{n,N}$$\phi^{(h)}_{n+1,N}$$\phi^{(h)}_{n-1,N-1}$$\phi^{(h)}_{n,N-1}$$\phi^{(h)}_{n+1,N-1}$$\phi^{(h)}_{n-1,2}$$\phi^{(h)}_{n,2}$$\phi^{(h)}_{n+1,2}$$\phi^{(v)}_{n-1,1}$$\phi^{(v)}_{n,1}$$\phi^{(v)}_{n+1,1}$$\phi^{(h)}_{n-1,1}$$\phi^{(h)}_{n,1}$$\phi^{(h)}_{n+1,1}$$X$$Y$$I_{B}$$I_{B}$$I_{B}$$I_{B}$$I_{B}$$I_{B}$$I_{B}$$I_{B}$$I_{B}$$I_{B}$
Figure 1: Schematic view of the Q1D JJA with $N$ rows. Only two top and two
bottom rows are shown. The crosses $\times$ denote locations of the respective
junctions. Thick arrows show the direction and location of the incoming and
outcoming dc bias $I_{B}$.

The respective phase of the junction will be denoted as $\phi^{(v,h)}_{n,k}$
where the superscript $v$ or $h$ will denote whether the junction belongs to
the row (a horizontal one) or to the bridge (a vertical one). The set of
subscripts $(n,k)$ enumerates the junction along the $X$ and $Y$ axis,
respectively. The array is uniformly biased by the dc current $I_{B}$ at each
$(n,N)$th junction and the same current is extracted from the each$(n,N)$th
junction. The equations for the time evolution of the Josephson phases for
each of the junctions within the resistively and capacitatively shunted (RCSJ)
model [23] are given by the following set of equations

$\displaystyle\frac{C_{v,h}\hbar}{2e}\frac{d^{2}}{dt^{2}}\phi^{(v,h)}_{n,k}+\frac{\hbar}{2eR_{v,h}}\frac{d}{dt}{\phi}^{(v,h)}_{n,k}+I_{c}^{(v,h)}\sin\phi^{(v,h)}_{n,k}=I^{(v,h)}_{n,k},\;\;$
$\displaystyle n\in{\mathbb{Z}},\;v:k=\overline{1,N-1},\;h:k=\overline{1,N}.$
(1)

Here $C_{v,h}$ is the capacitance of the vertical or horizontal junction,
$R_{v,h}$ is its resistance and $I^{(v,h)}_{c}$ is its critical current,
respectively. Finally, $I^{(v,h)}_{n,k}$ is the current that flows through the
$(n,k)$th junction.

For the derivation of the evolution equations one can consult paper [24]. Here
only the main points of the derivation will be repeated. For each of these
currents we need to write the Kirchhoff’s equations and the flux quantization
law. The latter connects the mesh currents $I^{(m)}_{n,k}$ in the cell formed
by the vertical junctions $(n,k)$ and $(n+1,k)$ and horizontal junctions
$(n,k)$ and $(n,k+1)$ and the magnetic flux through this cell:

$I^{(m)}_{n,k}=-\frac{\Phi_{n,k}}{L}=-{\Phi_{0}\over{2\pi
L}}\left(\phi^{(v)}_{n+1,k}-\phi^{(v)}_{n,k}+\phi^{(h)}_{n,k+1}-\phi^{(h)}_{n,k}\right).$
(2)

Here $L$ is the self-inductance of the cell and $\Phi_{0}=\pi\hbar/e$ is the
magnetic flux quantum. It is convenient to introduce the dimensionless
variables in the following way:

$\tau=\omega_{p}t,\,\gamma=\frac{I_{B}}{I_{c}^{(v)}},\,\beta_{L}=\frac{2\pi
I_{c}^{(v)}L}{\Phi_{0}},\,\eta=\frac{I_{C}^{(h)}}{I_{C}^{(v)}}=\frac{C_{h}}{C_{v}}=\frac{R_{v}}{R_{h}}.$
(3)

The coupling constant $\beta_{L}$ measures the discreteness of the array. The
dimensionless dissipation parameter is then $\alpha=\Phi_{0}\omega_{p}/(2\pi
I_{c}^{(v)}R_{v})$, and the time is normalized to the inverse Josephson plasma
frequency $\omega_{p}^{-1}=\sqrt{C_{v}\Phi_{0}/(2\pi I_{c}^{(v)})}$. Finally,
the parameter $\eta$ measures the anisotropy between the junctions placed in
the rows and columns of the array. For the sake of convenience we also
introduce the nonlinear operator

${\cal N}(x)\equiv\ddot{x}+\alpha\dot{x}+\sin x.$ (4)

With the help of this operator the equations of motion are written inside the
Q1D JJA

$\displaystyle{\cal
N}(\phi^{(v)}_{n,k})=\gamma+\frac{1}{\beta_{L}}\left(\hat{\Delta}_{x}\phi^{(v)}_{n,k}+\hat{\nabla}_{x}\phi^{(h)}_{n-1,k+1}-\hat{\nabla}_{x}\phi^{(h)}_{n-1,k}\right),$
$\displaystyle\;\;n\in{\mathbb{Z}}~{},\;k=\overline{1,N-1},\;$ (5)
$\displaystyle{\cal
N}(\phi^{(h)}_{n,k})=\frac{1}{\eta\beta_{L}}\left(\hat{\Delta}_{y}\phi^{(h)}_{n,k}+\hat{\nabla}_{x}\phi^{(v)}_{n,k}-\hat{\nabla}_{x}\phi^{(v)}_{n,k-1}\right),$
$\displaystyle n\in{\mathbb{Z}},~{}\,k=\overline{2,N-1},$ (6)

and on the border rows ($k=1,N$)

$\displaystyle{\cal
N}(\phi^{(h)}_{n,1})=\frac{1}{\eta\beta_{L}}\left(\hat{\nabla}_{x}\phi^{(v)}_{n,1}+\hat{\nabla}_{y}\phi^{(h)}_{n,1}\right),\;n\in{\mathbb{Z}}~{}$
(7) $\displaystyle{\cal
N}(\phi^{(h)}_{n,N})=-\frac{1}{\eta\beta_{L}}\left(\hat{\nabla}_{x}\phi^{(v)}_{n,N-1}+\hat{\nabla}_{y}\phi^{(h)}_{n,N-1}\right),\;n\in{\mathbb{Z}}~{}.$
(8)

These equations represent a system of $2N-1$ coupled discrete sine-Gordon
equations. Here the difference operators introduced for the sake of simplicity
are given by the following expressions:

$\displaystyle{\hat{\triangle}}_{x}\phi_{n,k}\equiv\phi_{n+1,k}-2\phi_{n,k}+\phi_{n-1,k},\;$
(9)
$\displaystyle{\hat{\triangle}}_{y}\phi_{n,k}\equiv\phi_{n,k+1}-2\phi_{n,k}+\phi_{n,k-1},\;$
$\displaystyle{\hat{\nabla}}_{x}\,\phi_{n}\equiv\phi_{n+1,k}-\phi_{n,k},\;{\hat{\nabla}}_{y}\,\phi_{n,k}\equiv\phi_{n,k+1}-\phi_{n,k}.$

In real JJAs the dimensionless dissipation parameter $\alpha$ is rather small,
$\alpha\lesssim 0.1$ [25]. Moreover, this parameter provides us with
information on how fast the Josephson plasmons decay. Our aim is to compute
the plasmon spectrum, and, therefore, $\alpha$ will be neglected throughout
the next sections.

##  3 Plasmon bands and their properties

###  3.1 Dispersion law derivation

In order to calculate the spectrum of the Josephson plasma waves one has to
expand the equations of motion (5)-(8) around the steady state

$\displaystyle\phi^{(v)}_{n,k}=\arcsin\gamma,\;k=\overline{1,N-1};\;\phi^{(h)}_{n,k}=0,\;k=\overline{1,N},\;n\in{\mathbb{Z}}\,,$
(10)

that corresponds to the spatially uniform superconducting state of the whole
array. We will study linear waves that propagate in the $X$ direction. The
plane wave ansatz for the small deviations from the steady state

$\displaystyle\left(\delta\phi^{(v)}_{n,1},\cdots,\delta\phi^{(v)}_{n,N-1},\delta\phi^{(h)}_{n,1},\cdots,\delta\phi^{(h)}_{n,N}\right)^{T}=$
(11)
$\displaystyle=\left(A^{(v)}_{1},\cdots,A^{(v)}_{N-1},A^{(h)}_{1},\cdots,A^{(h)}_{N}\right)^{T}e^{\imath(qn+\omega\tau)}+\mbox{c.c},$

is substituted into the equations of motion (5-8). The resulting
characteristic polynomial is given as a determinant of the respective
$2N-1\times 2N-1$ matrix. After some calculations (see A for details) the
characteristic polynomial can be written explicitly and, after some
manipulations, factorized as

$\chi(\omega^{2})=\frac{1}{\eta^{N-1}}(\omega^{2}-1)\prod_{n=2}^{N}[d_{0}-\alpha_{n}d_{1}],~{}~{}\alpha_{n}=1-2\cos\frac{\pi(n-1)}{N},$
(12)

where

$\displaystyle
d_{0}(\omega^{2})=\eta[\omega^{2}_{i}(q)-\omega^{2}]\left(1+\frac{1}{\eta\beta_{L}}-\omega^{2}\right)-\frac{2(1-\cos{q})}{\beta_{L}^{2}},$
(13) $\displaystyle
d_{1}(\omega^{2})=-\frac{1}{\beta_{L}}[\omega^{2}_{i}(q)-\omega^{2}]+\frac{2(1-\cos{q})}{\beta_{L}^{2}},$
(14)
$\displaystyle\omega^{2}_{i}(q)\equiv\sqrt{1-\gamma^{2}}+\frac{2}{\beta_{L}}(1-\cos
q).$ (15)

From here one concludes that there is one flat band with $\omega^{2}=1$. The
expression $\omega_{i}(q)$ is the dispersion law of the biased standard one-
dimensional JJA where only the vertical Josephson junctions (see [26, 27]) are
present. The rest of the dispersion curves is derived from the equality
$d_{0}=\alpha_{n}d_{1}$. Thus, the whole set of the $2N-1$ dispersion branches
can be written as

$\displaystyle\omega_{0}^{2}=1,$ (16) $\displaystyle\omega_{\pm
n}^{2}(q)=\frac{1}{2}\left[1+\omega_{i}^{2}(q)+\frac{1+\alpha_{n+1}}{\eta\beta_{L}}\right]\pm$
(17)
$\displaystyle\pm\sqrt{\frac{1}{4}\left[\omega_{i}^{2}-\left(1+\frac{1+\alpha_{n+1}}{\eta\beta_{L}}\right)\right]^{2}+2(1+\alpha_{n+1})\frac{1-\cos
q}{\eta\beta_{L}^{2}}},$ $\displaystyle n=\overline{1,N-1}\,.$

The constant $\alpha_{n}$ is given in equation (12) and of importance are the
values for $n=\overline{2,N}$. For $N=2$ rows there is just one value
$\alpha_{2}=1$. For $N=3$ rows there are two values, $\alpha_{2}=0$,
$\alpha_{3}=1$. For larger $N$ the values of $\alpha_{n}$ will pack the
interval $]-1,3[$ more and more densely.

For example, for the particular case of $N=3$ rows, the dispersion law
consists of 5 branches. All of them are shown in figure 2 for different values
of dc bias. The branches are indexed in such a way that the dispersionless
branch $\omega_{0}$ is placed in the middle, the branches with positive
subscript lie above the $\omega_{0}$ branch and the branches with the negative
subscript lie below.

(a) $\gamma=0$

(b) $\gamma=0.4$

(c) $\gamma=0.8$

Figure 2: Plasmon bands for $N=3$ row array, with $\eta=1,~{}\beta_{L}=1$ and
different values of the dc bias $\gamma$.

In the $N=2$ case the formulae (16-17) repeat the already known result [22].
Thus, generally speaking, the following inequality is valid for all plasmon
branches (16)-(17):
$\omega_{-N+1}<\cdots\omega_{-1}<\omega_{0}=1<\omega_{1}<\cdots\omega_{N-1}$.
The set of branches with $\omega_{n>0}$ appear above the branch
$\omega_{0}=1$. They are significantly dispersive and are well separated from
each other. The branches $\omega_{n<0}$ depend crucially on the presence of
the dc bias. When $\gamma=0$ they are all degenerate
$\omega_{-N+1}=\cdots=\omega_{-1}=1$. When the bias is applied the degeneracy
is lifted. In that case the branches with $n<0$ detach from the $\omega_{0}=1$
branch and lie below it. If the bias is small these branches still remain
almost flat [see figure 2(b)]. Thus, in the general case of the dc biased Q1D
array its plasmon spectrum consists of

  * •

one flat band with $\omega=1$;

  * •

$N-1$ strongly dispersive bands;

  * •

$N-1$ weakly dispersive bands that become completely flat at $\gamma=0$.

###  3.2 Dispersion law properties for the different model parameters

In the long wave limit $q\to 0$ the frequencies of all modes satisfy the
following relations:

$\displaystyle\omega_{n}(0)=\left\\{\begin{array}[]{cc}\sqrt{1+\frac{1+\alpha_{n+1}}{\eta\beta_{L}}},&n>0\\\
(1-\gamma^{2})^{1/4},&n<0.\end{array}\right.$ (20)

Thus, even when the degeneracy is lifted at $\gamma\neq 0$ the lower branches
remain degenerate at the point $q=0$ while the upper branches are completely
separated. In the limit of strong discreteness ($\beta_{L}\gg 1$) the
difference between the neighboring branches reads
$\Delta\omega_{n}(0)=\omega_{n+1}(0)-\omega_{n}(0)\approx
2\sin[\pi/(2N)]\sin[(2n+1)\pi/(2N)](\eta\beta_{L})^{-1}+{\cal
O}[(\eta\beta_{L})^{-2}]$.

Next we discuss the properties of the plasmon spectrum as a function of the
discreteness parameter $\beta_{L}$. The respective plots for the different
values of $\beta_{L}$ are given in figures 3(a)-3(c). Naturally, the increase
of $\beta_{L}$ means larger discreteness effects thus the absolute values of
the plasmon frequencies (for modes with $\omega_{n>0}$) increase greatly when
$\beta_{L}$ decreases [see figure 3(a)]. In the opposite limit
$\beta_{L}\to\infty$ the interaction between the cells dies out, hence the
dispersion laws that lay above the $\omega_{0}=1$ branch flatten as can be
seen in figure 3(c).

(a) $\beta_{L}=0.1$

(b) $\beta_{L}=1$

(c) $\beta_{L}=10$

Figure 3: Dispersion relation for the 3-row JJA at fixed values of the dc bias
($\gamma=0.4$) and anisotropy ($~{}\eta=1$) and for the different values of
the dimensionless inductance $\beta_{L}$.

On the other hand, the properties of the lower (almost flat) bands depend
mostly on the value of the dc bias as these branches must satisfy
$(1-\gamma^{2})^{1/4}\leq\omega_{n}<1$ for all $n<0$.

The role of the anisotropy constant $\eta$ is demonstrated in figures
4(a)-4(e). This parameter controls the redistribution of the phase
oscillations between the horizontal and vertical subsystems.

(a) $\eta=0.1$

(b) $\eta=10$

(c) $\eta=0.1$

(d) $\eta=1$

(e) $\eta=10$

Figure 4: Dispersion relation for the 3-row JJA at the fixed values of the
bias and inductance ($\gamma=0.4,\beta_{L}=0.5$) and for the different values
of the anisotropy $\eta$. Figures (c)-(e) illustrate the detailed view for the
respective cases (see the subcations).

In the limit $\eta\to 0$ the horizontal oscillations dominate over the
vertical ones. First of all, the absolute values of the plasmon frequencies
for the branches with $\omega_{n>0}>1$ increase significantly due to the
presence of the $1/\eta$ singularity in the dispersion law. Moreover, the
different branches strongly separate from each other (see figure 4(a)). It is
interesting that in the limit of small $\eta$ and intermediate $\beta_{L}$ the
dispersive branches (those that lie above $\omega_{0}=1$) have significant
gaps between them. The value of the gap between the $\omega_{n}$ and
$\omega_{n+1}$ branches equals

$\displaystyle\Delta\omega_{n}=\omega_{n+1}(0)-\omega_{n}(\pi)\simeq\sqrt{1+\frac{1+\alpha_{n+2}}{\eta\beta_{L}}}-$
$\displaystyle-\sqrt{\frac{1+\alpha_{n+1}}{\eta\beta_{L}}}+{\cal
O}(\eta^{1/2}).$ (21)

This means that we have very weakly interacting horizontal rows of junctions
and the dispersion laws for them are strongly separated from each other. In
the opposite limit $\eta\to\infty$ the junctions in the vertical subsystem are
coupled much stronger as compared to the horizontal direction. Thus, there is
very small difference between the oscillations within one vertical column of
junctions. As a result, this can be viewed as each column oscillating as a
whole, and, consequently the branches $\omega_{n>0}$ in figure 4(b) are very
weakly separated and look almost identical.

Finally we discuss the behavior of the almost flat bands for the different
values of anisotropy. This behavior is illustrated in figures 4(c)-4(e). The
total width of the almost flat band ($\omega_{n<0}$) is defined as a
difference between the highest frequency value of the highest lying branch
which is $\omega_{-1}(q)$ and the smallest value of the lowest branch

$\Delta_{-}=\max_{q\in[-\pi,\pi]}\omega_{-1}(q)-\min_{q\in[-\pi,\pi]}\omega_{-N+1}(q)=\omega_{-1}(\pi)-(1-\gamma^{2})^{1/4}.$
(22)

In the limit of small $\eta$ or small $\beta_{L}$ one can obtain the following
asymptotic behavior of the plasmon frequency at the Brillouin zone edge:

$\omega_{n<0}(\pi)=\left\\{\begin{array}[]{cc}\left[1-\frac{1-\sqrt{1-\gamma^{2}}}{4\eta+1+\alpha_{n+1}}(1+\alpha_{n+1})\right]^{1/2},&\beta_{L}\to
0,\\\ (1-\gamma^{2})^{1/4},&\eta\to 0.\end{array}\right.$ (23)

The limit $\eta\to 0$ is illustrated in figure 4(c). In this limit
$\Delta_{-}\to 0$, hence, the whole band becomes flat. If we keep the
anisotropy $\eta$ constant and decrease the inductance the band width will
depend on the number of rows, $N$, because $\alpha_{2}=1-2\cos{\pi/N}$. As a
result, $\Delta_{-}$ does not tend to $0$.

In the limit of large $\beta_{L}$ or $\eta$ the limiting frequency value is
given by

$\omega_{n<0}(\pi)=\left\\{\begin{array}[]{c}(1-\gamma^{2})^{1/4},\;\;\;\;\;\beta_{L}\to\infty,\\\
\left\\{\begin{array}[]{c}1,\;\;\sqrt{1-\gamma^{2}}+\frac{4}{\beta_{L}}>1\\\
\left[\sqrt{1-\gamma^{2}}+\frac{4}{\beta_{L}}\right]^{1/2},\sqrt{1-\gamma^{2}}+\frac{4}{\beta_{L}}<1\end{array}\right.,\eta\to\infty.\end{array}\right.$
(24)

In the limit when $\beta_{L}\to\infty$ for $\eta$ fixed we again expect
complete flattening of the $\omega_{n<0}$ band. The situation becomes more
subtle if $\beta_{L}$ is finite and $\eta\to\infty$. If discreteness is quite
significant, or, alternatively, the bias is strong enough to guarantee the
inequality $\beta_{L}>4/(1-\sqrt{1-\gamma^{2}})$ one can obtain flattening
when the strong discreteness limit is taken:

$\Delta_{-}=\left[\sqrt{1-\gamma^{2}}+\frac{4}{\beta_{L}}\right]^{1/2}-(1-\gamma^{2})^{1/4}\to_{\beta_{L}\to\infty}0.$
(25)

In the opposite case the band does not flatten because
$\Delta_{-}=1-(1-\gamma^{2})^{1/4}$. This particular case is illustrated in
figure 4(e) where the plasmon branches have relatively sharp minimum for
$|q|<1$ and flatten only when $q$ approaches the Brillouin zone edge.

###  3.3 Spatial distribution of the Josephson phase vibrations

The respective eigenvectors
$\left(\mathbf{A}^{(v)},\mathbf{A}^{(h)}\right)=\left(A^{(v)}_{1},\cdots,A^{(v)}_{N-1},A^{(h)}_{1},\cdots,A^{(h)}_{N}\right)$
can be computed from the linear set of equations that emanates from the
equations of motion (5)-(8). With the help of the matrices $L$, $U$ and
$D_{v,h}$ defined in A it is possible to write the equations for the
eigenvectors in the following form:

$\displaystyle[\omega_{i}^{2}(q)-\omega^{2}]A_{n}^{(v)}+\frac{1-e^{-iq}}{\beta_{L}}\left(A_{n}^{(h)}-A_{n+1}^{(h)}\right)=0,\;n=\overline{1,N-1},$
(26)
$\displaystyle({1-e^{iq}})\left(A_{n}^{(v)}-A_{n-1}^{(v)}\right)-A_{n-1}^{(h)}-A^{(h)}_{n+1}+$
(27)
$\displaystyle+[2\beta_{L}\eta(1-\omega^{2})+1]A_{n}^{(h)}=0,\;n=\overline{2,N-1},$
$\displaystyle({1-e^{iq}})A_{1}^{(v)}+[\beta_{L}\eta(1-\omega^{2})+1]A_{1}^{(h)}-A_{2}^{(h)}=0,$
(28)
$\displaystyle({1-e^{iq}})A_{N}^{(v)}+A_{N-1}^{(h)}-[\beta_{L}\eta(1-\omega^{2})+1]A_{N}^{(h)}=0~{}.$
(29)

#### Long-wave limit $q=0$.

In this limit regardless of the particular eigenfrequency the plasmon modes of
the vertical and horizontal sublattices decouple. As a result, the amplitudes
of the vertical subsystem come as a solution of the strongly degenerate system
of equations and are basically the eigenvectors of the diagonal matrix:
$\mathbf{A}^{(v)}=[(1,0,\dots,0)$,$(0,1,\dots,0)$,$\ldots$, $(0,0,\dots,1)]$.
In the horizontal subsystem all the junctions are excited and their amplitudes
satisfy more complex conditions as they are eigenvectors of the tridiagonal
matrix. One of the eigenfrequencies coincides with the Josephson plasma
frequency ($\omega_{0}=1$) and is dispersionless. The respective eigenvector
consists of only horizontal junctions exited while all vertical ones are at
rest as shown schematically in figure 5(a). In the most degenerate case when
the external bias is absent the general picture shown in figures 5(a)-5(e)
does not change.

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (a) $\omega_{0}$

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (b) $\omega_{-2}$

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (c) $\omega_{-1}$

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (d) $\omega_{+1}$

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (e) $\omega_{+2}$

Figure 5: Schematic representation of the plasmon amplitude distribution for
the array with $N=3$ rows in the long wave limit $q=0$. Color arrows
correspond to the excited junctions where the red arrow (pointing right) and
blue arrow (pointing left) represent positive (negative) amplitude of the
respective junction. Arrows directed upwards describe excited vertical
junctions. Unexcited junctions have no arrows. Subcaptions under each figure
denote to which eigenvalue this eigenvector belongs.

#### Beyond the long-wave limit ($q\neq 0$).

If one departs slightly from the center of the Brillouin zone ($q=0$) the
vibrations in the horizontal and vertical subsystems start to mix. For small
values of dc bias the terms $\propto(1-\cos q)$ start to appear in the
components that were unexcited in the $q=0$ limit. In the unbiased case there
is $(N-1)$-fold degeneracy of the $\omega=1$ eigenfrequency. The mode which
initially was uniform with respect to horizontal vibrations and with no
excited vertical vibrations now has both vertical and horizontal junctions
excited. Similarly, the $\omega_{0}=1$ mode has also both vertical and
horizontal junctions excited.

If the external bias $\gamma$ is applied the degeneracy for the modes
$\omega_{-N+1},\ldots,\omega_{-1}$ is lifted (except the central point $q=0$).
As a result, the $\omega_{0}=1$ mode retains the same structure as it had in
the long-wave limit: $A_{n}^{(v)}=0$, $n=\overline{1,N-1}$; $A_{n}^{(h)}=A\neq
0$, $n=\overline{1,N}$ (see also figure 5(a)). The modes that lie below the
flat branch $\omega_{n}(q)<1,n<0$ have the same spatial structure as their
counterparts above the flat branch. In other words, the spatial structure of
$\omega_{-n}$ and $\omega_{n}$ modes will be the same for each $n$. The
particular case for $N=3$ is shown in figures 6(a)-6(c). The modes with
$\omega_{\neq 0}$ have certain symmetry with respect to the symmetry line that
is parallel to the $OX$ axis and cuts the array in halves. If the number of
rows $N$ is odd this line coincides with the $(N+1)\,$th row while if $N$ is
even this line lies in the middle between the $N/2\,$th and $(N+1)/2\,$th
rows. According to this symmetry, in the amplitude distribution of each mode
either the vertical phases are excited in the antisymmetric way while the
horizontal phases are excited in the symmetric way or vice versa. The figures
6(b)-6(c) clearly demonstrate that.

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (a) $\omega_{0}=1$

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (b) $\omega_{\pm 1}$

$\times$$\times$$\times$$\times$$\times$$\times$$\times$ (c) $\omega_{\pm 2}$

Figure 6: Schematic representation of the plasmon waves that correspond to
different eigenfrequencies (see the respective subcaption) for $N=3$ rows and
$\gamma\neq 0$. The meaning of the color arrows is the same as in figures
5(a)-5(e).

At the edges of the Brillouin zone ($q=\pm\pi$) the structure of the non-flat
bands becomes more symmetric with all the amplitudes of the horizontal
junctions having the same absolute value.

##  4 Discussion and conclusions

In this article the linear wave spectrum of the Q1D JJA is studied. The Q1D
JJA is a multi-ladder array that is considered to be infinite in $X$ direction
and consists of $N$ rows in $Y$ direction. The array is uniformly biased by
the dc current applied along the every vertical column. The main result can be
summarized in the following way. The Josephson plasmon spectrum of the array
consists of $2N-1$ branches. In the unbiased case $N$ branches are completely
flat and the respective eigenfrequency coincides with the Josephson plasma
frequency. The remaining $N-1$ branches are strongly dispersive and their
dispersion laws are similar to the standard 1D JJA dispersion relations. When
the dc bias is applied the $N$-fold degeneracy is lifted and only one flat
branch remains. This flat branch corresponds to the plasmon mode where all
vertical junctions are not excited and all horizontal junctions oscillate in
phase. The rest of the branches become weakly dispersive if the dc bias is
small: $\gamma=I_{B}/I_{c}^{(v)}\ll 1$. In that case the width of all flat
bands is confined by the limits $(1-\gamma^{2})^{1/4}\leq\omega<1$.

The obtained Q1D JJA spectra are important for the studies of discrete
breathers in these structures. While the discrete breather properties in the
simple JJ ladder are well understood, their counterparts in more complex
ladder-like structures have not been studied yet. The current studies of the
linear spectra are the necessary first step in that direction. In particular,
we would like to point out that in the limit of small horizontal-vertical
anisotropy $\eta\ll 1$ there are large gaps between the plasmon modes. This
opens a possibility of the gap discrete breathers [28] appearing in different
gaps of the linear spectrum. Finally, the phenomenon of Fano resonances in the
simlest JJ ladder [29] can be investigated for more complex Q1D JJAs.

Both the authors acknowledge the support by the National Research Foundation
of Ukraine grant (2020.02/0051) ”Topological phases of matter and excitations
in Dirac materials, Josephson junctions and magnets”.

##  Appendix A Computation of the characteristic polynomial

This Appendix is devoted to the details of the characteristic polynomial
computation. The characteristic polynomial is given by the determinant of the
$2N-1\times 2N-1$ matrix

$\chi(\omega^{2})=\det(A-\mathbb{I}_{2N-1}\omega^{2}),\;A=\left[\begin{array}[]{c|c}D_{v}&U\\\
\hline\cr L&D_{h}\end{array}\right].$ (30)

Here $D_{v}=\omega^{2}_{i}(q)\mathbb{I}_{N-1}$, $\mathbb{I}_{N}$ is the
$N\times N$ identity matrix, $U$ is $N\times N-1$ matrix, $L$ is $N-1\times N$
matrix and $D_{h}$ is tridiagonal $N-1\times N-1$. These three matrices are
given by the following expressions:

$\displaystyle{D_{h}}\equiv\left[\begin{array}[]{ccccccc}\vspace{2mm}1+b&-b&0&\cdots&0&0&0\\\
\vspace{2mm}-b&1+2b&-b&\cdots&0&0&0\\\ \vspace{2mm}0&-b&1+2b&\cdots&0&0&0\\\
\vspace{2mm}\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
\vspace{2mm}0&0&0&\cdots&-b&1+2b&-b\\\ \vspace{2mm}0&0&0&\cdots&0&-b&1+b\\\
\end{array}\right],$ (37) $\displaystyle
U\equiv\frac{1-e^{-iq}}{\beta_{L}}~{}S,\;S=\left[\begin{array}[]{ccccccc}\vspace{2mm}1&-1&0&\cdots&0&0&0\\\
\vspace{2mm}0&1&-1&\cdots&0&0&0\\\ \vspace{2mm}0&0&1&\cdots&0&0&0\\\
\vspace{2mm}\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
\vspace{2mm}0&0&0&\cdots&1&-1&0\\\ \vspace{2mm}0&0&0&\cdots&0&1&-1\\\
\end{array}\right],$ (44) $\displaystyle
L\equiv\frac{1-e^{iq}}{\beta_{L}\eta}~{}S^{T},\;b\equiv\frac{1}{\beta_{L}\eta}.$
(45)

The term $\omega_{i}(q)$ is the dispersion law of the one-dimensional array of
the biased vertical Josephson junctions (see [25, 27])

$\omega^{2}_{i}(q)\equiv\sqrt{1-\gamma^{2}}+\frac{2}{\beta_{L}}(1-\cos q).$
(46)

With the help of Schur complement [30] it is possible to simplify the
determinant of the matrix $A-\mathbb{I}_{2N-1}\omega^{2}$ [see equation (30)]:

$\displaystyle\chi(\omega^{2})=\det(D_{v}-\omega^{2}\mathbb{I}_{N-1})\det[D_{h}-\omega^{2}\mathbb{I}_{N}-L(D_{v}-\omega^{2}\mathbb{I}_{N-1})^{-1}U]=$
$\displaystyle=[\omega_{i}^{2}(q)-\omega^{2}]^{N-1}\det\left[D_{h}-\omega^{2}\mathbb{I}_{N}-\frac{1}{(\omega_{i}^{2}(q)-\omega^{2})}LU\right],$
(47) $\displaystyle
LU=\frac{|1-e^{iq}|^{2}}{\eta\beta_{L}^{2}}\left[\begin{array}[]{ccccccc}\vspace{2mm}1&-1&0&\cdots&0&0&0\\\
\vspace{2mm}-1&2&-1&\cdots&0&0&0\\\ \vspace{2mm}0&-1&2&\cdots&0&0&0\\\
\vspace{2mm}\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
\vspace{2mm}0&0&0&\cdots&-1&2&-1\\\ \vspace{2mm}0&0&0&\cdots&0&-1&1\\\
\end{array}\right].$ (54)

The matrix in $[\cdots]$ is an $N\times N$ tridiagonal matrix. The resulting
characteristic polynomial can be written as a determinant of the tridiagonal
matrix $D_{N}$:

$\displaystyle\chi(\omega^{2})=\frac{\det{D_{N}}}{\eta^{N}[\omega_{i}(q)-\omega]},\;$
(55) $\displaystyle D_{2}=\left[\begin{array}[]{cc}d_{0}&d_{1}\\\
d_{1}&d_{0}\end{array}\right],\;D_{N}=\left[\begin{array}[]{ccccccc}d_{0}&d_{1}&0&\cdots&0&0&0\\\
d_{1}&d_{2}&d_{1}&\cdots&0&0&0\\\ 0&d_{1}&d_{2}&\cdots&0&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&d_{2}&d_{1}&0\\\ 0&0&0&\cdots&d_{1}&d_{2}&d_{1}\\\
0&0&0&\cdots&0&d_{1}&d_{0}\\\ \end{array}\right],$ (65)

with

$\displaystyle
d_{0}=\eta[\omega^{2}_{i}(q)-\omega^{2}]\left(1+\frac{1}{\eta\beta_{L}}-\omega^{2}\right)-\frac{2(1-\cos{q})}{\beta_{L}^{2}},$
(66) $\displaystyle
d_{1}=-\frac{1}{\beta_{L}}[\omega^{2}_{i}(q)-\omega^{2}]+\frac{2(1-\cos{q})}{\beta_{L}^{2}},$
(67) $\displaystyle
d_{2}=d_{0}-d_{1}=\eta[\omega^{2}_{i}(q)-\omega^{2}]\left(1+\frac{2}{\eta\beta_{L}}-\omega^{2}\right)-\frac{4(1-\cos{q})}{\beta_{L}^{2}},$
(68)

Note that the matrix $D_{2}$ does not contain the $d_{2}$ element

In general, the determinant $\det D_{N}$ can be factorized if the matrix
$D_{N}$ can be diagonalized. The eigenvalues of this tridiagonal matrix are
known [31]:

$\lambda_{n}=d_{0}+\left[-1+2\cos\frac{\pi(n-1)}{N}\right]d_{1},\;n=\overline{1,N}\,.$
(69)

Therefore, the determinant of the matrix $D_{N}$ can be written explicitly.
Also it should be taken into accound that there always exists the first
eigenvalue, $\lambda_{1}=d_{0}+d_{1}$. This will help to remove the singular
term in the characteristic polynomial.

$\displaystyle\det
D_{N}=\prod_{n=1}^{N}\left\\{d_{0}+\left[-1+\cos\frac{\pi(n-1)}{N}\right]d_{1}\right\\}=(d_{0}+d_{1})\times$
$\displaystyle\times\prod_{n=2}^{N}\left\\{d_{0}+\left[-1+\cos\frac{\pi(n-1)}{N}\right]d_{1}\right\\}.$
(70)

As a result, the characteristic polynomial for the squared frequency can be
written as a product:

$\displaystyle\chi(\omega^{2})\propto(\omega^{2}-1)\prod_{n=2}^{N}[d_{0}-\alpha_{n}d_{1}],~{}~{}\alpha_{n}=1-2\cos\frac{\pi(n-1)}{N}.$
(71)

The roots of this polynomial are given by the equation $\omega^{2}=1$ and
equation $d_{0}=\alpha_{n}d_{1}$. Note that $d_{0}$ and $d_{1}$ are
polynomials for $\omega^{2}$ given by equations (66)-(67). Equation
$d_{0}=\alpha_{n}d_{1}$ transforms (the subscript n has been dropped for the
sake of simplicity) into the quadratic equation for $\omega^{2}$:

$\displaystyle\omega^{4}$ $\displaystyle-$
$\displaystyle\left(1+\omega_{i}^{2}+\frac{1+\alpha}{\eta\beta_{L}}\right)\omega^{2}+\omega_{i}^{2}\left(1+\frac{1+\alpha}{\eta\beta_{L}}\right)-$
(72) $\displaystyle-$ $\displaystyle
2(1+\alpha)\frac{1-\cos{q}}{\eta\beta_{L}^{2}}=0.$

The roots of the abovementioned equation are

$\displaystyle\omega^{2}(q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+\omega_{i}^{2}+\frac{1+\alpha}{\eta\beta_{L}}\right)\pm$
(73) $\displaystyle\pm$
$\displaystyle\sqrt{\frac{1}{4}\left[\omega_{i}^{2}-\left(1+\frac{1+\alpha}{\eta\beta_{L}}\right)\right]^{2}+2(1+\alpha)\frac{1-\cos
q}{\eta\beta_{L}^{2}}}.$

## ORCID iDs

Daryna Bukatova https://orcid.org/0000-0002-0522-851X

Yaroslav Zolotaryuk https://orcid.org/0000-0003-1079-0221

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