# Perturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer
systems of subwavelength resonators

Habib Ammari1 1ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092
Zürich, Switzerland<EMAIL_ADDRESS>,  Silvio Barandun1
<EMAIL_ADDRESS>and  Ping Liu1<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is fourfold: (i) to obtain explicit formulas for the
eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use
of such formulas in order to provide a mathematical justification of the non-
Hermitian skin effect in dimer systems by proving the condensation of the
system’s bulk eigenmodes at one of the edges of the system; (iii) to show the
topological origin of the non-Hermitian skin effect for dimer systems and (iv)
to prove localisation of the interface modes between two dimer structures with
non-Hermitian gauge potentials of opposite signs based on new estimates of the
decay of the entries of the eigenvectors of block matrices with mirrored
blocks.

Keywords. Block Toeplitz matrix, tridiagonal $2$-Toeplitz matrix, non-
Hermitian skin effect, dimer system, complex gauge potential, gauge
capacitance matrix, condensation of the eigenmodes, interface eigenmodes,
topological invariant, subwavelength physics.

AMS Subject classifications. 15A18, 15B05, 35B34, 35P25, 35C20, 81Q12.

##  1\. Introduction

The ultimate goal of subwavelength physics is to manipulate waves at
subwavelength scales in a robust way [14, 15, 23, 17, 16, 4]. Subwavelength
resonators are the building blocks of the resonant structures used in
subwavelength physics. Many spectacular phenomena in subwavelength physics
have been recently demonstrated and mathematically studied [4, 5, 6, 12, 2,
9]. The non-Hermitian skin effect is one of the most intriguing ones [21, 24].
Due to a complex gauge potential inside the resonators, for a system of
finitely many resonators the eigenmodes decay exponentially and condensate at
one of the edges of the structure. In [1], a mathematical theory of the non-
Hermitian skin effect arising in subwavelength physics in one dimension has
been derived from first principles. Through a gauge capacitance matrix
formulation, explicit asymptotic expressions for the subwavelength
eigenfrequencies and eigenmodes of systems of a single repeating resonator
have been obtained. This allowed the authors to characterise the system’s
fundamental behaviours and reveal the mechanisms behind them. In particular,
the exponential decay of eigenmodes (the so-called non-Hermitian skin effect)
was shown to be induced by the Fredholm index of an associated (tridiagonal)
Toeplitz operator. The explicit theory developed in [1] was only possible
because of the simple structure of the gauge capacitance matrix and the rich
literature on (tridiagonal) Toeplitz matrices and perturbations thereof [19,
22]. The theory of systems with periodically repeated cells of $K$ resonators
($K\geq 2$) remains incomplete as no similar results to those used in [1] have
been known for block-Toeplitz matrices. However, some numerical illustrations
of the non-Hermitian skin effect in systems of finitely many dimers with
imaginary gauge potentials are presented. These numerical results clearly show
strongly localised eigenmodes at one edge of the dimer system and suggest that
the non-Hermitian skin effect holds for systems with multiple resonators in
the unit cell. Nevertheless, compared to the single resonator case, the
physics for dimer systems is much richer as the system eigenvalues are grouped
into two families corresponding to eigenmodes with two different physical
natures (monopole and dipole behaviors). Consequently, the mathematical
analysis of dimer systems is much harder.

In this paper, we obtain for the first time explicit formulas for the
eigenpairs of perturbed tridiagonal block Toeplitz matrices, which have their
own interest and may found applications in other fields such as quantum
mechanics and condensed matter theory. Applying these formulas in the field of
subwavelength physics, we provide a mathematical justification of the non-
Hermitian skin effect in dimer systems by proving the condensation of the
eigenvectors of the associated gauge capacitance matrix. Moreover, we show the
topological origin of the non-Hermitian skin effect for dimer systems. In
contrast with the single resonator case, the determinant of the symbol of the
$2$-Toeplitz operator associated with the semi-infinite structure has a zero
on the unit circle, and therefore its winding is not equal to the Fredholm
index of the operator. Nevertheless, since the system eigenvalues are grouped
into two families corresponding to eigenmodes with monopole and dipole
behaviors, we can show that each group corresponds to negative winding of one
of the eigenvalues of the symbol of the $2$-Toeplitz operator. On the other
hand, we consider interface modes between two structures where the sign of the
complex gauge potential changes and prove that all but few eigenmodes are
localised at the interface between the two structures.

The paper presents a number of original results and findings: (i) a general
strategy for deriving formulas for the eigenvalues and eigenvectors of
tridiagonal $2$-Toeplitz matrices with perturbations on the diagonal corners;
(ii) an estimate of the decay of the entries of the eigenvectors of block
matrices with mirrored blocks; and (iii) mathematical foundations of the non-
Hermitian skin effect in dimer systems, its topological origin and non-
Hermitian interface modes between opposing signs of the gauge potentials.

The paper is organised as follows. In Section 2, we first recall some known
results on Chebyshev polynomials and then characterize the eigenvalues of
tridiagonal $2$-Toeplitz matrices with perturbations on the diagonal corners.
Section 3 is dedicated to the construction of the eigenvectors of tridiagonal
2-Toeplitz matrices with perturbations on the diagonal corners. In Section 4,
we prove the condensation of the eigenvectors of perturbed $2$-Topelitz
matrices and block matrices with mirrored $2$-Toeplitz matrices. In Section 5,
we formulate the physical model for dimer systems of subwavelength resonators
with a complex gauge potential inside only the resonators. Without exciting
the structure’s subwavelength resonances, the effect of the complex gauge
potential would be negligible. We also show how to apply the general results
obtained in the previous sections to prove the non-Hermitian skin effect for
dimer systems and the eigenmode condensation at the interface between two
structures with opposite signs of gauge potentials. Furthermore, we provide
the topological origin of the non-Hermitian skin effect for dimer systems. In
Section 6, we draw some conclusions and state some open problems and
extensions to our present work.

##  2\. Eigenvalues of tridiagonal $2$-Toeplitz matrices with perturbations on
the diagonal corners

In this section, we first present some well-known results on Chebyshev
polynomials and then characterize the eigenvalues of tridiagonal $2$-Toeplitz
matrices with perturbations on the diagonal corners.

###  2.1. Tridiagonal $2$-Toeplitz matrices with perturbations on the diagonal
corners

Let $A_{2m+1}^{(a,b)}$ be the tridiagonal $2$-Toeplitz matrix of order $2m+1$
with perturbations $(a,b)$ in the diagonal corners, that is,

$A_{2m+1}^{(a,b)}=\left(\begin{array}[]{ccccccc}\alpha_{1}+a&\beta_{1}&&&&&\\\
\gamma_{1}&\alpha_{2}&\beta_{2}&&&&\\\ &\gamma_{2}&\alpha_{1}&\beta_{1}&&&\\\
&&\gamma_{1}&\alpha_{2}&\ddots&&\\\ &&&\ddots&\ddots&\ddots&\\\
&&&&\gamma_{1}&\alpha_{2}&\beta_{2}\\\ &&&&&\gamma_{2}&\alpha_{1}+b\\\
\end{array}\right).$ (2.1)

Here, $\beta_{i},\gamma_{i},\alpha_{i}$, $i=1,2$ and $a,b$ are in
$\mathbb{R}$. Let $A_{2m}^{(a,b)}$ be the tridiagonal $2$-Toeplitz matrix of
order $2m$ with perturbations $(a,b)$ in the diagonal corners, that is,

$A_{2m}^{(a,b)}=\left(\begin{array}[]{ccccccc}\alpha_{1}+a&\beta_{1}&&&&&\\\
\gamma_{1}&\alpha_{2}&\beta_{2}&&&&\\\ &\gamma_{2}&\alpha_{1}&\beta_{1}&&&\\\
&&\gamma_{1}&\alpha_{2}&\ddots&&\\\ &&&\ddots&\ddots&\ddots&\\\
&&&&\gamma_{2}&\alpha_{1}&\beta_{1}\\\ &&&&&\gamma_{1}&\alpha_{2}+b\\\
\end{array}\right).$ (2.2)

######  Remark 2.1.

We assume throughout that the off-diagonal elements in (2.1) and (2.2) are
nonzero and satisfy the following condition:

$\gamma_{i}\beta_{i}>0,\quad i=1,2.$

###  2.2. Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the
cosine and sine functions, denoted respectively by $T_{n}(x)$ and $U_{n}(x)$.
In particular, the Chebyshev polynomials of the first kind are obtained from
the recurrence relation

$\displaystyle T_{0}(x)$ $\displaystyle=1,$ $\displaystyle T_{1}(x)$
$\displaystyle=x,$ $\displaystyle T_{n+1}(x)$
$\displaystyle=2xT_{n}(x)-T_{n-1}(x),$

and the Chebyshev polynomials of the second kind are obtained from the
recurrence relation

$\displaystyle U_{0}(x)$ $\displaystyle=1,$ $\displaystyle U_{1}(x)$
$\displaystyle=2x,$ $\displaystyle U_{n+1}(x)$
$\displaystyle=2xU_{n}(x)-U_{n-1}(x).$

The roots of $T_{n}(x)$ are

$x_{k}=\cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1,$

and the roots of $U_{n}(x)$ are

$x_{k}=\cos\left(\frac{k\pi}{n+1}\right),\quad k=1,\cdots,n.$ (2.3)

It is also well-known that for $-1\leq x\leq 1$ and $k=0,1,\ldots$, we have
the upper bounds

$\left|T_{n}(x)\right|\leq\left|T_{n}(1)\right|=1,\quad\left|U_{k}(x)\right|\leq\left|U_{k}(1)\right|=k+1.$
(2.4)

###  2.3. Eigenvalues of $A_{2m+1}^{(a,b)}$ and $A_{2m}^{(a,b)}$

In this subsection, we present a detailed characterisation of the eigenvalues
of $A_{2m+1}^{(a,b)}$ and $A_{2m}^{(a,b)}$.

Let us first define the polynomials

$\pi_{2}(x)=\left(x-\alpha_{1}\right)\left(x-\alpha_{2}\right)$

and

$P_{k}^{*}(x)=\left(\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}\right)^{k}U_{k}\left(\frac{x-\gamma_{1}\beta_{1}-\gamma_{2}\beta_{2}}{2\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}}\right),$
(2.5)

where $U_{k}$ is the Chebyshev polynomial of the second kind. It is well known
(see [13, 10, 18] ) that the characteristic polynomials of the $2$-Toeplitz
matrices $A_{2m+1}^{(0,0)},A_{2m}^{(0,0)}$ are respectively

$Q_{2k+1}(x)=\left(x-\alpha_{1}\right)P_{k}^{*}\left(\pi_{2}(x)\right)$ (2.6)

and

$Q_{2k}(x)=P_{k}^{*}\left(\pi_{2}(x)\right)+\gamma_{2}\beta_{2}P_{k-1}^{*}\left(\pi_{2}(x)\right).$
(2.7)

It is also shown in [11] that the characteristic polynomials of
$A_{2m+1}^{(a,b)},A_{2m}^{(a,b)}$ are respectively

$\displaystyle P_{2m+1}(x)=$
$\displaystyle\left(x-\alpha_{1}-a-b\right)P_{m}^{*}\left(\pi_{2}(x)\right)$
(2.8)
$\displaystyle+\left(ab\left(x-\alpha_{2}\right)-a\gamma_{1}\beta_{1}-b\gamma_{2}\beta_{2}\right)P_{m-1}^{*}\left(\pi_{2}(x)\right)$

and

$\displaystyle P_{2m}(x)=$ $\displaystyle
P_{m}^{*}\left(\pi_{2}(x)\right)+\left(a\left(\alpha_{2}-x\right)+b\left(\alpha_{1}-x\right)+a+b+\gamma_{2}\beta_{2}\right)P_{m-1}^{*}\left(\pi_{2}(x)\right)$
(2.9) $\displaystyle+ab\gamma_{1}\beta_{1}P_{m-2}^{*}\left(\pi_{2}(x)\right).$

For more results on the eigenproblem of tridiagonal $K$-Toeplitz matrices, we
refer the readers to [11] and the references therein.

To help to demonstrate the non-Hermitian skin effect, we first present a
detailed characterisation of the eigenvalues of $A_{2m+1}^{(a,b)}$ and
$A_{2m}^{(a,b)}$. In particular, we remark that one may have better results
through a delicate analysis on the roots of (2.8) and (2.9), but here we
choose a simple method using the Cauchy interlacing theorem.

######  Theorem 2.2.

Let $m$ be large enough. The eigenvalues $\lambda_{r}$’s of $A_{2m+1}^{(a,b)}$
are all real numbers. Except for at most $11$ eigenvalues, we can reindex the
$\lambda_{r}$’s to have

$\lambda_{3}^{l}<\lambda_{4}^{l}<\cdots<\lambda_{m-3}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq\lambda_{m-3}^{r}<\cdots<\lambda_{4}^{r}<\lambda_{3}^{r}.$

In particular, for $k=3,\cdots,m-3$,

$\cos\left(\frac{k\pi}{m}\right)\leq\min\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\max\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\cos\left(\frac{(k-2)\pi}{m}\right),$
(2.10)

with

$y(x)=\frac{(x-\alpha_{1})(x-\alpha_{2})-\gamma_{1}\beta_{1}-\gamma_{2}\beta_{2}}{2\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}}.$
(2.11)

###### Proof.

The proof is divided into two steps.

Step 1. Note that $A_{2m+1}^{(a,b)}$ has the same eigenvalues as the Hermitian
matrix

$H:=\left(\begin{array}[]{ccccccc}\alpha_{1}+a&\sqrt{\gamma_{1}\beta_{1}}&&&&&\\\
\sqrt{\gamma_{1}\beta_{1}}&\alpha_{2}&\sqrt{\gamma_{2}\beta_{2}}&&&&\\\
&\sqrt{\gamma_{2}\beta_{2}}&\alpha_{1}&\sqrt{\gamma_{1}\beta_{1}}&&&\\\
&&\sqrt{\gamma_{1}\beta_{1}}&\alpha_{2}&\ddots&&\\\
&&&\ddots&\ddots&\ddots&\\\
&&&&\sqrt{\gamma_{1}\beta_{1}}&\alpha_{2}&\sqrt{\gamma_{2}\beta_{2}}\\\
&&&&&\sqrt{\gamma_{2}\beta_{2}}&\alpha_{1}+b\\\ \end{array}\right)$ (2.12)

which can seen from computing $\left|{xI-
A_{2m+1}^{(a,b)}}\right|=\left|{xI-H}\right|$ using Laplace expansion on the
last row. Thus the eigenvalues of $A_{2m+1}^{(a,b)}$ are real numbers. To
demonstrate (2.10), we first analyse the eigenvalues of $A_{2m+1}^{(0,0)}$,
i.e., the case when $a=b=0$. Note that by (2.6) the eigenvalues
$\\{\lambda_{r}\\}$ of $A_{2m+1}^{(0,0)}$ are the roots of the polynomial

$Q_{2m+1}(x)=\left(x-\alpha_{1}\right)P_{m}^{*}\left(\pi_{2}(x)\right).$

By the definition of $P_{k}^{*}(x)$ in (2.5), to find the roots of
$Q_{2m+1}(x)$ (except the trivial root $x=\alpha_{1}$), we only need to find
the solutions of

$\displaystyle U_{m}(y)=0,$ (2.13)

where $y$ is defined by (2.11). By (2.3), the solutions are given by

$y_{k}=\cos\left(\frac{k\pi}{m+1}\right),\ k=1,2,\cdots,m.$

From (2.11), it is not hard to see that the $\lambda_{r}$’s corresponding to
$y_{k}=\cos\left(\frac{k\pi}{m+1}\right)$ should belong to
$\left(-\infty,\min\\{\alpha_{1},\alpha_{2}\\}\right]$ or
$\left[\max\\{\alpha_{1},\alpha_{2}\\},+\infty\right)$. Therefore, the
$\lambda_{r}$’s can be reindexed in order to have

$\lambda_{1}^{l}<\lambda_{2}^{l}<\cdots<\lambda_{m}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq\lambda_{m}^{r}<\cdots<\lambda_{2}^{r}<\lambda_{1}^{r}$

with

$y\left(\lambda_{k}^{l}\right)=y\left(\lambda_{k}^{r}\right)=\cos\left(\frac{k\pi}{m+1}\right),\quad
k=1,2,\cdots,m.$

Step 2. We now turn to the case when $(a,b)\neq(0,0)$. Consider
$A_{2m+1}^{(a,b)}$’s principal submatrices $A_{2m}^{(a,0)}$ and

$D_{2m-1}^{(0,0)}=\left(\begin{array}[]{ccccc}\alpha_{2}&\beta_{2}&&&\\\
\gamma_{2}&\alpha_{1}&\beta_{1}&&\\\ &\gamma_{1}&\alpha_{2}&\ddots&\\\
&&\ddots&\ddots&\beta_{2}\\\ &&&\gamma_{2}&\alpha_{1}\end{array}\right).$

Denote the eigenvalues of $A_{2m}^{(a,0)}$ by $t_{1},\cdots,t_{2m}$, assuming
that they are distributed in decreasing order and the eigenvalues of
$D_{2m-1}^{(0,0)}$ by $g_{1},\cdots,g_{2m-1}$, assuming that they are
distributed in decreasing order. Since the eigenvalues of $A_{2m}^{(a,0)}$ and
$D_{2m-1}^{(0,0)}$ are the same as those of some Hermitian matrices like
(2.12), by the Cauchy interlacing theorem, we thus obtain that

$t_{2m}\leq g_{2m-1}\leq t_{2m-1}\leq g_{2m-2}\leq\cdots\leq t_{2}\leq
g_{1}\leq t_{1}.$ (2.14)

Applying the same arguments as those in Step 1 to the eigenvalues
$\\{g_{r}\\}$ of $D_{2m-1}^{(0,0)}$, we have that the $g_{r}$’s can be
reindexed to have

$g_{1}^{l}<g_{2}^{l}<\cdots<g_{m-1}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq
g_{m-1}^{r}<\cdots<g_{2}^{r}<g_{1}^{r}$

with

$y\left(g_{k}^{l}\right)=y\left(g_{k}^{r}\right)=\cos\left(\frac{k\pi}{m}\right),\quad
k=1,\cdots,m-1.$ (2.15)

By (2.14), except for at most $6$ $t_{r}$’s, the eigenvalues of
$A_{2m}^{(a,0)}$ can be reindexed to have

$t_{2}^{l}<t_{3}^{l}<\cdots<t_{m-2}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq
t_{m-2}^{r}<\cdots<t_{3}^{r}<t_{2}^{r}.$

In particular, since (2.11) is decreasing on the left of
$\frac{\alpha_{1}+\alpha_{2}}{2}$ and increasing on the right, by (2.15) we
have for $k=2,3,\cdots,m-2$,

$\cos\left(\frac{k\pi}{m}\right)\leq\min\left\\{y\left(t_{k}^{l}\right),y\left(t_{k}^{r}\right)\right\\}\leq\max\left\\{y\left(t_{k}^{l}\right),y\left(t_{k}^{r}\right)\right\\}\leq\cos\left(\frac{(k-1)\pi}{m}\right).$

Similarly, as $A_{2m}^{(a,0)}$ is a principal submatrix of $A_{2m+1}^{(a,b)}$,
by the Cauchy interlacing theorem, we have that, except for at most $11$
eigenvalues, we can arrange the eigenvalues in such a way that

$\lambda_{3}^{l}<\lambda_{4}^{l}<\cdots<\lambda_{m-3}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq\lambda_{m-3}^{r}<\cdots<\lambda_{4}^{r}<\lambda_{3}^{r}.$

In particular, for $k=3,\cdots,m-3$,

$\cos\left(\frac{k\pi}{m}\right)\leq\min\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\max\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\cos\left(\frac{(k-2)\pi}{m}\right).$

This completes the proof.

∎

We now prove the following result.

######  Theorem 2.3.

Let $m$ be large enough. The eigenvalues $\\{\lambda_{r}\\}$ of
$A_{2m}^{(a,b)}$ are all real numbers. Except for at most $12$ eigenvalues, we
can reindex the $\lambda_{r}$’s to have

$\lambda_{3}^{l}<\lambda_{4}^{l}<\cdots<\lambda_{m-4}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq\lambda_{m-4}^{r}<\cdots<\lambda_{4}^{r}<\lambda_{3}^{r}.$

In particular, for $k=3,\cdots,m-4$,

$\cos\left(\frac{(k+1)\pi}{m}\right)\leq\min\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\max\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\cos\left(\frac{(k-2)\pi}{m}\right)$
(2.16)

with $y(x)$ being defined by (2.11).

###### Proof.

The proof is divided into two steps.

Step 1. Similar to the case of $A_{2m+1}^{(a,b)}$, the eigenvalues of
$A_{2m}^{(a,b)}$ are all real numbers. To demonstrate (2.16), we first analyse
the eigenvalues of $A_{2m}^{(0,0)}$. Note that by (2.7) the eigenvalues
$\\{\lambda_{r}\\}$ of $A_{2m}^{(0,0)}$ are the roots of

$Q_{2m}(x)=P_{m}^{*}\left(\pi_{2}(x)\right)+\gamma_{2}\beta_{2}P_{m-1}^{*}\left(\pi_{2}(x)\right).$

Similarly, in order to find the roots of $Q_{2m}(x)$, we only need to find the
solutions $\\{y_{k}\\}$ to

$U_{m}(y)+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}U_{m-1}(y)=0.$
(2.17)

Define

$f(y)=U_{m}(y)+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}U_{m-1}(y).$

Without loss of generality, we suppose that $m$ is an odd number. Note that
the roots of $U_{m-1}(y)$ are $\cos\frac{k\pi}{m},\ k=1,2,\cdots,m-1$ and

$\begin{array}[]{ll}U_{m-1}(y)>0&\text{for
$y\in\left(\cos\left(\frac{(2k+1)\pi}{m}\right),\
\cos\left(\frac{2k\pi}{m}\right)\right),\ k=0,\cdots,\frac{m-1}{2},$}\\\
U_{m-1}(y)<0&\text{for $y\in\left(\cos\left(\frac{(2k+2)\pi}{m}\right),\
\cos\left(\frac{(2k+1)\pi}{m}\right)\right),\
k=0,\cdots,\frac{m-3}{2}.$}\end{array}$ (2.18)

Note also that

$\frac{k-1}{m}<\frac{k}{m+1}<\frac{k}{m},\quad k=1,\cdots,m$

and

$\cos\left(\frac{k\pi}{m}\right)<\cos\left(\frac{k\pi}{m+1}\right)<\cos\left(\frac{(k-1)\pi}{m}\right),k=1,\cdots,m.$

Recalling that the roots of $U_{m}(y)$ are $\cos\frac{k\pi}{m+1},\
k=1,2,\cdots,m$, we are now ready to estimate $f(y)$. We have

$f\left(\cos\left(\frac{\pi}{m+1}\right)\right)>0,\
f\left(\cos\left(\frac{2\pi}{m+1}\right)\right)<0,\
f\left(\cos\left(\frac{3\pi}{m+1}\right)\right)>0,\ \cdots$

Thus, the solutions $\\{y_{k}\\}$ to $f(y)=0$ satisfy that, except for only at
most two $y_{k}$’s, after reindexation,

$y_{k}=\cos\theta_{k},\quad\theta_{k}\in\left(\frac{k\pi}{m+1},\frac{(k+1)\pi}{m+1}\right),\
k=1,\cdots,m-1.$

Then, similarly to the discussions in Step 1 of the proof of Theorem 2.2, we
can reindex the $\lambda_{r}$’s to obtain

$\lambda_{1}^{l}<\lambda_{2}^{l}<\cdots<\lambda_{m-1}^{l}\leq\min\\{\alpha_{1},\alpha_{2}\\}\leq\max\\{\alpha_{1},\alpha_{2}\\}\leq\lambda_{m-1}^{r}<\cdots<\lambda_{2}^{r}<\lambda_{1}^{r}$

with

$\cos\left(\frac{(k+1)\pi}{m+1}\right)\leq\min\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\max\left\\{y\left(\lambda_{k}^{l}\right),y\left(\lambda_{k}^{r}\right)\right\\}\leq\cos\left(\frac{k\pi}{m+1}\right)$

for $k=1,2,\cdots,m$. Analogously, we can prove the result for the case when
$m$ is an even number.

Step 2. Similarly to Step 2 in the proof of Theorem 2.2, by considering
principal submatrices of $A_{2m}^{(a,b)}$ and utilizing the result in Step 1
together with the Cauchy interlacing theorem, we can prove the statement. ∎

##  3\. Eigenvectors of tridiagonal $2$-Toeplitz matrices with perturbations
on the diagonal corners

This section serves to construct the formula of the eigenvectors of
tridiagonal $2$-Toeplitz matrices with perturbations on the diagonal corners
through a general strategy. It generalizes the results obtained in [13].

###  3.1. Preliminaries

We start by introducing the following two families of polynomials.

######  Definition 3.1.

We define the two families of polynomials
$q_{k}^{(\xi_{p},\xi_{q})},p_{k}^{(\xi_{p},\xi_{q})}$ by

$q_{0}^{(\xi_{p},\xi_{q})}(\nu)=\xi_{q},\qquad
p_{0}^{\xi_{p},\xi_{p}}(\nu)=\xi_{p},$

and the recurrence formulas

$q_{k}^{(\xi_{p},\xi_{q})}(\nu)=\nu
p_{k-1}^{(\xi_{p},\xi_{q})}(\nu)-q_{k-1}^{(\xi_{p},\xi_{q})}(\nu)$ (3.1)

and

$p_{k}^{(\xi_{p},\xi_{q})}(\nu)=q_{k}^{(\xi_{p},\xi_{q})}(\nu)-\zeta
p_{k-1}^{(\xi_{p},\xi_{q})}(\nu),$ (3.2)

where

$\zeta=\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}.$ (3.3)

Then we observe that the recurrence formulas (3.1) and (3.2) can be
simplified.

######  Proposition 3.2.

If $p_{k}^{(\xi_{p},\xi_{q})}(\nu)$ and $q_{k}^{(\xi_{p},\xi_{q})}(\nu)$
satisfy (3.1) and (3.2) respectively, then

$p_{k+1}^{(\xi_{p},\xi_{q})}(\nu)=[\nu-(1+\zeta)]p_{k}^{(\xi_{p},\xi_{q})}(\nu)-\zeta
p_{k-1}^{(\xi_{p},\xi_{q})}(\nu)$ (3.4)

and

$q_{k+1}^{(\xi_{p},\xi_{q})}(\nu)=[\nu-(1+\zeta)]q_{k}^{(\xi_{p},\xi_{q})}(\nu)-\zeta
q_{k-1}^{(\xi_{p},\xi_{q})}(\nu)$ (3.5)

with

$\displaystyle p_{0}^{(\xi_{p},\xi_{q})}(\nu)=\xi_{p},\quad
p_{1}^{(\xi_{p},\xi_{q})}(\nu)=\left(\nu-\zeta\right)\xi_{p}-\xi_{q},$ (3.6)
$\displaystyle q_{0}^{(\xi_{p},\xi_{q})}(\nu)=\xi_{q},\quad
q_{1}^{(\xi_{p},\xi_{q})}(\nu)=\nu\xi_{p}-\xi_{q},$

where $\zeta$ is defined in (3.3).

###### Proof.

From (3.1) and (3.2), we have

$p_{k+1}^{(\xi_{p},\xi_{q})}(\nu)=(\nu-\zeta)p_{k}^{(\xi_{p},\xi_{q})}(\nu)-q_{k}^{(\xi_{p},\xi_{q})}(\nu),$

which gives

$q_{k}^{(\xi_{p},\xi_{q})}(\nu)=(\nu-\zeta)p_{k}^{(\xi_{p},\xi_{q})}(\nu)-p_{k+1}^{(\xi_{p},\xi_{q})}(\nu).$

Substituting the above identity into (3.2) yields (3.4). Similarly, from
(3.1), it follows that

$\nu
p_{k-1}^{(\xi_{p},\xi_{q})}(\nu)=q_{k}^{(\xi_{p},\xi_{q})}(\nu)+q_{k-1}^{(\xi_{p},\xi_{q})}(\nu),$

which together with (3.2) yields (3.5). By Definition 3.1, (3.6) can be easily
verified. ∎

###  3.2. Normalisation

To help to explain the skin effect later, in this section we normalise the
polynomials $p_{k}^{(\xi_{p},\xi_{q})}(\nu),q_{k}^{(\xi_{p},\xi_{q})}(\nu)$.
This can be achieved by setting

$\mu=\frac{\nu-\left(1+\beta^{2}\right)}{2\beta}$ (3.7)

with

$\beta^{2}=\frac{\beta_{2}\gamma_{2}}{\beta_{1}\gamma_{1}},$

and

$\widehat{p}_{k}^{(\xi_{p},\xi_{q})}(\mu)=\frac{1}{\beta^{k}}p_{k}^{(\xi_{p},\xi_{q})}\left(1+2\beta\mu+\beta^{2}\right),\
\widehat{q}_{k}^{(\xi_{p},\xi_{q})}(\mu)=\frac{1}{\beta^{k}}q_{k}^{(\xi_{p},\xi_{q})}\left(1+2\beta\mu+\beta^{2}\right).$
(3.8)

Thus from (3.4) and (3.5), we get respectively,

$\widehat{p}_{k+1}^{(\xi_{p},\xi_{q})}(\mu)=2\mu\widehat{p}_{k}^{(\xi_{p},\xi_{q})}(\mu)-\widehat{p}_{k-1}^{(\xi_{p},\xi_{q})}(\mu),$
(3.9)

and

$\widehat{q}_{k+1}^{(\xi_{p},\xi_{q})}(\mu)=2\mu\widehat{q}_{k}^{(\xi_{p},\xi_{q})}(\mu)-\widehat{q}_{k-1}^{(\xi_{p},\xi_{q})}(\mu).$
(3.10)

Equations (3.9) and (3.10) are the Chebyshev three point recurrence formula.
Also, from (3.6) the initial polynomials are given by

$\displaystyle\widehat{p}_{0}^{(\xi_{p},\xi_{q})}(\mu)=\xi_{p},\quad\widehat{p}_{1}^{(\xi_{p},\xi_{q})}(\mu)=2\mu\xi_{p}+\frac{\xi_{p}-\xi_{q}}{\beta},$
(3.11)
$\displaystyle\widehat{q}_{0}^{(\xi_{p},\xi_{q})}(\mu)=\xi_{q},\quad\widehat{q}_{1}^{(\xi_{p},\xi_{q})}(\mu)=(2\mu+\beta)\xi_{p}+\frac{\xi_{p}-\xi_{q}}{\beta}.$

###  3.3. Eigenvectors of $A_{2m+1}^{(a,b)}$ and $A_{2m}^{(a,b)}$

In this section, we present a formula for the eigenvectors of tridiagonal
$2$-Toeplitz matrices with perturbations on diagonal corners through a direct
construction. This formula shows that, even though some of the elements of the
$2$-Toeplitz matrix were perturbed, the structure of the eigenvectors is
similar to the one in [13].

######  Theorem 3.3.

The eigenvector of $A_{2m+1}^{(a,b)}$ in (2.1) associated with the eigenvalue
$\lambda_{r}$ is given by

$\displaystyle\mathbf{x}=$
$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
(3.12)
$\displaystyle\qquad\left.\ldots,-\frac{1}{\beta_{1}}s^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where $\top$ denotes the transpose and
$\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(\cdot\right),\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(\cdot\right)$
are defined as in (3.8) with
$\xi_{q}=(\alpha_{1}-\lambda_{r}),\xi_{p}=(\alpha_{1}+a-\lambda_{r})$, and
$s,\mu_{r}$ are respectively given by

$s=\sqrt{\frac{\gamma_{1}\gamma_{2}}{\beta_{1}\beta_{2}}},\quad\mu_{r}=\frac{\left(\alpha_{1}-\lambda_{r}\right)\left(\alpha_{2}-\lambda_{r}\right)-(\gamma_{1}\beta_{1}+\gamma_{2}\beta_{2})}{2\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}}.$
(3.13)

###### Proof.

We first demonstrate that the eigenvector of $A_{2m+1}^{(a,b)}$ in (2.1)
associated with the eigenvalue $\lambda_{r}$ has the following form:

$\displaystyle\mathbf{x}=$
$\displaystyle\left(q_{0}^{(\xi_{p},\xi_{q})}\left(\nu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)p_{0}^{(\xi_{p},\xi_{q})}\left(\nu_{r}\right),\left(\frac{\gamma_{1}}{\beta_{2}}\right)q_{1}^{(\xi_{p},\xi_{q})}\left(\nu_{r}\right),\right.$
$\displaystyle-\frac{1}{\beta_{1}}\left(\frac{\gamma_{1}}{\beta_{2}}\right)\left(\alpha_{1}-\lambda_{r}\right)p_{1}^{(\xi_{p},\xi_{q})}\left(\nu_{r}\right),\ldots,-\frac{1}{\beta_{1}}\left(\frac{\gamma_{1}}{\beta_{2}}\right)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)p_{m-1}^{(\xi_{p},\xi_{q})}\left(\nu_{r}\right),$
$\displaystyle\qquad\left.\left(\frac{\gamma_{1}}{\beta_{2}}\right)^{m}q_{m}^{(\xi_{p},\xi_{q})}\left(\nu_{r}\right)\right)^{\top},$
(3.14)

where
$q_{k}^{(\xi_{p},\xi_{q})}\left(\cdot\right),p_{k}^{(\xi_{p},\xi_{q})}\left(\cdot\right)$
are the polynomials defined as in Definition 3.1 with
$\xi_{q}=(\alpha_{1}-\lambda_{r}),\xi_{p}=(\alpha_{1}+a-\lambda_{r})$, and
$\nu_{r}$ is given by

$\nu_{r}=\frac{\left(\alpha_{1}-\lambda_{r}\right)\left(\alpha_{2}-\lambda_{r}\right)}{\gamma_{1}\beta_{1}}.$
(3.15)

To prove (3.3), we consider

$\left(A_{2m+1}-\lambda_{r}I\right)\bm{\mathbf{x}}=0,$

where

$\bm{\mathbf{x}}=\left(x_{1},-\frac{1}{\beta_{1}}(\alpha_{1}-\lambda_{r})x_{2},\left(\frac{\gamma_{1}}{\beta_{2}}\right)x_{3},\cdots,-\frac{1}{\beta_{1}}\left(\frac{\gamma_{1}}{\beta_{2}}\right)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)x_{2m},\left(\frac{\gamma_{1}}{\beta_{2}}\right)^{m}x_{2m+1}\right)^{\top}.$

Considering the first row, we can choose

$x_{1}=(\alpha_{1}-\lambda_{r}),\quad x_{2}=(\alpha_{1}+a-\lambda_{r}).$

Then by the second row, we have

$\gamma_{1}x_{1}-\frac{1}{\beta_{1}}(\alpha_{1}-\lambda_{r})(\alpha_{2}-\lambda_{r})x_{2}+\gamma_{1}x_{3}=0,$

which gives

$x_{3}=\nu_{r}x_{2}-x_{1}.$

The third row is

$-\frac{\gamma_{2}}{\beta_{1}}(\alpha_{1}-\lambda_{r})x_{2}+(\alpha_{1}-\lambda_{r})\left(\frac{\gamma_{1}}{\beta_{2}}\right)x_{3}-(\alpha_{1}-\lambda_{r})\left(\frac{\gamma_{1}}{\beta_{2}}\right)x_{4}=0,$

and thus,

$\displaystyle
x_{4}=-\frac{\beta_{2}\gamma_{2}}{\beta_{1}\gamma_{1}}x_{2}+x_{3}=x_{3}-\zeta
x_{2},$

where $\zeta=\frac{\beta_{2}\gamma_{2}}{\beta_{1}\gamma_{1}}$. Continuing the
process, we can easily verify that

$\displaystyle x_{2k+1}$ $\displaystyle=\nu_{r}x_{2k}-x_{2k-1},\quad
k=1,\cdots,m,$ $\displaystyle x_{2k}$ $\displaystyle=x_{2k-1}-\zeta
x_{2k-2},\quad k=2,\cdots,m.$

By the definition of
$q_{k}^{(\xi_{p},\xi_{q})}(\nu_{r}),p_{k}^{(\xi_{p},\xi_{q})}(\nu_{r})$ in
Definition 3.1, we note that

$\displaystyle x_{2k+1}$
$\displaystyle=q_{k}^{(\xi_{p},\xi_{q})}(\nu_{r}),\quad k=0,\cdots,m,$
$\displaystyle x_{2k}$ $\displaystyle=p_{k}^{(\xi_{p},\xi_{q})}(\nu_{r}),\quad
k=1,\cdots,m,$

with $\xi_{q}=(\alpha_{1}-\lambda_{r}),\xi_{p}=(\alpha_{1}+a-\lambda_{r})$.
This proves (3.3).

Finally, by (3.7) and (3.8), we can write (3.3) as (3.15). This completes the
proof. ∎

In the same manner, we have the following theorem for the eigenvectors of
$A_{2m}^{(a,b)}$.

######  Theorem 3.4.

The eigenvector of $A_{2m}^{(a,b)}$ in (2.2) associated with the eigenvalue
$\lambda_{r}$ is given by

$\displaystyle\mathbf{x}=$
$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
(3.16)
$\displaystyle\qquad\left.\ldots,-\frac{1}{\beta_{1}}s^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where $\xi_{q}=(\alpha_{1}-\lambda_{r}),\xi_{p}=(\alpha_{1}+a-\lambda_{r})$,
$s$ and $\mu_{r}$ are defined as in (3.13).

Figure 3.1. Theorems 3.3 and 3.4 outperform numerical solvers. For
$\alpha_{1}=1,\alpha_{2}=2,\beta_{1}=3,\beta_{2}=4,\gamma_{1}=4,\gamma_{2}=5,a=9,b=10$
and $m=50$, the numerical solvers produce worse outputs due to floating point
arithmetic limitations. For the eigenvalue $\lambda_{2m-1}\approx 11.6217$,
the figure shows $A_{2m+1}v_{2m-1}^{(j)}/v_{2m-1}^{(j)}$ (where $v^{(j)}$ is
the $j$-th entry of the vector) for $v_{2m-1}$ computed with standard
numerical routines (dashed line) and computed with Theorem 3.3 (solid line).
The expected result is a constant line at $11.621$.

######  Remark 3.5.

When $a=0$, choosing $\xi_{p}=\xi_{q}=1$ in the above two theorems, the
results are reduced to the ones obtained in [13]. Thus, the method and
findings presented here are generalisations of [13]. We note that the authors
of [7] also proposed a way to derive the eigenvectors of some tridiagonal
$2$-Toeplitz matrices with perturbations on four corners, but their method
cannot be easily adapted to prove the condensation of eigenvectors of the
perturbed tridiagonal $K$-Toeplitz matrices arising in the non-Hermitian skin
effect.

######  Remark 3.6.

Note that the crucial idea in the proofs of Theorems 3.3 and 3.4 is that when
the $2$-Toeplitz structure appears in certain parts of the matrix, then we can
start to construct these parts of the eigenvector by the polynomials from the
recurrence formula in Definition 3.1 and ignore the irregular parts. This
strategy can be applied to construct the eigenvectors of $K$-Toeplitz matrix
with more complicated perturbations.

######  Remark 3.7.

By (3.11), in Theorems 3.3 and 3.4, the initial values of
$\widehat{p}_{k}^{(\xi_{p},\xi_{q})}(\mu_{r})$ and
$\widehat{q}_{k}^{(\xi_{p},\xi_{q})}(\mu_{r})$ are

$\begin{array}[]{ll}\widehat{p}_{0}^{(\xi_{p},\xi_{q})}(\mu_{r})=(\alpha_{1}-\lambda_{r}),&\widehat{p}_{1}^{(\xi_{p},\xi_{q})}(\mu_{r})=2\mu_{r}(\alpha_{1}-\lambda_{r})-\frac{a}{\beta},\\\
\widehat{q}_{0}^{(\xi_{p},\xi_{q})}(\mu_{r})=(\alpha_{1}+a-\lambda_{r}),&\widehat{q}_{1}^{(\xi_{p},\xi_{q})}(\mu_{r})=(2\mu_{r}+\beta)(\alpha_{1}-\lambda_{r})-\frac{a}{\beta}.\end{array}$
(3.17)

######  Remark 3.8.

Figure 3.1 shows that the formulas obtained in Theorems 3.3 and 3.4 outperform
the standard numerical routines. This is already noticeable in relatively
small matrices of size $101\times 101$ and is due to floating point arithmetic
limitations.

###  3.4. Other representations

To help to demonstrate the existence of interface modes later, we parameterise
$A_{2m+1}^{(a,b)}$ in (2.1) as
$A_{2m+1}^{(a,b)}(\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2})$.
Throughout the rest of the paper, if not specified, $A_{2m+1}^{(a,b)}$ is
$A_{2m+1}^{(a,b)}(\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2})$.

Note that

$\displaystyle
A_{2m+1}^{(b,a)}(\alpha_{1},\gamma_{2},\beta_{2},\alpha_{2},\gamma_{1},\beta_{1})=R_{2m+1}A_{2m+1}^{(a,b)}(\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2})R_{2m+1},$

where

$R_{2m+1}=\begin{pmatrix}0&0&0&\cdots&0&1\\\ 0&0&0&\cdots&1&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&1&\cdots&0&0\\\
0&1&0&\cdots&0&0\\\ 1&0&0&\cdots&0&0\end{pmatrix}.$ (3.18)

It follows that the eigenvalues $\\{\lambda_{r}\\}$ of
$A_{2m+1}^{(b,a)}(\alpha_{1},\gamma_{2},\beta_{2},\alpha_{2},\gamma_{1},\beta_{1})$
are the same as those of
$A_{2m+1}^{(a,b)}(\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2})$
and the eigenvectors of
$A_{2m+1}^{(b,a)}(\alpha_{1},\gamma_{2},\beta_{2},\alpha_{2},\gamma_{1},\beta_{1})$
are of the form

$\displaystyle\mathbf{x}=$
$\displaystyle\left(s^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\cdots,-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\left.\qquad
s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$
(3.19)

where $\xi_{q}=(\alpha_{1}-\lambda_{r}),\xi_{p}=(\alpha_{1}+a-\lambda_{r})$
and $s$, $\mu_{r}$ are defined as in (3.13).

##  4\. Exponential decay and localisation of eigenvectors

###  4.1. Exponential decay of the eigenvectors

In this section, we demonstrate the exponential decay for the entries of the
eigenvectors of matrices $A_{2m+1}^{(a,b)},A_{2m}^{(a,b)}$. To do so, by
Theorems 3.3 and 3.4, the only thing left is to control the polynomials
$\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$ and
$\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$. This requires
information on the eigenvalues and the Chebyshev polynomials in Section 2. The
main theorem is presented below.

######  Theorem 4.1.

Except for at most $11$ eigenvalues $\\{\lambda_{r}\\}$ of $A_{2m+1}^{(a,b)}$,
the corresponding eigenvectors $\bm{\mathbf{x}}$ in Theorem 3.3 satisfy that

$\left|{\bm{\mathbf{x}}^{(j)}}\right|\leq
Mj\left(\sqrt{\frac{\gamma_{1}\gamma_{2}}{\beta_{1}\beta_{2}}}\right)^{\lfloor\frac{j-1}{2}\rfloor}$
(4.1)

for some constant $M>0$ independent of the $\lambda_{r}$’s, where
$\bm{\mathbf{x}}^{(j)}$ is the $j$-th component of ${\bm{\mathbf{x}}}_{r}$.
The estimate (4.1) holds also for eigenvectors $\bm{\mathbf{x}}$ of
$A_{2m}^{(a,b)}$ associated with the eigenvalue $\lambda_{r}$ in Theorem 3.4,
except for at most $12$ $r$’s.

###### Proof.

By Theorems 2.2 and 2.3, all the $\lambda_{r}$’s of $A_{2m+1}^{(a,b)}$ and
$A_{2m}^{(a,b)}$ are real numbers, and except for at most $11$ $\lambda_{r}$’s
of $A_{2m+1}^{(a,b)}$, we have

$\displaystyle\mu_{r}=\frac{(\lambda_{r}-\alpha_{1})(\lambda_{r}-\alpha_{2})-\gamma_{1}\beta_{1}-\gamma_{2}\beta_{2}}{2\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}}=\cos\theta_{r}$
(4.2)

for certain $\theta_{r}\in[0,\pi]$. Now we are going to demonstrate that
$\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$ and
$\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$ are bounded for
$\mu_{r}=\cos\theta_{r}$. The idea is to represent them with Chebyshev
polynomials, although they are not Chebyshev polynomials straightforwardly. To
do so, we separate $\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(x\right)$ as
follows

$\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(x\right)=u_{k}(x)+v_{k-1}(x)$

with

$\begin{array}[]{ll}u_{0}(x)=(\alpha_{1}-\lambda_{r}),&\
u_{1}(x)=2x(\alpha_{1}-\lambda_{r}),\\\ v_{-1}(x)=0,&\
v_{0}(x)=-\frac{a}{\beta}.\end{array}$

Note that by (3.17), we have
$\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)=u_{0}(\mu_{r})+v_{-1}(\mu_{r}),\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)=u_{1}(\mu_{r})+v_{0}(\mu_{r})$.
By (3.9), it is not hard to check that

$\displaystyle u_{k+1}(x)=2xu_{k}(x)-u_{k-1}(x),$ $\displaystyle
v_{k+1}(x)=2xv_{k}(x)-v_{k-1}(x),$

and

$\begin{array}[]{ll}u_{0}(x)=(\alpha_{1}-\lambda_{r}),&\
u_{1}(x)=2x(\alpha_{1}-\lambda_{r}),\\\ v_{0}(x)=-\frac{a}{\beta},&\
v_{1}(x)=2x\left(-\frac{a}{\beta}\right).\end{array}$

Thus both the $u_{k}(x),v_{k}(x)$ are Chebyshev polynomials of the second kind
after scaling. By (2.4), we have for $-1\leq x\leq 1$ and $k=0,1,\cdots,$

$\left|{\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(x\right)}\right|\leq\left|{u_{k}(x)}\right|+\left|{v_{k-1}(x)}\right|\leq\left(k+1\right)\left|{\alpha_{1}-\lambda_{r}}\right|+k\left|{\frac{a}{\beta}}\right|.$

It is not hard to see that, for $\lambda_{r}$ satisfying (4.2),
$|\alpha_{1}-\lambda_{r}|$ and $\left|{\frac{a}{\beta}}\right|$ are uniformly
bounded. That is,

$\left|{\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(x\right)}\right|\leq
k\tilde{M},\quad x\in[-1,1],$

for some $\tilde{M}>0$.

To demonstrate the boundness of the quantities
$\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$ we separate them as

$\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(x\right)=u_{k}(x)+w_{k}(x)+v_{k-1}(x)$

with

$\displaystyle\begin{array}[]{ll}u_{0}(x)=(\alpha_{1}-a-\lambda_{r}),&u_{1}(x)=2x(\alpha_{1}-a-\lambda_{r}),\\\
w_{0}(x)=2a,&w_{1}(x)=x(2a),\\\
v_{-1}(x)=0,&v_{0}(x)=(\alpha_{1}-\lambda_{r})\beta+\frac{-a}{\beta}.\end{array}$

It is not hard to see that $u_{k}(x),v_{k}(x)$ are Chebyshev polynomials of
the second kind after scaling and $w_{k}(x)$ are Chebyshev polynomials of the
first kind after scaling. In the same fashion, by (2.4) we can show that
$\left|{\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)}\right|$ are
bounded by $k\tilde{M}$ for some $\tilde{M}$>0\. Considering the formula of
the eigenvector $\bm{\mathbf{x}}$ in Theorem 3.3, it is enough to demonstrate
(4.1). The result for $A_{2m}^{(a,b)}$ follows from Theorem 2.3. This
completes the proof. ∎

######  Remark 4.2.

It should be noted that the method and results presented in this section, as
well as the preceding ones, can be extended to tridiagonal $K$-Toeplitz
matrices with certain perturbations. This generalisation will be presented in
future work.

###  4.2. Localised eigenvectors

In this section, we present a theorem for demonstrating the localisation of
the coefficients of the eigenvectors of block matrices with mirrored
tridiagonal $2$-Toeplitz blocks. We define the matrix $C_{2m+1,2m+1}$ of order
$4m+2$ by

$\displaystyle C_{2m+1,2m+1}^{(a,b)}=\left(\begin{array}[]{ll}G_{11}&G_{12}\\\
G_{21}&G_{22}\end{array}\right),$ (4.3)

where $G_{11}=R_{2m+1}A_{2m+1}^{(0,a)}R_{2m+1}$ with $R_{k}$ being defined as
in (3.18), $G_{22}=A_{2m+1}^{(0,b)}$, and

$G_{12}=\begin{pmatrix}0&0&\cdots&0\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&0\\\ \gamma_{2}&0&\cdots&0\end{pmatrix},\quad
G_{21}=\begin{pmatrix}0&\cdots&0&\gamma_{2}\\\ \vdots&\vdots&\vdots&\vdots\\\
0&\cdots&0&0\\\ 0&\cdots&0&0\end{pmatrix}.$

Below is the main theorem for the interface modes. For the matrix $C_{2m,2m}$,
one can demonstrate similar results in the same manner.

######  Theorem 4.3.

Let $\\{\lambda_{r}\\}$ be the eigenvalues of $C_{2m+1,2m+1}$ in (4.3) and
define

$\displaystyle
s=\sqrt{\frac{\gamma_{1}\gamma_{2}}{\beta_{1}\beta_{2}}},\quad\mu_{r}=\frac{(\lambda_{r}-\alpha_{1})(\lambda_{r}-\alpha_{2})-(\gamma_{1}\beta_{1}+\gamma_{2}\beta_{2})}{2\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}}.$

The corresponding eigenvector of $C_{2m+1,2m+1}$ is given by

$\bm{\mathbf{x}}=\begin{pmatrix}\bm{\mathbf{x}}_{1}\\\ \bm{\mathbf{x}}_{2}\\\
\bm{\mathbf{x}}_{3}\end{pmatrix}.$ (4.4)

The $\bm{\mathbf{x}}_{1}$ part has the form

$\displaystyle\bm{\mathbf{x}}_{1}=$
$\displaystyle\left(\left(s\right)^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(s\right)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\cdots,-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\left.\qquad
s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

or

$\displaystyle\bm{\mathbf{x}}_{1}=$
$\displaystyle\left(-\frac{1}{\beta_{1}}\left(s\right)^{m-1}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\cdots,-\frac{1}{\beta_{1}}s\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\left.\qquad
s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where $\xi_{q}=C,\xi_{p}=C$ for a certain constant $C$.

The $\bm{\mathbf{x}}_{3}$ part has the form

$\displaystyle\bm{\mathbf{x}}_{3}=$
$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\qquad\left.\ldots,-\frac{1}{\beta_{1}}(s)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),(s)^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top}$

or

$\displaystyle\bm{\mathbf{x}}_{3}=$
$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{2}}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\qquad\left.-\frac{1}{\beta_{2}}s\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\ldots,-\frac{1}{\beta_{2}}(s)^{m-1}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top}$

with $\xi_{p}=C,\xi_{q}=C$ for a certain constant $C$.

Moreover, except for a few $\lambda_{r}$’s, in all the above cases, we have

$\left|{\widehat{q}_{k}^{(\xi_{p},\xi_{q})}(\mu_{r})}\right|\lesssim
k,\quad\left|{\widehat{p}_{k}^{(\xi_{p},\xi_{q})}(\mu_{r})}\right|\lesssim k.$
(4.5)

###### Proof.

The proof is divided into four steps.

Step 1. We consider

$\left(C_{2m+1,2m+1}-\lambda_{r}I\right)\bm{\mathbf{x}}=0.$ (4.6)

We separate $\bm{\mathbf{x}}$ as

$\bm{\mathbf{x}}=\begin{pmatrix}\bm{\mathbf{y}}_{1}\\\ \eta_{1}\\\ \eta_{2}\\\
\bm{\mathbf{y}}_{2}\end{pmatrix},$

where $\bm{\mathbf{y}}_{1}$ is of size $2m$, $\bm{\mathbf{y}}_{2}$ is of size
$2m$ and $\eta_{1}$ and $\eta_{2}$ are two constants. Considering the last
$2m+1$ rows in (4.6), we first have

$\left(\begin{array}[]{cccccc}\gamma_{2}&\alpha_{1}-\lambda_{r}&\beta_{1}&&&\\\
&\gamma_{1}&\alpha_{2}-\lambda_{r}&\ddots&&\\\ &&\ddots&\ddots&\ddots&\\\
&&&\gamma_{1}&\alpha_{2}-\lambda_{r}&\beta_{2}\\\
&&&&\gamma_{2}&\alpha_{1}-\lambda_{r}\\\
\end{array}\right)\begin{pmatrix}\eta_{1}\\\ \eta_{2}\\\
\bm{\mathbf{y}}_{2,1}\\\ \bm{\mathbf{y}}_{2,2}\\\ \vdots\\\
\bm{\mathbf{y}}_{2,2m}\end{pmatrix}=0.$ (4.7)

Considering the first row in the above equation, we obtain that

$\gamma_{2}\eta_{1}+(\alpha_{1}-\lambda_{r})\eta_{2}+\beta_{1}\bm{\mathbf{y}}_{2,1}=0.$

If $\eta_{2}\neq 0$, then this gives

$\left(\alpha_{1}+\frac{\gamma_{2}\eta_{1}}{\eta_{2}}-\lambda_{r}\right)\eta_{2}+\beta_{1}\bm{\mathbf{y}}_{2,1}=0.$

Therefore, it follows that

$\left(\begin{array}[]{cccccc}\alpha_{1}+\frac{\gamma_{2}\eta_{1}}{\eta_{2}}-\lambda_{r}&\beta_{1}&&&\\\
\gamma_{1}&\alpha_{2}-\lambda_{r}&\ddots&&\\\ &\ddots&\ddots&\ddots&\\\
&&\gamma_{1}&\alpha_{2}-\lambda_{r}&\beta_{2}\\\
&&&\gamma_{2}&\alpha_{1}-\lambda_{r}\\\
\end{array}\right)\begin{pmatrix}\eta_{2}\\\ \bm{\mathbf{y}}_{2,1}\\\
\bm{\mathbf{y}}_{2,2}\\\ \vdots\\\ \bm{\mathbf{y}}_{2,2m}\end{pmatrix}=0.$

By Theorem 3.3, we thus have

$\begin{aligned} &\left(\eta_{2},\ \bm{\mathbf{y}}_{2,1},\
\bm{\mathbf{y}}_{2,2},\ \cdots,\ \bm{\mathbf{y}}_{2,2n}\right)^{\top}\\\
=&\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.\\\
&\qquad\left.\ldots,-\frac{1}{\beta_{1}}(s)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),(s)^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},\end{aligned}.$

where
$\xi_{q}=(\alpha_{1}-\lambda_{r})C,\xi_{p}=\left(\alpha_{1}+\frac{\gamma_{2}\eta_{1}}{\eta_{2}}-\lambda_{r}\right)C$
for a certain constant $C$. Moreover, $\lambda_{r}$ should be an eigenvalue of
the above matrix. For the case when $\eta_{2}=0$, considering the second rows
to the last rows of (4.7), we get

$\left(\begin{array}[]{cccccc}\alpha_{2}-\lambda_{r}&\beta_{2}&&&\\\
\gamma_{2}&\alpha_{1}-\lambda_{r}&\beta_{1}&&\\\ &\ddots&\ddots&\ddots&\\\
&&\gamma_{1}&\alpha_{2}-\lambda_{r}&\beta_{2}\\\
&&&\gamma_{2}&\alpha_{1}-\lambda_{r}\\\
\end{array}\right)\begin{pmatrix}\bm{\mathbf{y}}_{2,1}\\\ \vdots\\\
\bm{\mathbf{y}}_{2,2m}\end{pmatrix}=0.$

By Theorem 3.4, we have

$\displaystyle\left(\bm{\mathbf{y}}_{2,1},\cdots,\bm{\mathbf{y}}_{2,2n}\right)^{\top}$
$\displaystyle=$
$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{2}}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{2}}s\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\qquad\left.\ldots,-\frac{1}{\beta_{2}}(s)^{m-1}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where $\xi_{p}=C,\xi_{q}=C$ for some constant $C$. Moreover, $\lambda_{r}$
should be an eigenvalue of the matrix above.

Step 2. Considering the first $2m+1$ rows in (4.6), we have

$\left(\begin{array}[]{cccccc}\alpha_{1}-\lambda_{r}&\gamma_{2}&&&&\\\
\beta_{2}&\alpha_{2}-\lambda_{r}&\ddots&&&\\\ &\ddots&\ddots&\ddots&&\\\
&&\beta_{2}&\alpha_{2}-\lambda_{r}&\gamma_{1}&\\\
&&&\beta_{1}&\alpha_{1}-\lambda_{r}&\gamma_{2}\\\
\end{array}\right)\begin{pmatrix}\bm{\mathbf{y}}_{1,1}\\\
\bm{\mathbf{y}}_{1,2}\\\ \vdots\\\ \bm{\mathbf{y}}_{1,2m}\\\ \eta_{1}\\\
\eta_{2}\end{pmatrix}=0.$ (4.8)

Considering the last row in the above equation, we have

$\gamma_{2}\eta_{2}+(\alpha_{1}-\lambda_{r})\eta_{1}+\beta_{1}\bm{\mathbf{y}}_{1,2m}=0.$

If $\eta_{1}\neq 0$, this gives

$\left(\alpha_{1}+\frac{\gamma_{2}\eta_{2}}{\eta_{1}}-\lambda_{r}\right)\eta_{1}+\beta_{1}\bm{\mathbf{y}}_{1,2m}=0,$

and therefore,

$\left(\begin{array}[]{ccccc}\alpha_{1}-\lambda_{r}&\gamma_{2}&&&\\\
\beta_{2}&\alpha_{2}-\lambda_{r}&\ddots&&\\\ &\ddots&\ddots&\ddots&\\\
&&\beta_{2}&\alpha_{2}-\lambda_{r}&\gamma_{1}\\\
&&&\beta_{1}&\alpha_{1}+\frac{\gamma_{2}\eta_{2}}{\eta_{1}}-\lambda_{r}\\\
\end{array}\right)\begin{pmatrix}\bm{\mathbf{y}}_{1,1}\\\
\bm{\mathbf{y}}_{1,2}\\\ \vdots\\\ \bm{\mathbf{y}}_{1,2m}\\\
\eta_{1}\end{pmatrix}=0.$

Note that the above equation corresponds to
$A_{2m+1}^{(0,\frac{\gamma_{2}\eta_{2}}{\eta_{1}})}(\alpha_{1},\gamma_{2},\beta_{2},\alpha_{2},\gamma_{1},\beta_{1})$,
which is

$R_{2m+1}A_{2m+1}^{(\frac{\gamma_{2}\eta_{2}}{\eta_{1}},0)}(\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2})R_{2m+1}.$

By Theorem 3.3, the eigenvector of
$A_{2m+1}^{(\frac{\gamma_{2}\eta_{2}}{\eta_{1}},0)}(\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2})$
has the following form:

$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\ldots,\right.$
$\displaystyle\qquad\left.-\frac{1}{\beta_{1}}\left(s\right)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\left(s\right)^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where
$\xi_{q}=(\alpha_{1}-\lambda_{r})C,\xi_{p}=\left(\alpha_{1}+\frac{\gamma_{2}\eta_{2}}{\eta_{1}}-\lambda_{r}\right)C$
for a constant $C$. Therefore, by (3.19), we have

$\displaystyle\left(\bm{\mathbf{y}}_{1,1},\bm{\mathbf{y}}_{1,2},\cdots,\bm{\mathbf{y}}_{1,2m},\eta_{1}\right)^{\top}$
$\displaystyle=$
$\displaystyle\left(\left(s\right)^{m}\widehat{q}_{m}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(s\right)^{m-1}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\cdots,-\frac{1}{\beta_{1}}s\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\left.\qquad
s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{1}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where
$\xi_{q}=(\alpha_{1}-\lambda_{r})C,\xi_{p}=\left(\alpha_{1}+\frac{\gamma_{2}\eta_{2}}{\eta_{1}}-\lambda_{r}\right)C$
for a certain constant $C$. Moreover, $\lambda_{r}$ should be an eigenvalue of
the above matrix.

For the case when $\eta_{1}=0$, considering the first row to the $2m$ rows of
(4.8), we get

$\left(\begin{array}[]{cccccc}\alpha_{1}-\lambda_{r}&\gamma_{2}&&\\\
\beta_{2}&\alpha_{2}-\lambda_{r}&\gamma_{1}&\\\ &\ddots&\ddots&\ddots\\\
&&\beta_{1}&\alpha_{2}-\lambda_{r}\\\
\end{array}\right)\begin{pmatrix}\bm{\mathbf{y}}_{1,1}\\\ \vdots\\\
\bm{\mathbf{y}}_{1,2m}\end{pmatrix}=0.$

The above equation corresponds to
$A_{2m}^{(0,0)}(\alpha_{1},\gamma_{2},\beta_{2},\alpha_{2},\gamma_{1},\beta_{1})$
which is

$R_{2m}A_{2m}^{(0,0)}(\alpha_{2},\beta_{1},\gamma_{1},\alpha_{1},\beta_{2},\gamma_{2})R_{2m}.$

By Theorem 3.4, the eigenvector of
$A_{2m}^{(0,0)}(\alpha_{2},\beta_{1},\gamma_{1},\alpha_{1},\beta_{2},\gamma_{2})$
has the following form:

$\displaystyle\left(\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}s\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\qquad\ldots,\left.-\frac{1}{\beta_{1}}\left(s\right)^{m-1}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where $\xi_{q}=C,\xi_{p}=C$ for a certain constant $C$. Therefore, analogously
to (3.19), we have

$\displaystyle\left(\bm{\mathbf{y}}_{1,1},\bm{\mathbf{y}}_{1,2},\cdots,\bm{\mathbf{y}}_{1,2m}\right)^{\top}$
$\displaystyle=$
$\displaystyle\left(-\frac{1}{\beta_{1}}\left(s\right)^{m-1}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{m-1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\cdots,-\frac{1}{\beta_{1}}s\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\right.$
$\displaystyle\left.\qquad
s\widehat{q}_{1}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),-\frac{1}{\beta_{1}}\left(\alpha_{2}-\lambda_{r}\right)\widehat{p}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right),\widehat{q}_{0}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)\right)^{\top},$

where $\xi_{q}=C,\xi_{p}=C$ for a certain constant $C$. Moreover,
$\lambda_{r}$ should be an eigenvalue of the above matrix.

Step 3. The rest of the proof is to control $\mu_{r}$ in order to control the
quantities $\widehat{q}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$,
$\widehat{p}_{k}^{(\xi_{p},\xi_{q})}\left(\mu_{r}\right)$. Although by the
above discussions, each $\lambda_{r}$ should be an eigenvalue of a tridiagonal
$2$-Toeplitz matrix with perturbations on the diagonal corners, we cannot
directly apply Theorems 2.2 and 2.3 to control the eigenvalues and the values
of the polynomials, as the perturbations on the corners vary for different
$\lambda_{r}$. Note that we only need to prove, except for a few
$\lambda_{r}$’s, that

$\mu_{r}=\cos\theta_{r},\quad\theta_{r}\in[0,\pi].$ (4.9)

The idea is to first consider the principal submatrix of $C_{2m+1,2m+1}$, that
is,

$D_{2m,2m}=\left(\begin{array}[]{ll}{\widehat{G}}_{11}&{\widehat{G}}_{12}\\\
{\widehat{G}}_{21}&{\widehat{G}}_{22}\end{array}\right),$

where ${\widehat{G}}_{11}=R_{2m}A_{2m}^{(0,0)}R_{2m}$,
${\widehat{G}}_{22}=A_{2m}^{(0,0)}$, and

${\widehat{G}}_{12}=\begin{pmatrix}0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&0\\\
\gamma_{2}&0&\cdots&0\end{pmatrix},\quad{\widehat{G}}_{21}=\begin{pmatrix}0&\cdots&0&\gamma_{2}\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&0&0\\\ 0&\cdots&0&0\end{pmatrix},$

and then to analyse its eigenvalues. In particular, expanding the determinant
$\left|{xI-D_{2m,2m}}\right|$ of $xI-D_{2m,2m}$ by the Laplace method of
expansion of determinants, we have that

$\displaystyle\left|{xI-A_{2m}^{(0,0)}}\right|\left|{xI-
A_{2m}^{(0,0)}}\right|$ $\displaystyle\quad-\gamma_{2}^{2}\left|{xI-
A_{2m-1}^{(0,0)}(\alpha_{2},\gamma_{1},\beta_{1},\alpha_{1},\gamma_{2},\beta_{2})}\right|\left|{xI-
A_{2m-1}^{(0,0)}(\alpha_{2},\gamma_{1},\beta_{1},\alpha_{1},\gamma_{2},\beta_{2})}\right|.$

By (2.6) and (2.7), the above determinant can be written as

$\displaystyle\left(P_{m}^{*}\left(\pi_{2}(x)\right)+\gamma_{2}\beta_{2}P_{m-1}^{*}\left(\pi_{2}(x)\right)\right)^{2}-\gamma_{2}^{2}\left(x-\alpha_{2}\right)^{2}P_{m-1}^{*}\left(\pi_{2}(x)\right)^{2},$

where $P_{k}^{*}$ is defined by (2.5). In particular, to find the roots of the
above polynomial, we only need to solve the equation

$\displaystyle\left(\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}U_{m}\left(y(x)\right)+\gamma_{2}\beta_{2}U_{m-1}\left(y(x)\right)\right)^{2}-\gamma_{2}^{2}\left(x-\alpha_{2}\right)^{2}U_{m-1}\left(y(x)\right)^{2}=0,$

where $y(x)$ is defined by (2.11). It is clear that $x$ such that
$U_{m-1}(y(x))=0$ is not the solution to the above equation. We consider
solving

$\displaystyle\left(\frac{U_{m}\left(y(x)\right)}{U_{m-1}(y(x))}+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}\right)^{2}=\frac{\gamma_{2}^{2}\left(x-\alpha_{2}\right)^{2}}{\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}}},$
(4.10)

which gives

$\frac{U_{m}\left(y(x)\right)}{U_{m-1}(y(x))}+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}=+\frac{\gamma_{2}\left(x-\alpha_{2}\right)}{\left(\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}\right)^{\frac{1}{4}}}$
(4.11)

or

$\frac{U_{m}\left(y(x)\right)}{U_{m-1}(y(x))}+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}=-\frac{\gamma_{2}\left(x-\alpha_{2}\right)}{\left(\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}\right)^{\frac{1}{4}}}.$
(4.12)

In the following, we analyse the solutions to (4.11). By (2.11), we can write

$x=g^{+}(y):=\frac{(\alpha_{1}+\alpha_{2})+\sqrt{(\alpha_{1}+\alpha_{2})^{2}+4((\gamma_{1}\beta_{1}+\gamma_{2}\beta_{2}-\alpha_{1}\alpha_{2})+2y\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}})}}{2}$

and

$x=g^{-}(y):=\frac{(\alpha_{1}+\alpha_{2})-\sqrt{(\alpha_{1}+\alpha_{2})^{2}+4((\gamma_{1}\beta_{1}+\gamma_{2}\beta_{2}-\alpha_{1}\alpha_{2})+2y\sqrt{\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}})}}{2}.$

Now, (4.11) becomes

$\left(\frac{U_{m}\left(y\right)}{U_{m-1}(y)}+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}\right)=\frac{\gamma_{2}\left(g^{+}(y)-\alpha_{2}\right)}{\left(\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}\right)^{\frac{1}{4}}}$
(4.13)

and

$\left(\frac{U_{m}\left(y\right)}{U_{m-1}(y)}+\sqrt{\frac{\gamma_{2}\beta_{2}}{\gamma_{1}\beta_{1}}}\right)=\frac{\gamma_{2}\left(g^{-}(y)-\alpha_{2}\right)}{\left(\gamma_{1}\beta_{1}\gamma_{2}\beta_{2}\right)^{\frac{1}{4}}}.$
(4.14)

We also suppose that $m$ is odd. The case when $m$ is even can be handled in a
similar way, as discussed below. Note that the roots of $U_{m-1}(y)$ are
$\cos\frac{k\pi}{m},\ k=1,2,\cdots,m-1$ and

$\begin{array}[]{ll}U_{m-1}(y)>0,&\text{for
$y\in\left(\cos\left(\frac{(2k+1)\pi}{m}\right),\
\cos\left(\frac{2k\pi}{m}\right)\right),\ k=0,\cdots,\frac{m-1}{2},$}\\\
U_{m-1}(y)<0,&\text{for $y\in\left(\cos\left(\frac{(2k+2)\pi}{m}\right),\
\cos\left(\frac{(2k+1)\pi}{m}\right)\right),\
k=0,\cdots,\frac{m-3}{2}.$}\end{array}$

The roots of $U_{m}(y)$ are $\cos\frac{k\pi}{m+1},\ k=1,2,\cdots,m$ and

$\begin{array}[]{ll}U_{m}(y)>0,&\text{for
$y\in\left(\cos\left(\frac{(2k+1)\pi}{m+1}\right),\
\cos\left(\frac{2k\pi}{m+1}\right)\right),\ k=0,\cdots,\frac{m-1}{2},$}\\\
U_{m}(y)<0,&\text{for $y\in\left(\cos\left(\frac{(2k+2)\pi}{m+1}\right),\
\cos\left(\frac{(2k+1)\pi}{m+1}\right)\right),\
k=0,\cdots,\frac{m-1}{2}.$}\end{array}$

Note also that

$\frac{k}{m+1}<\frac{k}{m}<\frac{k+1}{m+1},\quad k=1,\cdots,m-1,$

and

$\cos\left(\frac{(k+1)\pi}{m+1}\right)<\cos\left(\frac{k\pi}{m}\right)<\cos\left(\frac{k\pi}{m+1}\right),\quad
k=1,\cdots,m-1.$

Thus, we have

$\displaystyle\lim_{y\rightarrow\cos\left(\frac{(m-1)\pi}{m}\right)^{-}}\frac{U_{m}(y)}{U_{m-1}(y)}=-\infty,$
$\displaystyle\lim_{y\rightarrow\cos\left(\frac{(m-1)\pi}{m}\right)^{+}}\frac{U_{m}(y)}{U_{m-1}(y)}=+\infty,$
$\displaystyle\lim_{y\rightarrow\cos\left(\frac{(m-2)\pi}{m}\right)^{-}}\frac{U_{m}(y)}{U_{m-1}(y)}=-\infty,$
$\displaystyle\lim_{y\rightarrow\cos\left(\frac{(m-2)\pi}{m}\right)^{+}}\frac{U_{m}(y)}{U_{m-1}(y)}=+\infty,$
$\displaystyle\qquad\vdots$
$\displaystyle\lim_{y\rightarrow\cos\left(\frac{\pi}{m}\right)^{-}}\frac{U_{m}(y)}{U_{m-1}(y)}=-\infty,$
$\displaystyle\lim_{y\rightarrow\cos\left(\frac{\pi}{m}\right)^{+}}\frac{U_{m}(y)}{U_{m-1}(y)}=+\infty.$

Therefore, (4.13) must have at least one solution in each
$\left(\cos\left(\frac{(k+1)\pi}{m}\right),\cos\left(\frac{k\pi}{m}\right)\right),\
k=1,\cdots,m-2$ and so does (4.14). For $m$ even, we can prove the desired
result in the same way. Thus we can assert that there are at least $2m-4$
solutions $\widehat{\lambda}_{r}$ of (4.11) satisfying
$y(\widehat{\lambda}_{r})=\cos\theta_{r}$. In the same manner, we can prove
there are at least $2m-4$ solutions $\widehat{\lambda}_{r}$ of (4.12)
satisfying that $y(\widehat{\lambda}_{r})=\cos\theta_{r}$. Therefore, for at
least $4m-8$ eigenvalues of $D_{2m,2m}$, we have

$\left|{y(\widehat{\lambda}_{j})}\right|\leq 1,\quad j=1,\cdots,4m-8$

with $y(x)$ being defined by (2.11).

Step 4. Furthermore, by the Cauchy interlacing theorem, we can show that,
except for a few $\lambda_{r}$’s of $C_{2m+1,2m+1}$, we have
$\mu_{r}=y(\lambda_{r})=\cos\theta_{r}$. Then by the same arguments as those
in the proof of Theorem 4.1, we can show that, for except a few
$\lambda_{r}$’s, we have $\left|{\widehat{q}_{k}(\mu_{r})}\right|\lesssim
k,\left|{\widehat{p}_{k}(\mu_{r})}\right|\lesssim k$. This completes the
proof. ∎

##  5\. Non-Hermitian skin effect and localised interface modes

Recently, a mathematical model of the skin effect in non-Hermitian chains of
subwavelength resonators with an imaginary gauge potential has been developed
[1]. The authors were able to develop a rigorous theory for systems composed
by equally spaced identical subwavelength resonators using the simple
structure of the gauge capacitance matrix associated with such structures and
the rich literature on (tridiagonal) Toeplitz matrices and perturbations
thereof.

In this section, we briefly recall the setup and established new results on
dimer systems showing that the theory developed in the previous sections can
be applied to study them. In particular, we prove condensation of the
eigenmodes at one edge of the structure, find the topological origin of this
phenomena and show that an interface formed by adjoining two half-structures
with opposite signs of complex gauge potentials leads to wave localisation
along the interface.

###  5.1. Problem formulation

We consider a one-dimensional chain of $N$ disjoint subwavelength resonators
$D_{i}\coloneqq(x_{i}^{\operatorname{{\mathsf{L}}}},x_{i}^{\operatorname{{\mathsf{R}}}})\subset\mathbb{R}$,
where $(x_{i}^{\operatorname{{\mathsf{L},\mathsf{R}}}})_{1\leq i\leq
N}\subset\mathbb{R}$ are the $2N$ extremities satisfying
$x_{i}^{\operatorname{{\mathsf{L}}}}<x_{i}^{\operatorname{{\mathsf{R}}}}<x_{i+1}^{\operatorname{{\mathsf{L}}}}$
for any $1\leq i\leq N$. We fix the coordinates such that
$x_{1}^{\operatorname{{\mathsf{L}}}}=0$. We also denote by
$\ell_{i}=x_{i}^{\operatorname{{\mathsf{R}}}}-x_{i}^{\operatorname{{\mathsf{L}}}}$
the length of the $i$-th resonators, and by
$s_{i}=x_{i+1}^{\operatorname{{\mathsf{L}}}}-x_{i}^{\operatorname{{\mathsf{R}}}}$
the spacing between the $i$-th and $(i+1)$-th resonators. The system is
illustrated in Figure 5.1.

$x_{1}^{\operatorname{{\mathsf{L}}}}$$x_{1}^{\operatorname{{\mathsf{R}}}}$$\ell_{1}$$x_{2}^{\operatorname{{\mathsf{L}}}}$$x_{2}^{\operatorname{{\mathsf{R}}}}$$\ell_{2}$$x_{3}^{\operatorname{{\mathsf{L}}}}$$x_{3}^{\operatorname{{\mathsf{R}}}}$$\ell_{3}$…$x_{4}^{\operatorname{{\mathsf{L}}}}$$x_{4}^{\operatorname{{\mathsf{R}}}}$$\ell_{4}$$x_{N-3}^{\operatorname{{\mathsf{L}}}}$$x_{N-3}^{\operatorname{{\mathsf{R}}}}$$\ell_{N-3}$$x_{N-2}^{\operatorname{{\mathsf{L}}}}$$x_{N-2}^{\operatorname{{\mathsf{R}}}}$$\ell_{N-2}$$x_{N-1}^{\operatorname{{\mathsf{L}}}}$$x_{N-1}^{\operatorname{{\mathsf{R}}}}$$\ell_{N-1}$$x_{N}^{\operatorname{{\mathsf{L}}}}$$x_{N}^{\operatorname{{\mathsf{R}}}}$$\ell_{N}$$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{N-2}$$s_{N-1}$$s_{N}$

Figure 5.1. A chain of $N$ subwavelength resonators, with lengths
$(\ell_{i})_{1\leq i\leq N}$ and spacings $(s_{i})_{1\leq i\leq N-1}$.

We will use the notation $\displaystyle
D\coloneqq\bigcup_{i=1}^{N}(x_{i}^{\operatorname{{\mathsf{L}}}},x_{i}^{\operatorname{{\mathsf{R}}}})\subset\mathbb{R}$
to symbolise the set of subwavelength resonators and denote for a function $w$

$w|_{\operatorname{{\mathsf{R}}}}(x)=\lim_{s\rightarrow 0^{+}}w(x+s),\quad
w|_{\operatorname{{\mathsf{L}}}}(x)=\lim_{s\rightarrow 0^{+}}w(x-s).$

We consider the following system of ordinary differential equations:

$\displaystyle\begin{dcases}u^{\prime\prime}(x)+\gamma
u^{\prime}(x)+\frac{\omega^{2}}{v_{b}^{2}}u=0,&x\in D,\\\
u^{\prime\prime}(x)+\frac{\omega^{2}}{v^{2}}u=0,&x\in\mathbb{R}\setminus D,\\\
u|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{L},\mathsf{R}}}}_{i})-u|_{\operatorname{{\mathsf{L}}}}(x^{\operatorname{{\mathsf{L},\mathsf{R}}}}_{i})=0,&\text{for
all }1\leq i\leq N,\\\
\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{L}}}}_{{i}})=\delta\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{L}}}}(x^{\operatorname{{\mathsf{L}}}}_{{i}}),&\text{for
all }1\leq i\leq N,\\\
\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{R}}}}_{{i}})=\delta\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{L}}}}_{{i}}),&\text{for
all }1\leq i\leq N,\\\
\frac{\operatorname{d\\!}u}{\operatorname{d\\!}|x|}-\mathbf{i}\frac{\omega}{v}u=0,&x\in(-\infty,x_{1}^{\operatorname{{\mathsf{L}}}})\cup(x_{N}^{\operatorname{{\mathsf{R}}}},\infty).\end{dcases}$
(5.1)

Here, the parameter $\gamma\in\mathbb{R},\gamma\neq 0,$ models the complex
gauge potential, $0<\delta\ll 1$ is a small contrast material parameter, the
positive constants $v$ and $v_{b}$ are respectively the wave speeds outside
and inside the resonators and $\omega$ denotes the frequency. We refer the
reader to [1, Section 2] for the physical motivation of this model.

We are interested in the resonances $\omega\in\mathbb{C}$ such that (5.1) has
a non-trivial solution $u$. We look for the modes within the subwavelength
regime, which we characterise by imposing $\omega\to 0$ as $\delta\to 0$. This
regime will recover subwavelength resonances, while keeping the size of the
resonators fixed.

A central result of [1] is the approximation of the eigenfrequencies and
eigenmodes of (5.1) in the subwavelength regime with a finite dimensional
eigenvalue problem as we recall in 5.1. The matrix
$\mathcal{C}^{\gamma}=\bigr{(}\mathcal{C}_{i,j}^{\gamma}\bigr{)}_{i,j}$
involved in such finite-dimensional approximation of (5.1) is the so-called
_gauge capacitance matrix_ and is given by

$\displaystyle\mathcal{C}_{i,j}^{\gamma}\coloneqq\begin{dcases}\frac{\gamma}{s_{1}}\frac{\ell_{1}}{1-e^{-\gamma\ell_{1}}},&i=j=1,\\\
\frac{\gamma}{s_{i}}\frac{\ell_{i}}{1-e^{-\gamma\ell_{i}}}-\frac{\gamma}{s_{i-1}}\frac{\ell_{i}}{1-e^{\gamma\ell_{i}}},&1<i=j<N,\\\
-\frac{\gamma}{s_{i}}\frac{\ell_{i}}{1-e^{-\gamma\ell_{j}}},&1\leq i=j-1\leq
N-1,\\\ \frac{\gamma}{s_{j}}\frac{\ell_{i}}{1-e^{\gamma\ell_{j}}},&2\leq
i=j+1\leq N,\\\
-\frac{\gamma}{s_{N-1}}\frac{\ell_{N}}{1-e^{\gamma\ell_{N}}},&i=j=N,\\\
\end{dcases}$ (5.2)

while all the other entries are zero. The following result is from [1,
Corollary 2.6].

######  Proposition 5.1 (Discrete approximations of the eigenfrequencies and
eigenmodes).

The $N$ subwavelength eigenfrequencies $\omega_{i}$ satisfy, as $\delta\to 0$,

$\displaystyle\omega_{i}=v_{b}\sqrt{\delta\lambda_{i}}+\mathcal{O}(\delta),$

where $(\lambda_{i})_{1\leq i\leq N}$ are the eigenvalues of the eigenvalue
problem

$\mathcal{C}^{\gamma}\bm{\mathbf{a}}_{i}=\lambda_{i}V\bm{\mathbf{a}}_{i},\qquad
1\leq i\leq N,$ (5.3)

where $V=\operatorname{diag}(\ell_{1},\dots,\ell_{N})$. Furthermore, let
$u_{i}$ be a subwavelength eigenmode corresponding to $\omega_{i}$ and let
$\bm{\mathbf{a}}_{i}$ be the corresponding eigenvector of
$\mathcal{C}^{\gamma}$. Then

$\displaystyle
u_{i}(x)=\sum_{j}\bm{\mathbf{a}}_{i}^{(j)}V_{j}(x)+\mathcal{O}(\delta),$

where $V_{j}$ are the solutions of

$\displaystyle\begin{dcases}-\frac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}x^{2}}V_{j}=0,&x\in\mathbb{R}\setminus
D,\\\
V_{i}(x)=\delta_{ij},&x\in(x_{j}^{\operatorname{{\mathsf{L}}}},x_{j}^{\operatorname{{\mathsf{R}}}}),\\\
V_{i}(x)=\mathcal{O}(1)&\mbox{as }|x|\to\infty,\end{dcases}$ (5.4)

with $\delta_{ij}$ being the Kronecker symbol and $\bm{\mathbf{a}}^{(j)}$
denotes the $j$-th entry of the eigenvector.

######  Remark 5.2.

Since $V_{j}$ is piecewise linear, supported in
$(x_{j-1}^{\operatorname{{\mathsf{R}}}},x_{j+1}^{\operatorname{{\mathsf{L}}}})$
and $V_{j}(x)=1$ for
$x\in(x_{j}^{\operatorname{{\mathsf{L}}}},x_{j}^{\operatorname{{\mathsf{R}}}})$
the overall behaviour of the eigenmodes $u_{i}$ is captured by the
eigenvectors $\bm{\mathbf{a}}_{i}$.

###  5.2. Non-Hermitian skin effect in dimer systems and condensation of the
system’s eigenmodes

For a system of dimers — that is $s_{i}=s_{i+2}$ for all $1\leq i\leq N-3$ and
$\ell_{i}=\ell$ for all $1\leq i\leq N$ — the gauge capacitance matrix from
(5.2) takes the form

$\displaystyle\mathcal{C}^{\gamma}=\begin{pmatrix}\tilde{\alpha}_{1}&\beta_{1}&&&&&\\\
\eta_{1}&\alpha_{2}&\beta_{2}&&&&\\\ &\eta_{2}&\alpha_{1}&\beta_{1}&&&\\\
&&\eta_{1}&\alpha_{2}&\ddots&&\\\ &&&\ddots&\ddots&\ddots&\\\
&&&&\eta_{2}&\alpha_{1}&\beta_{1}\\\ &&&&&\eta_{1}&\tilde{\alpha}_{2}\\\
\end{pmatrix},$ (5.5)

where

$\displaystyle\alpha_{1}$
$\displaystyle=\frac{\gamma}{s_{1}}\frac{\ell}{1-e^{-\gamma\ell}}-\frac{\gamma}{s_{2}}\frac{\ell}{1-e^{\gamma\ell}},$
$\displaystyle\alpha_{2}$
$\displaystyle=\frac{\gamma}{s_{2}}\frac{\ell}{1-e^{-\gamma\ell}}-\frac{\gamma}{s_{1}}\frac{\ell}{1-e^{\gamma\ell}},$
$\displaystyle\tilde{\alpha}_{1}$
$\displaystyle=\frac{\gamma}{s_{1}}\frac{\ell}{1-e^{-\gamma\ell}},$
$\displaystyle\tilde{\alpha}_{2}$ $\displaystyle=-\tilde{\alpha}_{1},$
$\displaystyle\beta_{1}$
$\displaystyle=-\frac{\gamma}{s_{1}}\frac{\ell}{1-e^{-\gamma\ell}},$
$\displaystyle\beta_{2}$
$\displaystyle=-\frac{\gamma}{s_{2}}\frac{\ell}{1-e^{-\gamma\ell}},$
$\displaystyle\eta_{1}$
$\displaystyle=\frac{\gamma}{s_{1}}\frac{\ell}{1-e^{\gamma\ell}},$
$\displaystyle\eta_{2}$
$\displaystyle=\frac{\gamma}{s_{2}}\frac{\ell}{1-e^{\gamma\ell}}.$

We use $\eta_{i}$ instead of $\gamma_{i}$ (as done in the previous sections)
to denote the coefficients on the lower diagonal in order to prevent
confusion. One may remark that all rows of $\mathcal{C}^{\gamma}$ sum to 0 and
thus $\bm{1}\in\ker(\mathcal{C}^{\gamma})$. The next theorem states that
$\bm{1}$ is the only eigenvector of $\mathcal{C}^{\gamma}$ that is not
localised.

######  Theorem 5.3.

All but a few (independent of $N$) eigenvectors of the gauge capacitance
matrix $\mathcal{C}^{\gamma}$ satisfy the following inequality:

$\left|{\bm{\mathbf{x}}^{(j)}}\right|\leq
Mje^{-\ell\gamma\lfloor\frac{j-1}{2}\rfloor},$ (5.6)

where $\bm{\mathbf{x}}^{(j)}$ is the $j$-th component of the eigenvector
$\bm{\mathbf{x}}$.

Theorem 5.3 is a direct consequence of Theorem 3.4.

Numerically, we can verify that there is exactly one eigenvector which is not
localised. More precisely, we show in Figure 2(a) that only the eigenvalue
$\lambda_{1}=0$ satisfies $|y(\lambda_{i})|\geq 1$ for $y$ as in (2.11). In
Figure 2(b), we show the eigenmodes of a system of $25$ dimers.

(a) The black dots show the value $y(\lambda_{i})$ for the eigenvalues
$\lambda_{i}$ of $\mathcal{C}^{\gamma}$. The red line show the boundaries
stability zone $y=\pm 1$. Only for $\lambda=0$, $y(\lambda)$ lays outside of
this zone while Theorem 3.4 predicts that at most $10$ do.

(b) Eigenmodes superimposed on one another to portray the skin effect: all but
one modes are exponentially localised on the left edge of the system.

Figure 5.2. Eigenmode localisation for a system of $25$ dimers ($N=50$),
$\ell_{i}=1$, $s_{1}=1$, $s_{2}=2$ and $\gamma=1$.

It is interesting to remark that the eigenvalue $0$ is both the only outlier
of Figure 2(a) and the only point laying on the trace of the eigenvalues of
the symbol of the 2-Toeplitz operator in Figures 3(b) and 3(c).

###  5.3. Topological origin of the non-Hermitian skin effect in dimer systems

In the case of a perturbed Toeplitz matrix — as for equally spaced and
identical resonators — it is well known that the exponential decay of the
eigenvectors is due to the winding of the symbol of the corresponding Toeplitz
operator (or equivalently, its Fredholm index) [1, 20]. To the best of our
knowledge, no such result is known for $K$-Toeplitz matrices when $K\geq 2$.
However, it is known (see, for instance, [8]) that the Fredholm index of the
associated operator is given by the winding of the determinant of its symbol,
which in this case is a $K\times K$ matrix, provided that the determinant does
not vanish at any point on the unit circle. In Figure 3(a), we show the
spectrum and pseudospectrum of the gauge capacitance matrix of a system of
$25$ dimers together with the trace of the determinant of the symbol of the
associated $2$-Toeplitz operator in the complex plane. We observe that various
eigenvalues whose eigenvectors are localised lay in a region without winding.
This is due to the fact that the determinant of the symbol takes value zero at
some point on the unit circle.

In Figures 3(b) and 3(c), we consider systems of $25$ and $50$ dimers,
respectively. This time, the colored trace shows the winding of the two
eigenvalues of the symbol of the corresponding $2$-Toeplitz operator. This
winding predicts accurately the exponential decay of the eigenmodes and is the
limit of the pseudospectrum as $N\to\infty$ as in the simplest case studied in
[1].

(a) The winding of the determinant does not predict the exponential decay
correctly. Simulation performed with a system of $25$ dimers.

(b) The winding of the eigenvalues does predict the exponential decay
correctly. Simulation performed with a system of $25$ dimers.

(c) The region of non-trivial winding of the eigenvalues is the limit, as
$N\to\infty$, of the $\varepsilon$-pseudospectrum for any fixed $\varepsilon$.
Simulation performed with a system of $50$ dimers.

Figure 5.3. Spectra and pseudospectra for dimer systems. In all three figures
the black dots show the spectrum of $\mathcal{C}^{\gamma}$ and the solid
colored lines the $\varepsilon$-pseudospectrum for $\varepsilon=10^{k}$ for
$k=-1,\dots,-5$. In Figure 3(a), the graduated line shows the winding of the
determinant and in Figures 3(b) and 3(c), the winding of the union of the
eigenvalues of the symbol of the corresponding $2$-Toeplitz operator. In light
grey, the respective region with non-trivial winding. All simulations are
performed with $\ell_{i}=1$, $s_{1}=1$, $s_{2}=2$ and $\gamma=1$.

###  5.4. Non-Hermitian interface modes in systems with opposing signs of
$\gamma$.

We consider a system modelled by

$\displaystyle\begin{dcases}u^{\prime\prime}(x)+\gamma_{i}u^{\prime}(x)+\frac{\omega^{2}}{v_{b}^{2}}u=0,&x\in(x_{i}^{\operatorname{{\mathsf{L}}}},x_{i}^{\operatorname{{\mathsf{R}}}}),\\\
u^{\prime\prime}(x)+\frac{\omega^{2}}{v^{2}}u=0,&x\in\mathbb{R}\setminus D,\\\
u|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{L},\mathsf{R}}}}_{i})-u|_{\operatorname{{\mathsf{L}}}}(x^{\operatorname{{\mathsf{L},\mathsf{R}}}}_{i})=0,&\text{for
all }1\leq i\leq N,\\\
\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{L}}}}_{{i}})=\delta\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{L}}}}(x^{\operatorname{{\mathsf{L}}}}_{{i}}),&\text{for
all }1\leq i\leq N,\\\
\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{R}}}}_{{i}})=\delta\left.\frac{\operatorname{d\\!}u}{\operatorname{d\\!}x}\right|_{\operatorname{{\mathsf{R}}}}(x^{\operatorname{{\mathsf{L}}}}_{{i}}),&\text{for
all }1\leq i\leq N,\\\
\frac{\operatorname{d\\!}u}{\operatorname{d\\!}|x|}-\mathbf{i}\frac{\omega}{v}u=0,&x\in(-\infty,x_{1}^{\operatorname{{\mathsf{L}}}})\cup(x_{N}^{\operatorname{{\mathsf{R}}}},\infty).\end{dcases}$
(5.7)

From [1], we know that the subwavelength eigenfrequencies and eigenmodes of
the above system can be approximated by the eigenvalues and eigenvectors of
the following gauge capacitance matrix $(\mathcal{C}^{\gamma})_{i,j}$ defined
by

$\displaystyle\mathcal{C}_{i,j}^{\gamma}\coloneqq\begin{dcases}\frac{\gamma_{1}}{s_{1}}\frac{\ell_{1}}{1-e^{-\gamma_{i}\ell_{1}}},&i=j=1,\\\
\frac{\gamma_{i}}{s_{i}}\frac{\ell_{i}}{1-e^{-\gamma_{i}\ell_{i}}}-\frac{\gamma_{i}}{s_{i-1}}\frac{\ell_{i}}{1-e^{\gamma_{i}\ell_{i}}},&1<i=j<N,\\\
-\frac{\gamma_{i}}{s_{i}}\frac{\ell_{i}}{1-e^{-\gamma_{i}\ell_{j}}},&1\leq
i=j-1\leq N-1,\\\
\frac{\gamma_{i}}{s_{j}}\frac{\ell_{i}}{1-e^{\gamma_{i}\ell_{j}}},&2\leq
i=j+1\leq N,\\\
-\frac{\gamma_{i}}{s_{N-1}}\frac{\ell_{N}}{1-e^{\gamma_{i}\ell_{N}}},&i=j=N,\\\
\end{dcases}$ (5.8)

while all the other entries are zero. We are particularly interested in the
case where

$\displaystyle\gamma_{i}=\begin{dcases}-\gamma,&\text{for }1\leq i\leq m,\\\
\gamma,&\text{for }m+1\leq i\leq N,\end{dcases}$ (5.9)

for some $\gamma>0$ and $1<m<N$, where we typically choose $2m=N$ for $m$
even, creating an symmetric structure. The gauge capacitance matrix then
becomes

$\displaystyle\mathcal{C}^{\gamma}=\begin{pmatrix}\tilde{\alpha}_{1}&\eta_{1}&&&&&\\\
\beta_{1}&\alpha_{2}&\eta_{2}&&&&\\\ &\beta_{2}&\alpha_{1}&\eta_{1}&&&\\\
&&\ddots&\ddots&\ddots&\\\ &&&\beta_{1}&\alpha_{2}&\eta_{2}&&\\\
&&&&\eta_{2}&\alpha_{1}&\beta_{1}&&\\\ &&&&&\ddots&\ddots&\ddots&\\\
&&&&&&\eta_{2}&\alpha_{1}&\beta_{1}\\\ &&&&&&&\eta_{1}&\tilde{\alpha}_{2}\\\
\end{pmatrix}.$ (5.10)

Using the results from Subsection 4.2, it is now possible to show that all but
a few eigenmodes are localised around the interface.

######  Theorem 5.4.

Consider a system satisfying (5.7) for $\gamma_{i}$ satisfying (5.9). Then all
but a few eigenmodes are exponentially localised around the interface at the
site index $m$. More explicitly, we have

$\displaystyle|\bm{\mathbf{x}}^{(j)}|\leq
M|m-j|e^{-\gamma\ell\left|\frac{m-j}{2}\right|},$

for all $1\leq j\leq N$ but $m\neq j$, with $M$ being some positive constant.

In Figure 5.4, we show that the interface modes for a system of $2\times 30$
resonators. One immediately recognises that all but two eigenmodes are
exponentially localised around the interface. The non-localised eigenmodes
present once a monopole and once a dipole behavior.

Figure 5.4. Interface modes for a system of $30$ resonators with negative
$\gamma_{i}$ on the left and $30$ resonators with positive $\gamma_{i}$ on the
right. Simulations performed with $\ell_{i}=1$, $s_{1}=1$, $s_{2}=2$ and
$|\gamma_{i}|=1$.

##  6\. Concluding remarks

Based on new explicit formulas for the eigenpairs of perturbed tridiagonal
block Toeplitz matrices, we have analysed the non-Hermitian skin effect
arising in dimer systems of subwavelength resonators and shown its topological
origin. We have also proved the localisation of interface modes between
systems of resonators with imaginary gauge potentials with opposite signs.

This paper opens the door to the study of many-body non-Hermitian systems
where the non-Hermiticity arises from complex gauge potentials. The explicit
theory we have developed for tridiagonal $2$-Toeplitz matrices could be
extended to tridiagonal $K$-Toeplitz matrix and would lead to a generalisation
of the obtained results to systems with arbitrary number of periodically
repeated resonators. Another interesting problem is to estimate the stability
of the non-Hermitian skin effect with respect to small changes in either the
positions of the subwavelength resonators or their material parameters. This
will be the subject of a forthcoming publication [3].

## Acknowledgments

This work was partially supported by Swiss National Science Foundation grant
number 200021–200307.

## Code availability

The data that support the findings of this work are openly available at
https://doi.org/10.5281/zenodo.8171204.

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