Exponentially localised interface eigenmodes in finite chains of resonators

H. Ammari]Habib Ammari
Habib Ammari

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

S. Barandun]Silvio Barandun
Silvio Barandun

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

B. Davies]Bryn Davies
Bryn Davies

Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK

E.O. Hiltunen]Erik Orvehed Hiltunen
Erik Orvehed Hiltunen

Department of Mathematics, Yale University, 10 Hillhouse Ave,
New Haven, CT 06511, USA

T. Kosche]Thea Kosche
Thea Kosche

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

P. Liu]Ping Liu
Ping Liu

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

This paper studies wave localisation in chains of finitely many resonators. There is an extensive theory predicting the existence of localised modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analogue of the Su-Schrieffer-Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and then show the existence of an eigenvalue in the gap of the defect structure. We find a direct relationship between an eigenvalue being within the spectral gap and the localisation of its associated eigenmode, which we show is exponentially localised. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first  to characterise quantitatively the localised interface modes in systems of finitely many resonators.

Keywords.   Finite Hermitian systems, subwavelength resonators, capacitance matrix, topological protection, Chebyshev polynomials, wave localisation

AMS Subject classifications.

§ INTRODUCTION

Wave localisation at subwavelength scales has many important applications in nanophotonics and nanophononics <cit.>. Here, subwavelength means that the incident wavelengths are much larger than the size of the building blocks of the structure. When these relatively small building blocks are locally resonant (which, in the case studied here, will be due to large material contrasts) this allows for waves to be localised at subwavelength scales, thereby beating traditional diffraction limits <cit.>.  This principle has unlocked a wealth of novel nanotechnologies.

In this paper, we consider wave localisation at the interface between two systems of finitely many subwavelength resonators. These resonators are arranged in pairs or dimers, such that the model we consider shares many of the features of the Su-Schrieffer-Heeger (SSH) model in quantum mechanics <cit.>. We prove the existence of exponentially localised interface eigenmodes in this finite structure. These interface modes have been subject to numerous studies in the setting of infinite structures. A particular focus has been put on studying the topological properties of infinite periodic structures and then introducing carefully designed interfaces so as to create so-called topologically protected eigenmodes <cit.>. These modes have significant implications for applications since they are expected to be robust with respect to imperfections in the design. These concepts have been widely studied in a variety of settings, most notably in quantum mechanics for the Schrödinger operator <cit.> and more recently, for related continuous models of classical wave systems in <cit.>. In finite-sized systems, these eigenmodes have been observed both experimentally and numerically; see, for instance, <cit.> and references therein.

The present work considers the far less-explored but more realistic physical setting of interface modes in finite dimer structures. As far as we know, it is the first work to deal with the existence of interface modes in  finite structures. It provides a one-to-one correspondence between the position of the eigenfrequency in the spectrum of the corresponding infinite periodic structure (i.e., in the asymptotic spectral bulk, its boundary or in the asymptotic spectral gap; see <Ref>) and the behaviour (localised versus delocalised) of the corresponding eigenmode. Our results hold for any finite system of dimer structure with defect satisfying a mild condition on the size of the resonators. Furthermore, we show that the eigenfrequencies lying in the gap converge exponentially as the size of the structure goes to infinity and provide an explicit and simple formula for the limit. We also show that the Hermitian nature of the system together with the position of the interface eigenfrequency in the gap yields a very strong stability property when the geometry of the system is perturbed.

Our approach is based on the capacitance matrix, which in general is a powerful tool for characterising the subwavelength eigenfrequencies of a system of high-contrast resonators <cit.>. In essence, the capacitance formulation provides a discrete approximation to the continuous spectral problem of the differential model, valid in the high-contrast asymptotic limit. This approximation is based solely on first principles and provides a natural starting point for both theoretical analysis and numerical simulation of wave localisation in the subwavelength regime <cit.>. In the case of
our one-dimensional, finite system of dimer resonators with a geometric defect, the capacitance matrix is a perturbed tridiagonal block $2$-Toeplitz matrix; see (<ref>). Based on some properties of the Chebyshev polynomials, we prove existence and uniqueness of an eigenfrequency in the gap for the finite dimer structures with a geometric defect and show exponential localisation of the corresponding eigenmode. Our proof does not rely on any perturbation argument neither on any a prior assumption on the band gaps of the two dimer systems in the structure.

The paper is organised as follows. In <Ref>, we introduce the capacitance matrix approximation of a finite, dimer system with a geometric defect in order to approximate its eigenfrequencies and associated eigenmodes. In <Ref>, we first write the characteristic polynomial of the capacitance matrix associated to a defectless, finite dimer system in terms of Chebyshev polynomials. Then, we characterise the structure of the eigenvectors of the capacitance matrix associated with the perturbed dimer structure. <Ref> is devoted to the derivation of a direct relationship between an eigenvalue being within the spectral gap and the localisation of its corresponding eigenvector. In <Ref>, we analyse the limiting behavior of the system as its size goes to infinity and show that the eigenvalue of the capacitance matrix lying in the asymptotic spectral gap converges exponentially fast to a value in the gap. In <Ref>, we analyse the robustness of the interface localisation with respect to imperfections in the structure design. The papers ends with some concluding remarks in <Ref>. In <Ref>, we recall some basic definitions and results on pseudo-spectra of normal matrices. <Ref> is devoted to the discussion of the topological origin of the robustness of the interface eigenmodes.

§ ONE-DIMENSIONAL SUBWAVELENGTH RESONATOR SYSTEMS

We consider a one-dimensional chain of $N$ disjoint identical high-contrast resonators $D_i\coloneqq (x_i^{\iL},x_i^{\iR})$, where $(x_i^{\iLR})_{1\<i\<N} \subset \R$ are the $2N$ boundaries satisfying $x_i^{\iL} < x_i^{\iR} <  x_{i+1}^{\iL}$ for any $1\leq i \leq N-1$. We fix the coordinates such that $x_1^{\iL}=0$. We also denote by  $\ell_i = x_i^{\iR} - x_i^{\iL}$ the length of the each of the resonators, and by $s_i= x_{i+1}^{\iL} -x_i^{\iR}$ the spacing between the $i$-th and $(i+1)$-th inclusions. The system is illustrated in <ref>.

(x1l) at (1,0);
(x1l) +(1,0) coordinate (x1r);
(x1r) +(0.75,0.7) coordinate (s1);
(x1r) +(1.5,0) coordinate (x2l);
(x2l) +(1,0) coordinate (x2r);
(x2r) +(0.5,0.7) coordinate (s2);
(x2r) +(1,0) coordinate (x3l);
(x3l) +(1,0) coordinate (x3r);
(x3r) +(1,0.7) coordinate (s3);
(x3r) +(2,0) coordinate (x4l);
(x4l) +(1,0) coordinate (x4r);
(x4r) +(0.4,0.7) coordinate (s4);
(x4r) +(1,0) coordinate (dots);
(dots) +(1,0) coordinate (x5l);
(x5l) +(1,0) coordinate (x5r);
(x5r) +(1.75,0) coordinate (x6l);
(x5r) +(0.875,0.7) coordinate (s5);
(x6l) +(1,0) coordinate (x6r);
(x6r) +(1.25,0) coordinate (x7l);
(x6r) +(0.525,0.7) coordinate (s6);
(x7l) +(1,0) coordinate (x7r);
(x7r) +(1.5,0) coordinate (x8l);
(x7r) +(0.75,0.7) coordinate (s7);
(x8l) +(1,0) coordinate (x8r);
[ultra thick] (x1l) – (x1r);
[anchor=north] (label1) at (x1l) $x_1^{\iL}$;
[anchor=north] (label1) at (x1r) $x_1^{\iR}$;
[anchor=south] (label1) at ($(x1l)!0.5!(x1r)$) $\ell_1$;
[dotted,|-|] ($(x1r)+(0,0.25)$) – ($(x2l)+(0,0.25)$);
[ultra thick] (x2l) – (x2r);
[anchor=north] (label1) at (x2l) $x_2^{\iL}$;
[anchor=north] (label1) at (x2r) $x_2^{\iR}$;
[anchor=south] (label1) at ($(x2l)!0.5!(x2r)$) $\ell_2$;
[dotted,|-|] ($(x2r)+(0,0.25)$) – ($(x3l)+(0,0.25)$);
[ultra thick] (x3l) – (x3r);
[anchor=north] (label1) at (x3l) $x_3^{\iL}$;
[anchor=north] (label1) at (x3r) $x_3^{\iR}$;
[anchor=south] (label1) at ($(x3l)!0.5!(x3r)$) $\ell_3$;
[dotted,|-|] ($(x3r)+(0,0.25)$) – ($(x4l)+(0,0.25)$);
(dots) at (dots) …;
[ultra thick] (x4l) – (x4r);
[anchor=north] (label1) at (x4l) $x_4^{\iL}$;
[anchor=north] (label1) at (x4r) $x_4^{\iR}$;
[anchor=south] (label1) at ($(x4l)!0.5!(x4r)$) $\ell_4$;
[dotted,|-|] ($(x4r)+(0,0.25)$) – ($(dots)+(-.25,0.25)$);
[ultra thick] (x5l) – (x5r);
[anchor=north] (label1) at (x5l) $x_{N-3}^{\iL}$;
[anchor=north] (label1) at (x5r) $x_{N-3}^{\iR}$;
[anchor=south] (label1) at ($(x5l)!0.5!(x5r)$) $\ell_{N-3}$;
[dotted,|-|] ($(x5r)+(0,0.25)$) – ($(x6l)+(0,0.25)$);
[ultra thick] (x6l) – (x6r);
[anchor=north] (label1) at (x6l) $x_{N-2}^{\iL}$;
[anchor=north] (label1) at (x6r) $x_{N-2}^{\iR}$;
[anchor=south] (label1) at ($(x6l)!0.5!(x6r)$) $\ell_{N-2}$;
[dotted,|-|] ($(x6r)+(0,0.25)$) – ($(x7l)+(0,0.25)$);
[ultra thick] (x7l) – (x7r);
[anchor=north] (label1) at (x7l) $x_{N-1}^{\iL}$;
[anchor=north] (label1) at (x7r) $x_{N-1}^{\iR}$;
[anchor=south] (label1) at ($(x7l)!0.5!(x7r)$) $\ell_{N-1}$;
[dotted,|-|] ($(x7r)+(0,0.25)$) – ($(x8l)+(0,0.25)$);
[ultra thick] (x8l) – (x8r);
[anchor=north] (label1) at (x8l) $x_{N}^{\iL}$;
[anchor=north] (label1) at (x8r) $x_{N}^{\iR}$;
[anchor=south] (label1) at ($(x8l)!0.5!(x8r)$) $\ell_N$;
[anchor=north] (label1) at (s1) $s_1$;
[anchor=north] (label1) at (s2) $s_2$;
[anchor=north] (label1) at (s3) $s_3$;
[anchor=north] (label1) at (s4) $s_4$;
[anchor=north] (label1) at (s5) $s_{N-3}$;
[anchor=north] (label1) at (s6) $s_{N-2}$;
[anchor=north] (label1) at (s7) $s_{N-1}$;

A chain of $N$ resonators, with lengths
$(\ell_i)_{1\leq i\leq N}$ and spacings $(s_{i})_{1\leq i\leq N-1}$ (which will be chosen to alternate between two distinct values, as depicted).

We use

\begin{align*}
D\coloneqq \bigcup_{i=1}^N(x_i^{\iL},x_i^{\iR})
\end{align*}

to denote the set of subwavelength resonators. In this paper, we only consider systems of identically sized resonators, that is

\begin{align*}
\ell_i = \ell \in \R_{>0}\text{ for all } 1\leq i\leq N.
\end{align*}

This will simplify the formulas in our subsequent analysis and is sufficient to observe the physical phenomena we are interested in.

In this work, we consider the one-dimensional wave equation propagating in a heterogeneous medium with space-dependent material parameters:

\begin{align}
\frac{\omega^{2}}{\kappa(x)}u(x) +\frac{\dd}{\dd x}\left( \frac{1}{\rho(x)}\frac{\dd}{\dd
x}  u(x)\right) =0,\qquad x \in\R.
\label{eq: gen Strum-Liouville}
\end{align}

The material parameters $\kappa(x)$ and $\rho(x)$ are piecewise constant in the interior and exterior of the resonators

\begin{align*}
\kappa(x)=
\begin{dcases}
\kappa_b, & x\in D,\\
\kappa, &  x\in\R\setminus D,
\end{dcases}\quad\text{and}\quad
\rho(x)=
\begin{dcases}
\rho_b, & x\in D,\\
\rho, &  x\in\R\setminus D,
\end{dcases}
\end{align*}

where the constants $\rho_b, \rho, \kappa, \kappa_b \in \R_{>0}$. The wave speeds inside the set $D$ of resonators  and inside the background medium $\R\setminus D$, are denoted respectively by $v_b$ and $v$, the wave numbers respectively by $k_b$ and $k$, and the contrast between the densities of the resonators and the background medium by $\delta$:

\begin{align}
v_b:=\sqrt{\frac{\kappa_b}{\rho_b}}, \qquad v:=\sqrt{\frac{\kappa}{\rho}},\qquad
k_b:=\frac{\omega}{v_b},\qquad k:=\frac{\omega}{v},\qquad
\delta:=\frac{\rho_b}{\rho}.
\end{align}

For these step-wise defined material parameters, the wave problem determined by (<ref>) reduces to the following system of coupled one-dimensional Helmholtz equations:

\begin{align}
\label{eq: system of coupled equations}
\begin{dcases}
\frac{\dd{^2}}{\dd x^2}u(x)+ \frac{\omega^2}{v^2}u(x) = 0, & x\in \R \setminus D ,\\
\frac{\dd{^2}}{\dd x^2}u(x)+ \frac{\omega^2}{v_b^2}u(x) = 0, & x\in D ,\\
u\vert_{\iR}(x^{\iLR}_{{i}}) - u\vert_{\iL}(x^{\iLR}_{{i}}) = 0, &  1\leq i\leq N ,\\
\left.\frac{\dd u}{\dd x}\right\vert_{\iR}(x^{\iL}_{{i}}) - \delta\left.\frac{\dd u}{\dd x}\right\vert_{\iL}(x^{\iL}_{{i}}) = 0, & 1\leq i\leq N ,\\
\delta\left.\frac{\dd u}{\dd x}\right\vert_{\iR}(x^{\iR}_{{i}}) - \left.\frac{\dd u}{\dd x}\right\vert_{\iL}(x^{\iR}_{{i}}) = 0, & 1\leq i\leq N,
\end{dcases}
\end{align}

where for a one-dimensional function $w$ we denote by

\begin{align*}
w\vert_{\iL}(x) \coloneqq \lim_{\substack{s\to 0\\ s>0}}w(x-s) \quad \mbox{and} \quad  w\vert_{\iR}(x) \coloneqq \lim_{\substack{s\to 0\\ s>0}}w(x+s)
\end{align*}

if the limits exist.

We are interested in the high-contrast regime characterised by the contrast parameter $\delta$ being small. In this case, there exist resonant modes of the system at subwavelength frequencies, for which the size of the resonators is smaller than the wavelength in the background medium.

A resonance $\omega(\delta)\in\C$ for which there exists a non trivial solution to (<ref>) is called subwavelength in the high-contrast regime if

\begin{align*}
\omega(\delta)\to 0 \quad \text{as}\quad \delta \to 0.
\end{align*}

One consequence of this asymptotic ansatz is that it lends itself to characterisation using asymptotic analysis <cit.>. This limit recovers subwavelength resonances, while keeping the size of the resonators fixed.

In <cit.>, an asymptotic analysis in the subwavelength limit was performed on the system of one-dimensional subwavelength resonators considered here. It was shown that the leading-order behavior of the resonances is given by the eigenpairs of the capacitance matrix:

\begin{align}\label{eq: general capacitance matrix}
\mathcal{C}\coloneqq \begin{pmatrix}
\frac{1}{s_1} & -\frac{1}{s_1}\\
-\frac{1}{s_1} & \frac{1}{s_1} + \frac{1}{s_2} & -\frac{1}{s_2}\\
& -\frac{1}{s_2} & \frac{1}{s_2} + \frac{1}{s_3} & -\frac{1}{s_3}\\
&& \ddots & \ddots & \ddots \\
&&& -\frac{1}{s_{N-2}} & \frac{1}{s_{N-2}} + \frac{1}{s_{N-1}} & -\frac{1}{s_{N-1}}\\
&&&& -\frac{1}{s_{N-1}} & \frac{1}{s_{N-1}}
\end{pmatrix}.
\end{align}

This is a modified version of the conventional capacitance matrix that is often used to characterise many-body low-frequency resonance problems. We summaries the findings of <cit.> here below.

Consider a system of $N$ subwavelength resonators with size $\ell$ and spacings $s_i$ for $1\leq i \leq N-1$. Assume that the eigenvalues of $\mathcal{C}$ are simple. Then, the $N$ subwavelength resonant frequencies $\omega_i$ of (<ref>) satisfy to the first order

\begin{align*}
\omega_i =  v_b\sqrt{\delta\lambda_i} + \BO(\delta),
\end{align*}

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

\begin{equation}
\label{eq:eigevalue problem capacitance matrix}
\mathcal{C} \bm a_i = \lambda_i \ell \bm a_i\qquad 1\leq i\leq N.
\end{equation}

Furthermore, let $u_i(x)$ be a subwavelength eigenmode corresponding to $\omega_i$ and let $\bm a_i$ be the corresponding eigenvector of $\mathcal{C}$. Then

\begin{align*}
u_i(x) = \sum_{j=1}^N\bm a_i^{(j)}V_j(x) + \BO(\delta)%\quad x\in(x_j^{\iL},x_j^{\iR}),
\end{align*}

where $\bm a^{(j)}_i$ denotes the $j$-th entry of the eigenvector $\bm a_i$ and $V_j(x)$ is piecewise linear, supported in $(x_{j-1}^{\iR},x_{j+1}^{\iL})$ and $V_j(x)=1$ for $x\in(x_{j}^{\iL},x_{j}^{\iR})$.

§.§ Dimer chains with a defect

Systems of repeated dimers (that is $s_i=s_{i-2}$ for $3\leq i\leq N$) are of particular interest as the corresponding infinite structure can be studied with Floquet–Bloch band theory <cit.>; when the two repeating separation distances are distinct, this  provides a non-trivial example of a band gap between the subwavelength spectral bands. Here, we consider systems of dimers with a defect in the geometric structure, so that at some point the repeating pattern of alternating separation distances is broken. This is inspired by the famous Su-Schrieffer-Heeger (SSH) model from quantum settings and is the canonical example of a topologically protected interface mode <cit.>. It is a system of $N=4m+1$ for $m\in\N$ resonators such that

\begin{align*}
s_i &= s_{i-2} \quad \text{for } 3\leq i \leq 2m,\\
s_i &= s_{i+2} \quad \text{for } 2m+1\leq i \leq 4m-2,
\end{align*}

where we typically assume $s_1=s_{2m+2}$ and $s_2=s_{2m+1}$ so that the system is symmetric with respect to the center of the $(2m+1)$-th resonator with spacings $s_1$ and $s_2$. A sketch of this system with a defect is shown in <ref>.

[-,thick,dotted] (-.5,-1) – (-.5,2);
[|-|,dashed] (0,1) – (1,1);
[above] at (0.5,1) $s_2$;
[ultra thick] (1,0) – (2,0);
[below] at (1.5,0) $D_{2m+2}$;
[-,dotted] (1,0) – (1,1);

[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_1$;
[ultra thick] (3,0) – (4,0);
[below] at (3.5,0) $D_{2m+3}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);
[-,dotted] (4,0) – (4,1);

[|-|,dashed] (0,1) – (1,1);
[above] at (0.5,1) $s_2$;
[ultra thick] (1,0) – (2,0);
[below] at (1.5,0) $D_{2m+4}$;
[-,dotted] (1,0) – (1,1);
at (2.5,.5) …;

[ultra thick] (1,0) – (2,0);
[below] at (1.5,0) $D_{4m-2}$;

[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_2$;
[ultra thick] (3,0) – (4,0);
[below] at (3.5,0) $D_{4m}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);
[-,dotted] (4,0) – (4,1);
[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_1$;
[ultra thick] (3,0) – (4,0);
[below] at (3.5,0) $D_{4m+1}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);

[|-|,dashed] (0,1) – (1,1);
[above] at (0.5,1) $s_1$;
[ultra thick] (1,0) – (2,0);
[below] at (1.5,0) $D_{2m}$;
[-,dotted] (1,0) – (1,1);

[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_2$;
[ultra thick] (3,0) – (4,0);
[below,fill=white] at (3.5,0) $D_{2m+1}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);
[-,dotted] (4,0) – (4,1);

at (.5,.5) …;
[ultra thick] (1,0) – (2,0);
[below] at (1.5,0) $D_{2m-2}$;

[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_2$;
[ultra thick] (3,0) – (4,0);
[below] at (3.5,0) $D_{2m-3}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);
[-,dotted] (4,0) – (4,1);

[ultra thick] (1,0) – (2,0);
[below] at (1.5,0) $D_{1}$;

[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_1$;
[ultra thick] (3,0) – (4,0);
[below] at (3.5,0) $D_{2}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);
[-,dotted] (4,0) – (4,1);
[|-|,dashed] (2,1) – (3,1);
[above] at (2.5,1) $s_2$;
[ultra thick] (3,0) – (4,0);
[below] at (3.5,0) $D_{3}$;
[-,dotted] (2,0) – (2,1);
[-,dotted] (3,0) – (3,1);

Dimer structure with a defect.

For such systems the geometric symmetries mean that the capacitance matrix from (<ref>) has the following tridiagonal block structure:

\begin{align}
\label{eq: strucutre capacitance matrix}
\mathcal{C} = \begin{pNiceMatrix}
\Block[draw,fill=blue!40,rounded-corners]{7-7}{} \tilde{\alpha} & \beta_{1} &&&&&&&&&\\
\beta_{1} & \alpha & \beta_{2}  &&&&&&&&\\
& \beta_{2} & \alpha & \beta_{1}  &&&&&&&\\
&& \ddots     & \ddots     & \ddots     &&&&&&\\
&&& \beta_{2} & \alpha & \beta_{1}  &&&&&\\
&&&& \beta_{1} & \alpha & \beta_{2}  &&&&\\
&&&&& \beta_{2} & \Block[draw,fill=red!40,rounded-corners]{7-7}{}\eta & \beta_{2}  &&&\\
&&&&&& \beta_{2} & \alpha & \beta_{1}  &&\\
&&&&& && \beta_{1} & \alpha & \beta_{2}  &\\
&&&&&&&& \ddots     & \ddots & \ddots     &\\
&&&&&&&&& \beta_{1} & \alpha & \beta_{2}  \\
&&&&&&&&& & \beta_{2} & \alpha & \beta_{1}  \\
&&&&&&&&&&& \beta_{1} & \tilde{\alpha}
\end{pNiceMatrix}
\end{align}

\begin{align}
\label{eq: translation alpha beta to s1 s2}
\beta_1 = -s_1\inv,\quad \beta_2 = -s_2\inv,\quad \alpha = s_1\inv + s_2\inv,\quad \eta=2s_2\inv,\quad \tilde{\alpha} = s_1^{-1}.
\end{align}

It will be useful to have notation for the top left $(2m+1)\times(2m+1)$-submatrix $C_1%=\mathcal{C}[1:2m+1, 1:2m+1]
$ and the bottom right $(2m+1)\times(2m+1)$-submatrix $C_2%=\mathcal{C}[2m+1:4m+1, 2m+1:4m+1]
$ (as highlighted by the shading in (<ref>)).
<ref> shows the spectrum and the eigenvectors of the capacitance matrix (<ref>). It is immediately clear that there exists a spectral gap and there exists a localised eigenmode associated to an eigenvalue in the gap.

Eigenvalues of (<ref>). As red cross a specific eigenvalue lying isolated from the others.

A selection of eigenvectors of $\mathcal{C}$ from (<ref>). All eigenvectors are superimposed and with unit norm. The red solid eigenvector corresponds to the red cross eigenvalue in (<ref>).

Eigenvalues and eigenvectors of the capacitance matrix (<ref>) for $N=41, s_1=1$ and $s_2=3$.

§ PERTURBED TRIDIAGONAL $2$-TOEPLITZ MATRICES

In this section we will briefly recall results established in <cit.> about eigenvalues and eigenvectors of tridiagonal $2$-Toeplitz matrices with perturbations on the corners.

Denote by

\begin{equation}\label{equ:odd2toeplitzcornerperturbed1}
A_{2k+1}^{(a, b)}(\alpha, \beta_1, \beta_2)\coloneqq\left(\begin{array}{ccccccc}
\alpha+a & \beta_1 & 0 & 0 & \ldots & 0 & 0\\
\beta_1 & \alpha & \beta_2 & 0 & \ldots & 0 & 0 \\
0 & \beta_2 & \alpha & \beta_1 & \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
\ldots & \ldots & \ldots & \ldots & \ldots & \alpha & \beta_2 \\
0 & 0 & 0 & 0 & \ldots & \beta_2 & \alpha+b
\end{array}\right) \in \R^{(2k+1)\times (2k+1)}
\end{equation}

\begin{equation}\label{equ:even2toeplitzcornerperturbed1}
A_{2k}^{(a, b)}(\alpha, \beta_1, \beta_2)\coloneqq\left(\begin{array}{ccccccc}
\alpha+a & \beta_1 & 0 & 0 & \ldots & 0 & 0\\
\beta_1 & \alpha & \beta_2 & 0 & \ldots & 0 & 0 \\
0 & \beta_2 & \alpha & \beta_1 & \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
\ldots & \ldots & \ldots & \ldots & \ldots & \alpha & \beta_1 \\
0 & 0 & 0 & 0 & \ldots & \beta_1 & \alpha+b
\end{array}\right)\in \R^{2k\times 2k}
\end{equation}

the prototypical tridiagonal $2$-Toeplitz matrices with perturbations on the corners.

§.§ Eigenvalues

We define the polynomials

\begin{equation}\label{equ:defiofpkstar1}
P_k^*(x):=\left(\beta_1 \beta_{2}\right)^k U_k\left(\frac{(x-\alpha)^2-\beta_1^2-\beta_2^2}{2 \beta_{1} \beta_2}\right),
\end{equation}

where $U_k$ is the Chebyshev polynomial of the second kind, as well as the function

\begin{align}
\label{eq: def y}
y(z) \coloneqq \frac{z^2-\beta_1^2-\beta_2^2}{2 \beta_{1} \beta_2}.
\end{align}

The following proposition holds.

The characteristic polynomials of $A_{2k+1}^{(a, b)}$ and $A_{2k}^{(a, b)}$ are respectively

\begin{align}\label{equ:eigenpolynomial3}
\chi_{A_{2k+1}^{(a, b)}}(x) = \left(x-\alpha-a-b\right) P_k^*\left(x\right)+\left( ab\left(x-\alpha\right)  -a \beta_{1}^2-b \beta_{2}^2\right) P_{k-1}^*\left(x\right)
\end{align}

\begin{align}\label{equ:eigenpolynomial4}
\chi_{A_{2k}^{(a, b)}}(x)=P_k^*\left(x\right)+\left((a+b)\left(\alpha-x\right)+ab+\beta_{2}^2\right) P_{k-1}^*\left(x\right)+a b \beta_{1}^2 P_{k-2}^*\left(x\right).
\end{align}

§.§ Eigenvectors

We start by defining two families of polynomials $\widehat p_{k+1}^{(\xi_{p}, \xi_{q})}(x)$ and $\widehat q_{k+1}^{(\xi_{p}, \xi_{q})}(x)$ as solutions to the recursion relations

\begin{equation}
\begin{aligned}\label{equ:Chebyshevrecurrence1}
&\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},\\
&\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),
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}\label{equ:Chebyshevrecurrence2}
& \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},\\
&\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),
\end{aligned}
\end{equation}

where $\beta=y\beta_2/\beta_1$.
Then we state the following result.

Let $\lambda$ be an eigenvalue of $A_{2k+1}^{(a, b)}(\alpha, \beta_1, \beta_2)$. Then, using $\mu\coloneqq y(\lambda)$ where $y$ is as in (<ref>), the eigenvector corresponding to $\lambda$ is given by

\begin{align}\label{eq: eigenvector A2k+1}
\bm x =& \left( \hat q_{0}^{(\xi_p, \xi_{q})}\left(\mu\right),-\frac{1}{\beta_1}\left(\alpha-\lambda\right) \hat p_0^{(\xi_p, \xi_{q})}\left(\mu\right), \hat  q_1^{(\xi_p, \xi_{q})}\left(\mu\right), -\frac{1}{\beta_1} \left(\alpha-\lambda\right) \hat p_1^{(\xi_p, \xi_{q})}\left(\mu\right), \right. \nonumber\\
&\qquad  \left.  \ldots, -\frac{1}{\beta_1} \left(\alpha-\lambda\right) \hat p_{k-1}^{(\xi_p, \xi_{q})}\left(\mu\right),
\hat  q_{k}^{(\xi_p, \xi_{q})}\left(\mu\right)\right ).
\end{align}

If $\lambda$ is an eigenvalue of $A_{2k}^{(a, b)}(\alpha, \beta_1, \beta_2)$, then the corresponding eigenvector is given by

\begin{align}\label{eq: eigenvector A2k}
\bm x =& \left( \hat q_{0}^{(\xi_p, \xi_{q})}\left(\mu\right),-\frac{1}{\beta_1}\left(\alpha-\lambda\right) \hat p_0^{(\xi_p, \xi_{q})}\left(\mu\right), \hat  q_1^{(\xi_p, \xi_{q})}\left(\mu\right), -\frac{1}{\beta_1} \left(\alpha-\lambda\right) \hat p_1^{(\xi_p, \xi_{q})}\left(\mu\right), \right. \nonumber\\
&\qquad  \left.  \ldots, -\frac{1}{\beta_1} \left(\alpha-\lambda\right) \hat p_{k-1}^{(\xi_p, \xi_{q})}\left(\mu\right)\right ).
\end{align}

In both cases, $\xi_{q}=(\alpha-\lambda), \xi_{p}= (\alpha+a-\lambda)$.

The last result of this subsection is on the structure of the eigenvectors for the capacitance matrix $\mathcal{C}$ defined by (<ref>). <ref> is an adapted version of <cit.>.

Let $(\lambda,\bm v)$ be an eigenpair of $\mathcal{C}$ and let $\mu\coloneqq y(\lambda)$. Then $\bm v$ is given by

\begin{equation}\label{eq: structure eigenvector defect matrix}
\bm v = (\bm x^{(1)},\bm x^{(2)},\dots,\bm x^{(2m)},\bm x^{(2m+1)},(-1)^\sigma\bm x^{(2m)},\dots,(-1)^\sigma\bm x^{(2)},(-1)^\sigma\bm x^{(1)}),
\end{equation}

where $\bm x\in \R^{2m+1}$ is as in (<ref>) with $\xi_{q}=(\alpha-\lambda), \xi_{p}= (\alpha+a-\lambda)$ and $\sigma\in\{0,1\}$ except for $\bm x \in \sspan \{\bm 1\}$ where $\sigma=1$.

We will show (<ref>) by showing that $(\mathcal{C}-\lambda I)\bm v=\bm 0$ in two steps.

Step 1. In the first step, we consider the first $2m+1$ rows of $(\mathcal{C}-\lambda I)\bm v$:

\begin{equation}\label{eq: eigenvectors C setp 1 base}
\left(\begin{array}{cccccc}
\alpha+a-\lambda  & \beta_{1} & & &&\\
\beta_{1} & \alpha-\lambda  & \ddots & &&\\
& \ddots & \ddots&\ddots&&\\
&  &\beta_{1} & \alpha-\lambda  & \beta_2&\\
&  & & \beta_{2} & \eta-\lambda&\beta_2
\end{array}\right)\begin{pmatrix}
\bm x\\
\bm *
\end{pmatrix}= \bm 0,
\end{equation}

where we put $\bm *$ instead of $\bm x^{(2m)}$ to show that we will not use this entry in what follows as we only need the first $2m$ rows of (<ref>) to determine the $2m+1$ entries of $\bm x$. We write $\bm x$ as

\begin{align}\label{equ:proofeigenvectoreq1}
\bm x=\left(x_1,\ -\frac{1}{\beta_1}(\alpha-\lambda)x_2,\ \frac{\beta_1}{\beta_2}x_3,\ \cdots,\ -\frac{1}{\beta_1}\left(\frac{\beta_1}{\beta_2}\right)^{m-1} \left(\alpha-\lambda\right)x_{2m}, \left(\frac{\beta_1}{\beta_2}\right)^{m}x_{2m+1}\right).
\end{align}

Considering the first row of (<ref>), we can choose

\begin{align*}
x_1 =(\alpha-\lambda), \quad x_2 =  (\alpha+a -\lambda).
\end{align*}

Then by the second row, we have

\begin{align*}
\beta_{1} x_1 -\frac{1}{\beta_{1}} (\alpha-\lambda)^2 x_2 +\beta_1x_3 =0,
\end{align*}

which gives

\[
x_3 =  \frac{(\alpha-\lambda)^2}{\beta_1^2}x_2-x_1.
\]

The third row is

\[
-\frac{\beta_{2}}{\beta_{1}} (\alpha-\lambda)x_2 +(\alpha-\lambda)\left(\frac{\beta_1}{\beta_2}\right)x_3 -(\alpha-\lambda)\left(\frac{\beta_1}{\beta_2}\right)x_4=0,
\]

and thus,

\begin{align*}
x_4 = - \frac{\beta_{2}^2}{\beta_{1}^2}x_2+x_3,
\end{align*}

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

\begin{align}\label{eq: recurrence relations xis}
x_{2k+1} &= \frac{(\alpha-\lambda)^2}{\beta_1^2} x_{2k}- x_{2k-1}, \quad 1\leq k\leq m,\\
x_{2k}&=x_{2k-1} -\frac{\beta_{2}^2}{\beta_{1}^2} x_{2k-2},\quad \quad 2\leq k\leq m.
\end{align}

Applying to (<ref>) the same manipulations as those used in the  proof of <cit.>, it is now easy to see that $\bm x$ is of the form (<ref>).

Step 2. We conclude the proof by considering the last $2m$ rows. By the fact that $\mathcal{C}$ is $S=\antidiag(1,\dots,1)\in\R^{(4m+1)\times (4m+1)}$ symmetric, i.e., $S\mathcal{C}S=\mathcal{C}$, and $\mathcal{C}$ has only simple eigenvalues <cit.> we can immediately conclude that $S\bm v = \pm \bm v$ and thus obtain the desired result.

§ LOCALISED INTERFACE MODES

In this section we will prove the existence of a spectral gap for a defectless structure of dimers and then show the existence of an eigenvalue in the gap for the defect structure. Furthermore, we will demonstrate that a direct relationship exists between an eigenvalue being within the spectral gap and the localisation of its corresponding eigenvector.

§.§ Asymptotic spectral gap

We first show the existence of a spectral gap for the capacitance matrix of an unperturbed structure of dimers.

By physical considerations we assume $\beta_1,\beta_2<0$ and $\beta_1<\beta_2$. It will often be useful to shift $\mathcal{C}$ in order to have most diagonal entries zero, to that end we introduce $z(x)\coloneqq x - \alpha$.

Consider a finite structure of resonators. We define the asymptotic spectral bulk $\Sigma$ and asymptotic spectral gap $\Gamma$ of the structure as the spectral bulk and spectral gap of the associated infinite periodic system, respectively.

The spectral gap and spectral bulk of infinite periodic dimer systems have been computed in <cit.>.

Consider a system of repeated dimers (without defect) with $N=2m$ resonators. Denote by $C_{N}$ the associated capacitance matrix and let $\Sigma$ be the asymptotic spectral bulk. Then

\begin{align*}
\Sigma = \overline{\lim_{N\to\infty} \sigma(C_{N})} = \left[0,\frac{2}{s_2}\right] \cup \left[\frac{2}{s_1}, \frac{2}{s_1} + \frac{2}{s_2}\right],
\end{align*}

where $\lim$ denotes the Hausdorff limit. Consequently, the asymptotic spectral gap is

\begin{align*}
\Gamma = \left( \frac{2}{s_2}, \frac{2}{s_1} \right)\subset \R.
\end{align*}

From (<ref>) we see that any eigenvalue $\lambda$ of the capacitance matrix of a structure of dimer without defect — that is, a capacitance matrix of the form $C = A_{2m}^{(\beta_2,\beta_2)}$ — satisfy

\begin{align*}
0 &= P^*_m(\lambda) + (2\beta_2z+2\beta_2^2+\beta_2^2)P_{m-1}^*(\lambda)+\beta_2^2\beta_1^2P^*_{m-2}\\
&= \beta_1^m\beta_2^m U_m(y) + (2\beta_2z+2\beta_2^2)\beta_1^{m-1}\beta_2^{m-1} U_{m-1}(y)+\beta_1^{m}\beta_2^{m} U_{m-2}(y),
\end{align*}

where for ease of notation we use $z$ for $z(\lambda)$ and $y$ for $y(z(\lambda))$. Using the recurrence relation of the Chebyshev polynomials of the second kind we get that

\begin{align*}
0 &= \beta_1^m\beta_2^m[2yU_{m-1}(y) - U_{m-2}(y)] \\&+ (2\beta_2z+2\beta_2^2)\beta_1^{m-1}\beta_2^{m-1} U_{m-1}(y)+\beta_1^{m}\beta_2^{m} U_{m-2}(y)\\
\Leftrightarrow 0 &= U_{m-1}(y) [(2\beta_2z+2\beta_2^2)\beta_1^{m-1}\beta_2^{m-1} + \beta_1^m\beta_2^m2y]\\
\Leftrightarrow 0 &= U_{m-1}(y) [(z+\beta_2) + \beta_1y].
\end{align*}

So the eigenvalues are given by

\begin{align}
\lambda= \alpha+(\beta_1-\beta_2),\quad \lambda = \alpha+(-\beta_1-\beta_2)\quad \text{and} \quad \lambda = \alpha \pm \sqrt{\beta_1^2+2\beta_1\beta_2\cos\left(\frac{k\pi}{m}\right)+\beta_2^2},
\end{align}

for $1\leq k\leq m-1$. The sought result follows now
directly from inserting the definition of $\alpha, \beta_1,\beta_2$ in terms of $s_1$ and $s_2$ as in (<ref>).

§.§ Existence and uniqueness of eigenvalues in the gap

In this subsection, we will be working with a dimer structure with defect as in <ref>, whose subwavelength spectrum is given by the eigenvalues of $\mathcal{C}$ from (<ref>).

We start with a simple monotonicity result for Chebyshev polynomials of second kind.

Let $k\in\N$ then

\begin{align*}
\frac{U_{k-1}(x)}{U_k(x)}
\end{align*}

is strictly decreasing for $x\in(-\infty, -1)\cup(1,+\infty)$ for any $k\in\N$.

Considering the derivative of $\dfrac{U_{k-1}(x)}{U_k(x)}$, we have

\begin{equation}\label{equ:chebmonotoneequ1}
\left(\frac{U_{k-1}(x)}{U_k(x)}\right)\pri= \frac{U_{k}(x)U_{k-1}\pri(x)-U_{k}\pri(x)U_{k-1}(x)}{U_{k}^2(x)}.
\end{equation}

A well-known property of Chebyshev polynomials is that

\[
U_k\pri(x)=\frac{(k+1) T_{k+1}(x)-x U_k(x)}{x^2-1},
\]

where $T_{k}(x)$ are the Chebyshev polynomials of the first kind. Thus, to demonstrate the negativity of $R_k\pri(x)$ for $\vert x\vert>1$, we only need to show that

\begin{align}\label{eq: condition negativity}
0 &> U_k(x)[k T_{k}(x)-x U_{k-1}(x)] - [(k+1) T_{k+1}(x)-x U_k(x)]U_{k-1}(x)\nonumber \\&= U_k(x)k T_{k}(x) - (k+1) T_{k+1}(x)U_{k-1}(x).
\end{align}

It is also well-known that

\begin{equation}\label{equ:chebyshevrelation1}
T_{\ell}(x) U_n(x)= \begin{cases}\frac{1}{2}\left(U_{\ell+n}(x)+U_{n-\ell}(x)\right), & \text { if } n \geq \ell-1, \\ \frac{1}{2}\left(U_{\ell+n}(x)-U_{\ell-n-2}(x)\right), & \text { if } n \leq \ell-2 . \end{cases}
\end{equation}

So that (<ref>) becomes

\begin{align*}
\frac{k}{2}(U_{2k}(x)+U_{0}(x))-\frac{k+1}{2}(U_{2k}(x)+U_{0}(x)) = -\frac{U_{2k}(x)+1}{2}.
\end{align*}

By $U_{2k}(x)>0, x\in (-\infty, -1)\cup(1,+\infty)$ and the proof is complete.

In order to show the existence of a frequency in the asymptotic spectral gap $\Gamma$,  we first consider the submatrix $C_1$ from (<ref>) and prove the following result.

Let $\beta_1,\beta_2<0$ and $\beta_1<\beta_2$. Then, for $2m>\frac{\beta_1}{\beta_1-\beta_2}$, the matrix

\begin{align*}
C_1 = \begin{pmatrix}
\alpha + \beta_2 & \beta_{1}\\
\beta_{1} & \alpha & \beta_{2}\\
& \beta_{2} & \alpha & \beta_{1}\\
&& \ddots     & \ddots     & \ddots\\
&&& \beta_{2} & \alpha & \beta_{1}  \\
&&&& \beta_{1} & \alpha & \beta_{2}  \\
&&&&& \beta_{2} & \alpha +\beta_1-\beta_2 \\
\end{pmatrix} \in \R^{2m+1\times 2m+1}
\end{align*}

has exactly one eigenvalue $\lambda$ for which $y(z(\lambda))<-1$, where $y(x)$ is defined as in (<ref>). Furthermore, for all eigenvalues $\lambda$, $y(z(\lambda))<1$ holds.

Once again, to ease up the notation we will shorthand $z(\lambda)$ to $z$ and $y(z(\lambda))$ to $y$. By (<ref>) the characteristic polynomials of $C_1$ is given by

\begin{align*}
\chi_{A_{2m+1}^{(\beta_2,\ \beta_1-\beta_2)}}(x) = &\left(x-\alpha-\beta_2-(\beta_1-\beta_2))\right) P_m^*\left(x\right)\\&+\left( \beta_2(\beta_1-\beta_2)\left(x-\alpha\right)  -\beta_2 \beta_{1}^2-(\beta_1-\beta_2) \beta_{2}^2\right) P_{m-1}^*\left(x\right),
\end{align*}

showing that $\lambda$ is an eigenvalue if and only if

\begin{align}\label{eq: condition eigenvalue in gap C1}
\underbrace{-\beta_1\beta_2\frac{z-\beta_2-(\beta_1-\beta_2)}{\beta_2(\beta_1-\beta_2)z-\beta_2\beta_1^2-(\beta_1-\beta_2)\beta_2^2}}_{\eqqcolon L(z)=(L\circ z)(\lambda)}=\underbrace{\frac{U_{m-1}(y)}{U_m(y)}}_{\eqqcolon R(z)=(R\circ z)(\lambda)},
\end{align}

where we have used (<ref>). We first consider $R(z)$ and remark that $y = -1 \Leftrightarrow z^2 = (\beta_1-\beta_2)^2$ (by (<ref>)) and that $U_{k}(-1)= (-1)^k(k+1)$. So, on both ends of the asymptotic spectral gap $R(\pm(\beta_2-\beta_1))=-m/(m+1)$. Since the Chebyshev polynomials have all their roots in $(-1,1)$, $R(z)$ is continuous on the interval $\Gamma$. Furthermore, by <ref>, $R(z)$ is strictly monotonically increasing on $(\beta_1-\beta_2,0)$ and strictly monotonically decreasing on $(0,\beta_2-\beta_1)$.

Looking at $L(z)$, we notice a singularity at $z=\frac{\beta_1^2}{\beta_1-\beta_2}+\beta_2<\beta_1-\beta_2$ under the assumption $\beta_1<\beta_2<0$. So also $L(z)$ is continuous on $\Gamma$. Furthermore, $L(z)$ is the ratio of two polynomials of first degree and as such is monotone. At the extremes of the interval the two functions take the following values:

\begin{align*}
R(\beta_1-\beta_2) = -\frac{m}{m+1} &< L(\beta_1-\beta_2) = - \frac{\beta_1}{3\beta_1-2\beta_2} \\
L(\beta_2-\beta_1) = - 1 &< - \frac{m}{m+1} = R(\beta_2-\beta_1),
\end{align*}

showing that there is exactly one $z_0\in(\beta_1-\beta_2, \beta_2-\beta_1)$ satisfying (<ref>) when $m>\frac{1}{2}\frac{\beta_1}{\beta_1-\beta_2}$.

The second statement of the lemma is that $y(z)<1$ for all eigenvalues. As all capacitance matrices of the form (<ref>) are symmetric positive semi-definite <cit.>, it is left to show that $C_1$ has no eigenvalue $\lambda$ so that $z(\lambda)$ is in $(-\beta_1-\beta_2,\infty)$, which is a direct consequence of the Gershgorin circle theorem.

After this preparatory lemma we are ready to show existence and uniqueness of an eigenvalue in the gap.

A dimer structure with a defect as described by <ref> composed of $4m+1$ resonators with $2m>\dfrac{s_2}{s_2-s_1}$ has exactly one eigenfrequency in the asymptotic spectral gap of the
defectless structure, that is, one eigenvalue of $\mathcal{C}$ lies in $\Gamma = (2/s_2,2/s_1)$.

It should be noted that the condition on $m$ is very weak, for example when $s_1=1$ and $s_2=2$ it is equivalent to $m>1$.

We need to show that $\mathcal{C}$ from (<ref>) has exactly one eigenvalue in the gap.
<ref> shows that a compression of $\mathcal{C}$ has one eigenvalue in the gap, thus via the Cauchy interlacing theorem and the fact that $\mathcal{C}$ is positive semi-definite,  there must be at least one eigenvalue in the gap.

On the other hand, considering the compression of $\mathcal{C}$ obtained by removing the central row and column we obtain a block diagonal matrix with blocks given by

\begin{align*}
B = \begin{pmatrix}
\alpha + \beta_2 & \beta_{1}\\
\beta_{1} & \alpha & \beta_{2}\\
& \beta_{2} & \alpha & \beta_{1}\\
&& \ddots     & \ddots     & \ddots\\
&&& \beta_{2} & \alpha & \beta_{1}  \\
&&&& \beta_{1} & \alpha \\
\end{pmatrix}\quad \text{and}\quad PBP,
\end{align*}

where $P = \antidiag(1,\dots,1)$. Remark that the same assumptions as stated before <ref> on the coefficients of $B$ hold, except that now $b=0$, that is, the matrix $B$ is of the type $A^{(a,0)}_{2m}$ and its characteristic polynomials is thus given by

Consequently, we see from (<ref>) that $\lambda$ is an eigenvalue of $B$ if and only if

\begin{align}\label{eq: condition eigenvalue block}
\underbrace{\frac{-\beta_1\beta_2}{\beta_2z+\beta_2^2}}_{\eqqcolon L(z)}=\underbrace{\frac{U_{m-1}(y)}{U_m(y)}}_{\eqqcolon R(z)},
\end{align}

by using again the shorthand $z$ for $z(\lambda)$ and $y$ for $y(z(\lambda))$. Remark that $L(z)$ in (<ref>) is strictly monotonous decreasing with a pole of order one at $-\beta_2$. Furthermore, recall again that $R(z)$ is strictly monotonically increasing on $(\beta_1-\beta_2,0)$ and strictly monotonically decreasing on $(0,\beta_2-\beta_1)$

We first consider the case $2\beta_2 < \beta_1 < \beta_2$, which equates to the pole of $L(z)$ lying outside of the gap $\Gamma$. In this case we have

\begin{align*}
L(\beta_1-\beta_2) = -1 &< -\frac{m}{m+1}  = R(\beta_1-\beta_2),\\
L(\beta_2-\beta_1) = - \frac{\beta_1}{2\beta_2-\beta_1} &< -\frac{m}{m+1} = R(\beta_2-\beta_1),
\end{align*}

so that $L(z)<R(z)$ for all $z\in(\beta_1-\beta_2,\beta_2-\beta_1)$.

The case $\beta_1 \leq 2\beta_2$ is similar. The pole $-\beta_2$ is now in the interval of interest, but now $L(z)<R(z)$ for $z\in(\beta_1-\beta_2,-\beta_2)$ and $R(z)<0<L(z)$ for $z\in(-\beta_2, \beta_2-\beta_1)$ by the same argument as the one in the previous case. This concludes the proof as, by the Cauchy interlacing theorem, there can be at most
one eigenvalue in the gap.

§.§ Localised interface modes

In the previous subsection, we have shown the existence of a resonant frequency in the asymptotic band gap. In this subsection, we will prove that this eigenfrequency is associated with an eigenmode that is exponentially localised in the sense that it decays exponentially away from the interface in either direction, within the limits of the finite structure.

Let $v(x)$ be an eigenmode. Then we say that $v$ is a localised interface mode at the point $x_0$, if both $\vert v(x-x_0)\vert $ for $x_0<x\in D$ and $\vert v(x_0-x)\vert $  for $x_0>x\in D$ decay exponentially as a function of $x\in D$.

Let $\mathcal{C}\in\mathbb{R}^{4m+1 \times 4m+1}$ be the capacitance matrix of a defect structure as illustrated in <ref> and let $(\lambda,v)$ be an eigenpair of $\mathcal{C}$. Then, there exists $\vert r\vert \geq1$ independent of $m$ and $A,B,\tilde A,\tilde B\in\R$ dependent on $m$ such that

if $y(\lambda)^2>1$

\begin{align*}
v^{ (\vert 2m-2j\vert) } = Ar^j+Br^{-j},\\
v^{ (\vert 2m-2j-1\vert) } = \tilde A r^j+\tilde B r^{-j};
\end{align*}

with $A=\frac{r^{1-m} (c_1 r-c_2)}{r^2-1}=\mathcal{O}(\frac{1}{r^{m}})$ and $B = \frac{r^m (c_2 r-c_1)}{r^2-1}= \mathcal{O}(r^{m-1})$ as $m\to\infty$ for $c_1,c_2\in\R$ independent of $m$. The same asymptotics (with a slight different formula) hold for $\tilde A$ and $\tilde B$;

if $y(\lambda)^2<1$

\begin{align*}
v^{ (\vert 2m-2j\vert) } = A\cos(j\theta)+B\sin(j\theta),\\
v^{ (\vert 2m-2j-1\vert) } = \tilde A\cos(j\theta)+\tilde B\sin(j\theta),
\end{align*}

with $r=e^{\mathbf{i} \theta}$ and $A,B,\tilde A,\tilde B$ bounded as $m\to\infty$;

if $y(\lambda)^2=1$ $r=\pm 1$ and

\begin{align*}
v^{ (\vert 2m-2j\vert) } = Ar_1^j+Br_1^j\cdot j,\\
v^{ (\vert 2m-2j-1\vert) } = \tilde A r_1^j+\tilde B r_1^j\cdot j,
\end{align*}

$A=\frac{r^{1-m} (c_1 m r-c_1 r-c_2 m)}{m r^2-m-r^2}$ and $B = \frac{r^m (c_2 r-c_1)}{m r^2-m-r^2}$ as $m\to\infty$ for $c_1,c_2\in\R$ independent of $m$. The same asymptotics (with a slight different formula) hold for $\tilde A$ and $\tilde B$.

It is enough to consider the $\widehat p_k^{(\xi_{p}, \xi_{q})}$ and $\widehat q_k^{(\xi_{p}, \xi_{q})}$ part of the eigenvector in <ref> as the rest is a multiplicative factor. We only consider $\widehat p_k^{(\xi_{p}, \xi_{q})}$ as the procedure for $\widehat q_k^{(\xi_{p}, \xi_{q})}$ is exactly the same.
Let $(\lambda,v)$ be an eigenpair and recall from (<ref>) the recurrence formula

\begin{align*}
\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),
\end{align*}

where $\mu=y(\lambda)$. This is a two term recurrence formula with characteristic equation

\begin{align}\label{eq:characteristic equation}
X^2-2\mu X + 1 =0
\end{align}

having none, one or two roots $r_1$ and $r_2$ in exactly the cases delineated in the proposition. The formulas for the eigenvectors follow. By Vieta's formula, we know that the two roots of (<ref>) satisfy $r_1r_2=1$ so one of them must satisfy $\vert r_i \vert \geq 1$. The constants $A,B,\tilde A,\tilde B\in\R$ follow from the initial conditions <ref> with the values expressed in <ref>.

We remark that the three cases presented in <ref> correspond to $\lambda \in \Gamma$, $\lambda \in \Sigma$ and $\lambda \in \partial \Sigma$, respectively. We show this behaviour in <ref>.

Eigenvector associated to an eigenvalue in the asymptotic spectral bulk.

Eigenvector associated to an eigenvalue in the boundary of the asymptotic spectral bulk.

Eigenvector associated to an eigenvalue in the asymptotic spectral gap. $y$ axis in log-scale.

Eigenvector behaviour based on the location of eigenvalue. Computation performed with $N=81, s_1=1, s_2=3$.

Consider a dimer structure with a defect as described by <ref> composed of $4m+1$ resonators with $2m>\dfrac{s_2}{s_2-s_1}$. Then, there exists exactly one subwavelength interface mode.

From <ref>, we know that the capacitance matrix associated to the problem has exactly one eigenvalue in the spectral gap. <ref> shows that an eigenvector presents exponential decay if and only if the corresponding eigenvalue lies in the gap. We can then transpose the result to asymptotic behaviour of eigenmodes via <ref>.

§ CONVERGENCE OF THE RESONANT FREQUENCY IN THE GAP

In this section, we analyse the behaviour as the size of the system grows, specifically the limiting behaviour as $N\to\infty$. We will show that the eigenvalue of $\mathcal{C}$ lying in the asymptotic spectral gap converges exponentially fast to a value in the gap.

We start by remarking that by performing Laplace extension on the top block of rows $1\leq i\leq 2m+1$,  we obtain that the determinant of $\mathcal{C}\in\R^{(4m+1)\times (4m+1)}$ is given by

\begin{align}
\det \mathcal{C} &=  \det A_{2m+1}^{(a,\eta-\alpha)}(\alpha,\beta_1,\beta_2)\cdot\det A_{2m}^{(0,b)}(\alpha,\beta_1,\beta_2)\nonumber\\
&- \beta_2^2\det A_{2m}^{(a,0)}(\alpha,\beta_1,\beta_2)\cdot\det A_{2m-1}^{(0,b)}(\alpha,\beta_2,\beta_1),
\end{align}

where $a=b=\beta_2$. We observe that the last term has $\beta_1$ and $\beta_2$ switched.
Consequently, the characteristic polynomial $p(x)$ of $\mathcal{C}$ is given by

\begin{align}
p(x) &= (\left[\left(x-\alpha-\beta_2-(\beta_1 - \beta_2)\right) P_m^*\left(x\right)+\left( \beta_2(\beta_1 - \beta_2)(x-\alpha)  -\beta_2 \beta_{1}^2-(\beta_1 - \beta_2) \beta_{2}^2\right) P_{m-1}^*\left(x\right)\right] \nonumber\\
% % Here below there was an alpha - x that i subtituted with z, unsure if typo or not
% &\cdot \left[P_m^*\left(x\right)+\left(-\beta_2 z+\beta_{2}^2\right) P_{m-1}^*\left(x\right)\right] \nonumber\\
% &-\beta_2\left[P_m^*\left(x\right)+\left(-\beta_2z+\beta_{2}^2\right) P_{m-1}^*\left(x\right)\right] \nonumber\\
&- \beta_2^2\left[\left(x-\alpha-\beta_2\right) P_{m-1}^*\left(x\right)+\left( -\beta_2 \beta_{1}^2\right) P_{m-2}^*\left(x\right)\right])\chi_{A_{2m}^{(a,0)}(\alpha,\beta_1,\beta_2)}(x),
% p(x) &=
%     \det A_{2m+1}^{(a,\eta-\alpha)}(\alpha+x,\beta_1,\beta_2)\cdot\det A_{2m}^{(0,b)}(\alpha+x,\beta_1,\beta_2)\nonumber\\
%         &+ (-1)^\epsilon \beta_2^2\det A_{2m}^{(a,0)}(\alpha+x,\beta_1,\beta_2)\cdot\det A_{2m-1}^{(0,b)}(\alpha+x,\beta_2,\beta_1)\nonumber\\
%         &=\left[ \left(z-a-(\eta-\alpha)\right) P_m^*\left(x\right)+\left( a(\eta-\alpha)z  -a \beta_{1}^2-(\eta-\alpha) \beta_{2}^2\right) P_{m-1}^*\left(x\right)\right]\\
%         &\cdot\left[ P_m^*\left(x\right)+\left(bz+\beta_{2}^2\right) P_{m-1}^*\left(x\right)\right]\\
%         &+\left[\left(z-(\eta-\alpha)\right) P_{m-1}^*\left(x\right)+\left( -(\eta-\alpha) \beta_{2}^2\right) P_{m-1}^*\left(x\right)\right]\\
%         &\cdot\left[ P_m^*\left(x\right)+\left(a\left(z\right)+\beta_{2}^2\right) P_{m-1}^*\left(x\right)\right]
% \chi_{A_{2k+1}^{(a, b)}}(x) = \left(x-\alpha-a-b\right) P_k^*\left(x\right)+\left( ab\left(x-\alpha\right)  -a \beta_{1}^2-b \beta_{2}^2\right) P_{k-1}^*\left(x\right)
% \end{align}
% and
% \begin{align}\label{equ:eigenpolynomial4}
% \chi_{A_{2k}^{(a, b)}}(x)=P_k^*\left(x\right)+\left((a+b)\left(\alpha-x\right)+ab+\beta_{2}^2\right) P_{k-1}^*\left(x\right)+a b \beta_{1}^2 P_{k-2}^*\left(x\right).
\end{align}

as one swiftly remarks that $A_{2m}^{(0,b)}(\alpha,\beta_1,\beta_2)$ and $A_{2m}^{(a,0)}(\alpha,\beta_1,\beta_2)$ are similar matrices.

For the sake of brevity, we rewrite

\begin{align*}
p(x) &= \chi_{A_{2m}^{(a,0)}(\alpha,\beta_1,\beta_2)}(x)\left(\left[A P_m^*\left(x\right)+B P_{m-1}^*\left(x\right)\right]
% Here below there was an alpha - x that i subtituted with z, unsure if typo or not
% \cdot \left[P_m^*\left(x\right)+C P_{m-1}^*\left(x\right)\right] \nonumber\\
-\beta_2^2%\left[P_m^*\left(x\right)+D P_{m-1}^*\left(x\right)\right] \cdot
\left[E P_{m-1}^*\left(x\right)+F P_{m-2}^*\left(x\right)\right]\right),
\end{align*}

where $A,B,E,F$ do not depend on $m$. Thus, $\lambda\in\Gamma$ is an eigenvalue if and only if

\begin{align*}
\left[A P_m^*\left(\lambda\right)+B P_{m-1}^*\left(\lambda\right)\right]%\left[P_m^*\left(x\right)+C P_{m-1}^*\left(x\right)\right]
&= \beta_2^2%\left[P_m^*\left(x\right)+D P_{m-1}^*\left(x\right)\right]
\left[E P_{m-1}^*\left(\lambda\right)+F P_{m-2}^*\left(\lambda\right)\right]\\
\Leftrightarrow P_m^*\left(\lambda\right) \left[A+B\frac{P_{m-1}^*\left(\lambda\right)}{P_{m}^*\left(\lambda\right)}\right]%P_m^*\left(x\right) \left[1+C\frac{P_{m-1}^*\left(x\right)}{P_{m}^*\left(x\right)}\right]
&= \beta_2^2%P_m^*\left(x\right) \left[1+D\frac{P_{m-1}^*\left(x\right)}{P_{m}^*\left(x\right)}\right]
P_{m-1}^*\left(\lambda\right) \left[E+F\frac{P_{m-2}^*\left(\lambda\right)}{P_{m-1}^*\left(\lambda\right)}\right].
\end{align*}

In the above step we were able to divide by $\chi_{A_{2m}^{(a,0)}}(\lambda)$ and $P_{m-2}^*\left(\lambda\right)$ as we only consider $\lambda\in\Gamma$ so the latter term is non-zero because the Chebyshev polynomials only have roots in $(-1,1)$ and the former term was shown to be non-zero in $\Gamma$ in <ref>.
Let now $L\coloneqq \lim_{m\to\infty}\frac{P_{m-1}^*\left(\lambda\right)}{P_{m}^*\left(\lambda\right)}$. Then,

\begin{align*}
\left[A+BL\right]%\left[1+CL\right]
= \beta_2^2%\left[1+DL\right]
\left[E+FL\right]\\
\Leftrightarrow L^2(\beta_2^2F) + L(\beta_2^2E-B) - A = 0,
\end{align*}

from which we get the condition

\begin{align}\label{eq: condition L is root}
L &= \frac{B - E \beta_{2}^{2} \pm \sqrt{4 A F \beta_{2}^{2} + B^{2} - 2 B E \beta_{2}^{2} + E^{2} \beta_{2}^{4}}}{2 F \beta_{2}^{2}}. %\\
%  &=\frac{\beta_2 \left(\beta_1^2+\beta_1 (\beta_2-z)+2 \beta_2 (z-\beta_2)\right)-\sqrt{\beta_2^2
% \left(\beta_1^4+2 \beta_1^3 (\beta_2-z)-\beta_1^2 (\beta_2-z) (3 \beta_2+z)+4 \beta_1
% \beta_2 z (2 \beta_2-z)+4 \beta_2^2 \left(\beta_2^2-3 \beta_2 z+z^2\right)\right)}}{2 \beta_2^5}
\end{align}

On the other hand, by the recurrence formula of $U_m$, for $\lambda \in \Gamma$,

\begin{align}\label{eq: condition L is limit}
L = \lim_{m\to\infty}\frac{P_{m-1}^*\left(\lambda \right)}{P_{m}^*\left(\lambda\right)} = \frac{y+\sqrt{y^2-1}}{\beta_1\beta_2}=-\frac{-2 \beta_1 \beta_2 \sqrt{\frac{\left(\beta_1^2+\beta_2^2-z^2\right)^2}{4 \beta_1^2
\beta_2^2}-1}+\beta_1^2+\beta_2^2-z^2}{2 \beta_1^2 \beta_2^2},
\end{align}

by using again the shorthand $z$ for $z(\lambda)$ and $y$ for $y(z(\lambda))$. Now, an algebraic manipulation shows that conditions (<ref>) and (<ref>) have exactly one common solution in the gap given by

\begin{align}
z_0 = \frac{1}{2} \left(-\sqrt{9 \beta_1^2-14 \beta_1 \beta_2+9 \beta_2^2}-\beta_1-\beta_2\right).
\end{align}

We thus have shown the following result.

Consider a perturbed structure of dimers as illustrated in <ref>. Then the unique interface mode has frequency converging to

\begin{align}
\omega_{\mathsf{i}} = v_b\sqrt{\delta \frac{1}{2} \left(-\sqrt{\frac{9}{s_1^2}- \frac{14}{s_1s_2} + \frac{9}{s_2^2}}+\frac{3}{s_1}+\frac{3}{s_2}\right)}
\end{align}

as $N\to\infty$.
Furthermore, the convergence is exponential, i.e., if $\omega_{\mathsf{i}}^{(N)}$ denotes the interface frequency for a structure with $N$ resonators for $N$ big enough, then

\begin{align}\label{eq: error estimate convergence frequency in gap}
\vert \omega_{\mathsf{i}} - \omega_{\mathsf{i}}^{(N)}\vert
< Ae^{-BN},
\end{align}

for some $A,B>0$ independent of $N$.

The only part left to prove is the convergence rate. Denote by $0_d\in\R^d$ the zero element of that vector space and by $v_{\mathsf{i},N}$ the $l^2$-normalised localised mode associated to $\omega_{\mathsf{i}}^{(N)}$. Then, we remark that

\begin{align}
\label{eq: vgN0 is pseudoeigenvector}
\left\Vert(\mathcal{C}_{N+4}-\lambda_{\mathsf{i}}^{(N)})\begin{pmatrix}
\end{pmatrix}\right\Vert_2 = \left\Vert\begin{pmatrix}
\beta_2 v_{\mathsf{i},N}^{(1)}\\
\\
\beta_2 v_{\mathsf{i},N}^{(N)}\\
\end{pmatrix}\right\Vert_2 = 4\beta_2^2(v_{\mathsf{i},N}^{(1)})^2\eqqcolon \epsilon_{N}.
\end{align}

Remark also that the same estimation holds if we replace $\mathcal{C}_{N+4}$ in (<ref>) with $\mathcal{C}_{N+4k}$ for any $k\in\N$. This shows that $\lambda_{\mathsf{i}}^{(N)}$ is an $\epsilon$-pseudo-eigenvalue of $\mathcal{C}_{N+4k}$ for any $k\in\N$ for every $\epsilon>\epsilon_{N}$. Thus, $\vert \omega_{\mathsf{i}} - \omega_{\mathsf{i}}^{(N)}\vert\leq \epsilon_{N}$. The formulae of <ref> yield
\begin{align}\label{eq: epsionN0 goes to 0}
v_{\mathsf{i},N}^{(1)} = \mathcal{O}(1)\qquad \Vert v_{\mathsf{i},N} \Vert_2=\mathcal{O}(e^{C_1N})\qquad\text{as } {N\to\infty}
\end{align}

for some $C_1>0$. So if $v_{\mathsf{i},N}$ needs to be normalised $v_{\mathsf{i},N}^{(1)}$ decays exponentially.

Convergence of the eigenvalue in the gap ($y$-axis in log scale). We display the left-hand side of (<ref>) for a structure with $s_1=1$ and $s_2=2$.

§ STABILITY ANALYSIS

Interface modes of SSH-like structures are well known to be stable, i.e. perturbations of the system affect them only slightly. In this section we show that perturbation in the geometry have limited effect on both the resonant frequencies and the associated modes and we quantify this effect.

To this end we consider a system of $N=4m+1$ resonators as shown in <ref> but the spacings $s_i$ are now perturbed:

\begin{align}
\label{eq: perturbation si-s}
s_i = \begin{dcases}
s_1 + \tilde\epsilon_i,& 1\leq i \leq 2m, \text{ $i$ odd},\\
s_2 + \tilde\epsilon_i,& 1\leq i \leq 2m, \text{ $i$ even},\\
s_1 + \tilde\epsilon_i,& 2m+1\leq i \leq 4m, \text{ $i$ even},\\
s_2 + \tilde\epsilon_i,& 2m+1\leq i \leq 4m, \text{ $i$ odd}.
\end{dcases}
\end{align}

Furthermore, we denote

\begin{align}
\label{eq: eps_i}
\epsilon_i = \begin{dcases}
-\frac{\tilde\epsilon_i}{s_1(s_1+\tilde\epsilon_i)},& 1\leq i \leq 2m, \text{ $i$ odd},\\
-\frac{\tilde\epsilon_i}{s_2(s_2+\tilde\epsilon_i)},& 1\leq i \leq 2m, \text{ $i$ even},\\
-\frac{\tilde\epsilon_i}{s_1(s_1+\tilde\epsilon_i)},& 2m+1\leq i \leq 4m, \text{ $i$ even},\\
-\frac{\tilde\epsilon_i}{s_2(s_2+\tilde\epsilon_i)},& 2m+1\leq i \leq 4m, \text{ $i$ odd}.
\end{dcases}
\end{align}

The following proposition handles stability of the eigenvalues and is a direct application of the well-known Weyl theorem.

Let $\hat{\mathcal{C}}$ be the capacitance matrix associated to the structure described in (<ref>) and let

\begin{align}\label{equ:errorbound1}
\epsilon \coloneqq \max_{1\leq i\leq N-2}\vert \epsilon_i\vert+\vert\epsilon_{i+1}\vert.%+\vert\epsilon_i\vert+\vert\epsilon_{i+1}\vert.
\end{align}

Then, the eigenvalues $\hat{\lambda_k}$ (sorted increasingly) satisfy

\begin{align}
\label{eq: bound eigenvalues error}
\vert \hat{\lambda_k} - \lambda_k\vert \leq 2\epsilon, \quad 1\leq k\leq N,
\end{align}

where $\lambda_k$ are the eigenvalues of $\mathcal{C}$.

Weyl's theorem states the same result but with the bound of (<ref>) replaced by $\Vert \hat{\mathcal{C}} - \mathcal{C}\Vert$. However, for a tridiagonal matrix $M$ with $\bm 1$ in its kernel, $\Vert M\Vert\leq 2\max_{i}\vert M_{i(i-1)}\vert+\vert M_{i(i+1)}\vert$ by the Gershgorin circle theorem so we obtain the result.

Applying <ref> to our system of dimers where the perturbations $\vert\tilde\epsilon_i\vert\leq \eta$ are in the interval $(-\eta,\eta)$ for some $\eta>0$, we obtain that the eigenvalue perturbation is bounded by

\begin{align*}
\frac{2\eta}{s_1(s_1-\eta)}+\frac{2\eta}{s_2(s_2-\eta)}=2\eta\left(\frac{1}{s_1^2}+\frac{1}{s_2^2}\right) + \mathcal{O}(\eta^2)\quad \text{as }\eta\to 0.
\end{align*}

In order to analyse the stability of the eigenvectors, we will use the
Davis–Kahan theorem <cit.>, which needs some preliminary introduction. Let $E$ and $F$ be $d\times r$ matrices with orthonormal columns such that $\text{span}(E)=\mathcal{E}$ and $\text{span}(F)=\mathcal{F}$. The canonical angles between $\mathcal{E}$ and $\mathcal{F}$ are defined as

\begin{align*}
\theta_j = \cos\inv \sigma_j \quad 1\leq j\leq r,
\end{align*}

where $\sigma_j$ are the $r$ singular values of $E^TF$. We denote by

\begin{align*}
\Theta(\mathcal{E}, \mathcal{F}) = \diag(\theta_1,\dots,\theta_r)
\end{align*}

the canonical angles' matrix.

The Davis–Kahan theorem states the following.

Let $S$ and $\tilde S$ be two $d\times d$ symmetric matrices with eigenvalues

\begin{align*}
\lambda_1\geq \lambda_2\geq\dots\geq \lambda_d,\\
\tilde\lambda_1\geq \tilde\lambda_2\geq\dots\geq \tilde\lambda_d,
\end{align*}

respectively. Fix $1\leq r\leq s\leq d$ and let $V$ and $\tilde V$ be the matrices having as columns the eigenvectors corresponding to $\lambda_j$ and $\tilde \lambda_j$ for $r\leq j\leq s$. Let $\text{span}(V)=\mathcal{V}$ and $\text{span}(\tilde V)=\tilde{\mathcal{V}}$. Define

\begin{align*}
\delta \coloneqq \inf \{ \vert \lambda - \tilde\lambda\vert : \lambda\in[\lambda_s,\lambda_r], \tilde\lambda\in(-\infty,\tilde\lambda_{s+1}]\cup[\tilde\lambda_{r-1},\infty)\}.
\end{align*}

If $\delta>0$, then

\begin{align*}
\Vert \sin\Theta(\mathcal{V}, \tilde{\mathcal{V}})\Vert_2\leq \frac{\Vert S-\tilde S\Vert_2}{\delta},
\end{align*}

where $\sin\Theta(\mathcal{V}, \tilde{\mathcal{V}})_{ii}=\sin(\Theta(\mathcal{V}, \tilde{\mathcal{V}})_{ii})$.

As a direct consequence of Theorem <ref>, we have the following theorem for the stability of the interface eigemodes.

Let $\epsilon< \frac{1}{2}\left(\frac{1}{s_1}-\frac{1}{s_2}\right)$ in (<ref>). Let $\bm v$ and $\hat{\bm v}$ be the eigenvectors corresponding to the eigenvalues $\lambda_i$ and $\hat{\lambda}_i$ in the gap of $\mathcal{C}$ and $\hat{\mathcal{C}}$, respectively. Then

\begin{align}
\label{eq: stabiltiy v gap}
\Vert \bm v - \hat{\bm v}\Vert_2 &\leq \frac{2\sqrt{2}\epsilon}{\delta}\\&\leq \frac{2\sqrt{2}\epsilon}{\delta_0-2\epsilon},\label{eq: stabiltiy v gap apriori}
\end{align}

where $\delta\coloneqq\min\{\vert\lambda_i-\hat{\lambda}_{i+1}\vert,\vert\lambda_i-\hat{\lambda}_{i-1}\vert\}$ and $\delta_0 =\min\{\vert\lambda_i-\lambda_{i+1}\vert,\vert\lambda_i-\lambda_{i-1}\vert\}$. The a priori estimate (<ref>) holds for $\delta_0>2\epsilon$.

Remark that for $s_1=1$ and $s_2=2$ and $m$ large we have $\delta_0\approx0.219$ so that the a priori estimate (<ref>) holds for $\epsilon<0.1$. This a priori estimate is, however, suboptimal as <ref> shows.
The proof immediately follows from <ref> with $r=s$ by using the bound $\Vert \bm v - \hat{\bm v}\Vert_2\leq\sqrt{2}\sin\Theta(\bm v,\hat{\bm v})$ <cit.>.

In <ref> we show numerically the high stability of the interface modes. We consider perturbations of the type $\tilde\epsilon_i\sim U(-\eta,\eta)$ where we call $\eta$ the perturbation size, we display the latter as percentage relative to the resonator's size. <ref> shows that the interface eigenfrequency (lying in the gap) is only minimally perturbed even by perturbation in the size of $20\%$. We remark that numerically the bound of (<ref>) can even be sharpened to $\vert \hat{\lambda_k} - \lambda_k\vert \leq \frac{3}{2}\epsilon$. <ref> shows $\frac{\Vert\bm v - \hat{\bm v}\Vert}{\Vert\bm v\Vert\Vert\hat{\bm v}\Vert}$ for various perturbation sizes. The black lines shows the average over $10^4$ runs while the gray area encloses the range from the minimum to the maximum value of  $\frac{\Vert\bm v - \hat{\bm v}\Vert_2}{\Vert\bm v\Vert_2\Vert\hat{\bm v}\Vert_2}$.

Spectrum of the capacitance matrix with perturbations given by $\tilde\epsilon_i\sim U(-\eta,\eta)$. For every perturbation size the spectrum of one realisation is shown. The eigenvalue in red corresponds to the localised interface mode.

Stability of the interface mode. The solid black line shows the average dislocation over $10^4$ runs, while the gray area encloses the range from the minimum to the maximum dislocation observed over these realisations.

The interface eigenvalue and the corresponding interface mode are very stable also in presence of big perturbations. Simulations in a system of $N=41$ resonators with $s_1=1$ and $s_2=2$. Perturbations are uniformly distributed in $(-\eta,\eta)$ where we call $\eta$ the perturbation size, expressed relatively to the resonators' sizes.

§ CONCLUDING REMARKS

In this paper, we have quantitatively characterised interface eigenmodes in finite, dimer systems of subwavelength resonators with a geometric defect and proved the exponential decay of their associated eigenmodes. Our characterisation is based on (broken) translation invariance properties of the associated capacitance matrix together with  properties of Chebyshev polynomials. As we have shown in this paper, the  $3$-term recurrence relation satisfied by the  Chebyshev polynomials is shown to be useful for analysing spectra of tridiagonal (perturbed) $2$-Toeplitz matrices.

Following this line of research, it would be very interesting to generalise the results obtained in this paper to extended (known also as multi-band) SSH models, exhibiting  exponentially localised interface eigenmodes with corresponding eigenfrequencies inside the multiple subwavelength band gaps of the structure <cit.>. Another highly interesting direction would be to extend the current results to finite dimer systems of three-dimensional subwavelength resonators. In the case of three-dimensional systems of subwavelength resonators, the main difficulty  occurs from the long-range interactions between the resonators, which lead to a slow decay of the off-diagonal terms of the corresponding capacitance matrix. Nevertheless, in view of the recent results on  $k$-banded approximations of three-dimensional capacitance matrices <cit.> together with the convergence spectral results as the size of the structure goes to infinity in <cit.>,  it may be possible to prove existence and uniqueness of localised eigenmodes in finite chains of subwavelength resonators in three dimensions, as numerically shown in <cit.>.
Another very interesting problem is to relate the localisation effect at the interface to the statistics of the eigenvalues of the capacitance matrix under random perturbations in the parameters of the system. This would extend the well-known Thouless localisation criterion <cit.> to interface eigenmodes.

Data Availability

The data that support the findings of this study are openly available at <https://zenodo.org/doi/10.5281/zenodo.10361315>.

This work was supported by Swiss National Science Foundation grant number 200021–200307 and by the Engineering and Physical Sciences
Research Council (EPSRC) under grant number EP/X027422/1.

§ PSEUDO-SPECTRUM OF A NORMAL MATRIX

$\sigma_{\bar{\varepsilon}}(\mathbf{A})$ is the set of $z \in \mathbb{C}$ such that

\|(z-\mathbf{A}) \mathbf{v}\|<\varepsilon,

for some $\mathbf{v} \in \mathbb{C}^N$ with $\|\mathbf{v}\|=1$.

The next theorem expresses these facts with the aid of the following notation for an open $\varepsilon$-ball:

\Delta_{\varepsilon}=\{z \in \mathbb{C}:|z|<\varepsilon\} .

In this theorem, a sum of sets has the usual meaning:

\sigma(\mathbf{A})+\Delta_{\varepsilon}=\left\{z: z=z_1+z_2, z_1 \in \sigma(\mathbf{A}), z_2 \in \Delta_{\varepsilon}\right\},

which is equal to $\{z: \operatorname{dist}(z, \sigma(\mathbf{A}))<\varepsilon\}$.

For any $\mathbf{A} \in \mathbb{C}^{N \times N}$,

\sigma_{\varepsilon}(\mathbf{A}) \supseteq \sigma(\mathbf{A})+\Delta_{\varepsilon} \quad \forall \varepsilon>0,

and if $\mathbf{A}$ is normal and $\|\cdot\|=\|\cdot\|_2$, then

\sigma_{\varepsilon}(\mathbf{A})=\sigma(\mathbf{A})+\Delta_{\varepsilon} \quad \forall \varepsilon>0 .

Conversely, if $\|\cdot\|=\|\cdot\|_2$, then (2.17) implies that $\mathbf{A}$ is normal.

§ TOPOLOGICAL ORIGIN

Infinite SSH structures have long been known for their topological nature. In this section we show that a topological invariant can be defined also for finite structures in such a way that when the size of the system grows they converge to the infinite invariants. For one-dimensional crystals, the Zak phase has been shown to be related thought an if and only if condition to the existence of interface modes, see e.g. <cit.> and also <cit.>. Denoting by $u_j^\alpha$ a family of eigenmodes depending piecewise smoothly on the quasiperiodicity $\alpha$, the Zak phase is defined as

\begin{align}
\label{eq: def Zak}
\phi^{\text{zak}}_j \coloneqq \textbf{i}\int_{Y^*} \left\langle u_j^\alpha,\frac{\partial}{\partial \alpha} u^\alpha_j\right\rangle \dd ,\alpha
\end{align}

where $Y^*$ denotes the first Brillouin zone and $\langle \cdot, \cdot \rangle$ the usual $L^2$ inner product. The Zak phase is also related to the symmetries of the eigenfunctions <cit.>. Specifically, denoting by $u^+_j$ the eigenfunction associated to the quasifrequency that maximises the  $j$-th band function $\omega_j^\alpha$ we may define bulk topological index of the $j$-th band gap as

\begin{align}
\label{eq: def indicator}
\mathcal{J}_j \coloneqq \begin{dcases}
+1, & u^+_j(x) = \mathcal{P}(u^+_j),\\
-1, & u^+_j(x) = -\mathcal{P}(u^+_j),
\end{dcases}
\end{align}

for $j=1,2,$ where $\mathcal{P}$ denotes the mirroring of the unit cell $Y$ about its center. Then, the Zak phase is related to $\mathcal{J}_j$ via

\begin{align}
\label{eq: indicator and zak}
\mathcal{J}_j = (-1)^{j-1}\prod_{k=1}^{j}e^{\textbf{i}\phi^{\text{zak}}_k}.
\end{align}

Previous work <cit.> has shown that the Zak phase of a periodic system of dimers is quantised and depends on the inter- and intra-spacing between the cells. In particular,

\begin{align*}
\phi^{\text{zak}}_j = \begin{dcases}
\pi, &s_1 \geq s_2,\\
0, & s_1 < s_2,
\end{dcases}
\end{align*}

and thus

\begin{align*}
\mathcal{J}_1 = \begin{dcases}
-1, &s_1 \geq s_2,\\
1, & s_1 < s_2.
\end{dcases}
\end{align*}

We will show now that (<ref>) lends itself well to a reformulation in the finite case. We denote by $\mathcal{P}^{\text{d}}$ the discrete equivalent of $\mathcal{P}$ that is, for $\bm v \in \R^{2k}$,

\begin{align*}
\mathcal{P}^{\text{d}}(\bm v)^{(j)} \coloneqq \begin{dcases}
\bm v^{(j-1)},& 1\leq j\leq 2k,\text{ $j$ even},\\
\bm v^{(j+1)},& 1\leq j\leq 2k,\text{ $j$ odd},
\end{dcases}
\end{align*}

and define the discrete equivalent of (<ref>) by

\begin{align}
\label{eq: def indicator discrete}
\mathcal{J}^{\text{d}}(\bm v) \coloneqq \frac{1}{\Vert u\Vert_2^2} \left\langle \bm v,\mathcal{P}^{\text{d}}(\bm v)\right\rangle.
\end{align}

In <ref> we show the values of $\mathcal{J}^{\text{d}}(\bm v)$ for two structures composed of $40$ dimers. <ref> shows the case $s_1=1$ and $s_2=2$ while <ref> shows the case $s_1=2$ and $s_2=1$. We observe a very different behaviour, but in particular the value of interest in view of (<ref>) is the one at $\lambda=1$, where the two structures take values $\approx+1$ and $\approx-1$ respectively showing that through (<ref>) we may generalise (<ref>) to discrete structures.

Discrete indicator $\mathcal{J}^{\text{d}}(\bm v)$ $s_1=1$ and $s_2=2$.

Discrete indicator $\mathcal{J}^{\text{d}}(\bm v)$ $s_1=2$ and $s_2=1$.

The discrete indicator function of (<ref>) have very different behaviour for $s_1<s_2$ and $s_1>s_2$. The values at the end of the bands approximate accurately the values of the infinite periodic case of (<ref>).

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitlearXiv preprint arXiv:2311.10521
titleThe Non-Hermitian Skin Effect With Three-Dimensional Long-Range Coupling

journaltitleMultiscale Model. Simul.
titleEdge modes in subwavelength resonators in one dimension

journaltitlearXiv preprint arXiv:2307.13551
titlePerturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators

journaltitlearXiv preprint arXiv:2205.13337 (to appear in Comm. Math. Phys.)
titleAnderson localization in the subwavelength regime

journaltitleSIAM J. Math. Anal.
titleConvergence Rates for Defect Modes in Large Finite Resonator Arrays

titleFunctional Analytic Methods for Discrete Approximations of Subwavelength Resonator Systems
35C20,35J05,35P20,74J20,Analysis of PDEs (math.AP),FOS: Mathematics,FOS: Physical sciences,Mathematical Physics (math-ph)

journaltitlearXiv preprint arXiv:2305.16788
titleSpectral convergence in large finite resonator arrays: the essential spectrum and band structure

journaltitleJ. Math. Pures Appl. (9)
titleTopologically protected edge modes in one-dimensional chains of subwavelength resonators

titleSubwavelength integrated photonics

journaltitleJ. Acoust. Soc. Am.
titleSubwavelength Su-Schrieffer-Heeger topological modes in acoustic waveguides

titleSurface impedance and topologically protected interface modes in one-dimensional phononic crystals

journaltitleMultiscale Model. Simul.
titleAsymptotic characterization of localized defect modes: Su-Schrieffer-Heeger and related models

journaltitleSIAM J. Numer. Anal.
titleThe rotation of eigenvectors by a perturbation. III

journaltitleJ. Phys. C: Solid State Phys.
titleNumerical studies of localization in disordered systems

journaltitleMem. Amer. Math. Soc.
titleTopologically protected states in one-dimensional systems

journaltitleProc. Natl. Acad. Sci. USA
titleTopologically protected states in one-dimensional continuous systems and Dirac points

journaltitleSIAM Journal on Applied Mathematics
titleSubwavelength Resonances in One-Dimensional High-Contrast Acoustic Media

journaltitleNature Reviews Physics
title3D metamaterials

journaltitleJ. Phys. A
titleMathematical theory for topological photonic materials in one dimension
pagesPaper No. 49520345

titleLocally Resonant Sonic Materials

journaltitleNew Journal of Physics
titleTopological characterization of chiral models through their long time dynamics

titleMathematical theory for the interface mode in a waveguide bifurcated from a Dirac point

journaltitlePhys. Rev. Lett.
titleSolitons in Polyacetylene

journaltitleJ. Math. Phys.
titleTopological edge states of 1D chains and index theory
pagesPaper No. 06190117

journaltitleProc. A.
titleBulk-interface correspondences for one-dimensional topological materials with inversion symmetry
pagesPaper No. 2022067522

journaltitlenpj Quantum Information
titleTopological characterizations of an extended Su–Schrieffer–Heeger model
pagesPaper No. 55

titleA useful variant of the Davis-Kahan theorem for statisticians

journaltitlePhys. Rev. Appl.
titleObservation of Edge Waves in a Two-Dimensional Su-Schrieffer-Heeger Acoustic Network

§ BANDGAP OVERLAP
Change this to a section (Section 7 or 8) rather than in appendix. Give a complete theory for the band gap, the existence of the interface modes, the topological origin, and the stability analysis. Or a complete theory can be a new paper.
Consider a structure composed of two system of dimers merged on some interface. We denote by

\begin{align*}
s_{1}^{\iL},\ s_{2}^{\iL},\ s_{1}^{\iR},\ s_{2}^{\iR}
\end{align*}

the spacing intra-dimer and inter-dimer of the left and right structure,  respectively. In this section, we want to explore which are the values $s_{1}^{\iL},\ s_{2}^{\iL},\ s_{1}^{\iR},\ s_{2}^{\iR}$ that allow for a band gap in the total structure and thus open the possibility for the existence of an interface mode. We follow a similar procedure as in <ref>. Considering the compression of $\mathcal{C}$ obtained by removing the central row and column, we obtain a block diagonal matrix with blocks given by

\begin{align*}
B^{\iL} = \begin{pmatrix}
\tilde{\alpha}^{\iL} & \beta_{1}^{\iL}\\
\beta_{1}^{\iL} & \alpha^{\iL} & \beta_{2}^{\iL}\\
& \beta_{2}^{\iL} & \alpha^{\iL} & \beta_{1}^{\iL}\\
&& \ddots     & \ddots     & \ddots\\
&&& \beta_{2}^{\iL} & \alpha^{\iL} & \beta_{1}^{\iL}  \\
&&&& \beta_{1}^{\iL} & \alpha^{\iL} \\
\end{pmatrix}\quad \text{and} \quad
B^{\iR} =
\begin{pmatrix}
\alpha^{\iR} & \beta_{1}^{\iR}\\
\beta_{1}^{\iR} & \alpha^{\iR} & \beta_{2}^{\iR}\\
& \beta_{2}^{\iR} & \alpha^{\iR} & \beta_{1}^{\iR}\\
&& \ddots     & \ddots     & \ddots\\
&&& \beta_{2}^{\iR} & \alpha^{\iR} & \beta_{1}^{\iR}\\
&&&& \beta_{1}^{\iR} & \tilde{\alpha}^{\iR}\\
\end{pmatrix}.
\end{align*}

The argument presented in the proof of <ref> shows that

\begin{align*}
&\sigma(B^{\iL}) \subset (0, \alpha^{\iL} + \beta_1^{\iL} - \beta_2^{\iL}) \cup (\alpha^{\iL} + \beta_2^{\iL} - \beta_1^{\iL}, \alpha^{\iL}+\beta_2^{\iL}+\beta_1^{\iL}), \\&\sigma(B^{\iR}) \subset (0,\alpha^{\iR} + \beta_1^{\iR} - \beta_2^{\iR}) \cup (\alpha^{\iR} + \beta_2^{\iR} - \beta_1^{\iR}, \alpha^{\iR}+\beta_2^{\iR}+\beta_1^{\iR}).
\end{align*}

By the Cauchy interlacing theorem, there thus exists an asymptotic band gap if and only if

\begin{align*}
&\max(\alpha^{\iL} + \beta_1^{\iL} - \beta_2^{\iL}, \alpha^{\iR} + \beta_1^{\iR} - \beta_2^{\iR}) < \min(\alpha^{\iL} + \beta_2^{\iL} - \beta_1^{\iL}, \alpha^{\iR} + \beta_2^{\iR} - \beta_1^{\iR})\\
\Leftrightarrow\ & \max(s_1^{\iL},s_1^{\iR})<\min(s_2^{\iL},s_2^{\iR}).
\end{align*}

§ NO(?) TOPOLOGICAL PROTECTION

We quote the following theorem about the stability of eigenvalues from <cit.>. We assume that the eigenvalues of Hermitian matrices are sorted.

Let $A$ and $A+E$ be two $n\times n$ complex Hermitian matrices. Then for every normed eigenpair $(\lambda_i,v_i)$ of $A$, if $\min\{\lambda_{i}-\lambda_{i-1},\lambda_{i+1}-\lambda_i\}> 2 \Vert E\Vert$ we have

\begin{align}\label{eq: hermitian bound on eigenvalue perturbation}
\vert \lambda_i - \tilde \lambda_i \vert \leq \Vert Ev_i\Vert
\end{align}

where $\tilde \lambda_i$ is the $i$-th eigenvalue of $A+E$.

The immediate consequence of <ref> is that any geometrical perturbation satisfying $\Vert\hat{\mathcal{C}}-\mathcal{C}\Vert<\frac{d_\mathsf{i}}{2}$ where $d_\mathsf{i}$ is the distance $d_\mathsf{i}=d(\lambda_\mathsf{i},\Sigma)$ cannot push $\lambda_\mathsf{i}$ into the bulk as

\begin{align*}
\vert \lambda_\mathsf{i} - \tilde \lambda_i \vert \leq \Vert Ev_\mathsf{i}\Vert \leq \Vert E\Vert < \frac{d_\mathsf{i}}{2} <d_\mathsf{i}
\end{align*}

Silvio: To me this shows that the stability comes from the Hermitian structure of the system and the location of the interface frequency inside the gap, and not from some topological reason(?)