Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

Abstract

In this paper, we derive sufficient conditions for the localization of two-dimensional split-step quantum walks with a strong shift. For this purpose, we analyze the zero points of the function f introduced by Fuda et al. (Quantum Inf Process 16(8), 203, 2017) and make these zero points explicit. These zeros provide a concrete representation of the eigenvalues and eigenvectors of the evolution operator, and in particular, clarify where localization occurs. In addition, the eigenvalues obtained here asymptotically approach threshold resonance in special cases. We also describe the display of threshold resonances and generalized eigenfunctions.

1 Introduction

Quantum walks have been actively studied in recent years, being introduced to quantum information [16, 26], quantum simulations [22, 25], and topological phases [10, 11]. Split-step quantum walks (SSQWs), introduced by Kitagawa et al. [11] for the analysis of topological phases, have been studied in various fields including localized phenomena [4, 5, 7], weakly convergent limit [6], index theory [18, 19, 27], and so on [17, 21]. A necessary and sufficient condition for a quantum walk to induce localization is that the time evolution operator has eigenvalues [23]. In particular, non-zero components of eigenvectors are the points at which quantum walk localization occurs.

In the context of the discrete Schrödinger equation, a value called resonance is known [1, 8]. If a Hamiltonian satisfies the eigenequations but its eigenfunctions are not square integrable, its eigenvalues are called resonances and its eigenfunctions are called the generalized eigenfunctions corresponding to the resonances. The same thing can happen when considering the eigenequations of the time evolution operator described by quantum walks. As mentioned above, it is equivalent for a quantum walk to cause localization and for the time evolution operator of a quantum walk to have eigenvalues, but the eigenfunctions corresponding to those eigenvalues may not be square integrable. Therefore, it is natural that the time evolution operator of this quantum walk has resonance. Prior work on quantum walk resonances and generalized eigenfunctions can be found in [14, 15, 20].

Resonances for a quantum walk U also apply to stationary measures. If $\Vert (U^t\Psi _0)(x)\Vert ^2$ is independent with respect to time $t\in {\mathbb {Z}}_{\ge 0}$, then the measure $\mu (x):=\Vert (U^t\Psi _0)(x)\Vert ^2$ is called a stationary measure [3, 9, 12, 13], where $\Psi _0$ is the initial state. If the initial state $\Psi _0$ is a generalized eigenfunction of U, then $\Vert (U^t\Psi _0)(x)\Vert ^2$ is time independent. Therefore, the research of resonances is linked to the stationary state.

In fact, generalized eigenfunctions have been used to derive the stationary measure of quantum walks [3, 9, 12, 13]. In [9], the stationary measures corresponding to the generalized eigenfunctions of one-dimensional space-inhomogeneous quantum walk with finite defects are presented. The stationary measures of a multi-dimensional space-homogeneous quantum walk with cycles can be found in [12]. The eigenvalues of the quantum walk can be divided into two essentially different parts [24]. That is, the part derived from birth and the part inherited from the discriminant operator T. A multi-dimensional quantum walk with cycles has an infinite number of eigenvectors with finite support, which corresponds to the birth eigenspace. In [12], stationary measures are constructed by the superposition of an infinite number of eigenvectors in the birth eigenspace. On the other hand, in this paper, we derive the generalized eigenfunctions of the two-dimensional SSQWs not from the birth eigenspace, but from the inherited eigenspace. Thus, the eigenfunction in this paper differs from that in [12] in its origins.

In this paper, we also analyze eigenvalues derived from the inherited eigenspace of the SSQWs with strong shifts. On the other hand, [6] analyzed not eigenvalues but continuous spectra. Fuda et al. [4, 5, 7] also analyze eigenvalues of the SSQWs, but their conditions and results are different from those of this paper, respectively. In [5, 7], the birth eigenvalues are analyzed and not the inherited eigenvalues. Fuda et al. [4] focuses on the existence theorem for inherited eigenvalues of SSQWs. It was shown in [4] that in d-dimensional SSQWs, if the effect of the shift is sufficiently weak, i.e., transition probability $|q_j|^2$ is sufficiently small, the time evolution operator has eigenvalues in the inherited eigenspace. We are interested in whether the time evolution operator has eigenvalues when $|q_j|^2$ is sufficiently large. Therefore, in this paper, a detailed analysis is performed under the condition that the shift is extremely strong, i.e, $q_1 = q_2 = 1$ in two-dimensional SSQWs. In our model, we obtain not only an existence theorem for inherited eigenvalues but also specific displays of eigenvalues and eigenvectors. In addition, we derive resonances and corresponding generalized eigenfunctions in a special case.

The remainder of this paper is organized as follows. Section 2 first introduces the general form of a two-dimensional SSQW with one defect and present prior work on localization [4]. While the previous study dealt only with the case where the effect of the shift is weak, this study deals with a model in which the effect of the shift is strong. We introduce conditions for this purpose. The main results are presented in Sect. 3. In Sect. 3, we give sufficient conditions for SSQWs to cause localization. If that sufficient condition is satisfied, we also show that the time evolution operator has at least four eigenvalues. It also showed where exactly localization is occurring. Furthermore, $|\phi _{2,2}|<|\omega _{2,2}|$ is a sufficient condition for localization to occur, but U have resonances when $|\phi _{2,2}|=|\omega _{2,2}|$. In particular, these resonances are boundary values of the essential spectrum of U and are called threshold resonances.

2 Model definition and related results

In this section, we first define a general two-dimensional split-step quantum walks (SSQWs) and briefly introduce related results. For detailed background and discussion, see [4]. Next, we define a more specific model of SSQWs that can be interpreted as an “extremely strong shift effect”. The purpose of this paper is to shed light on a new aspect of SSQWs through the analysis of this specific model. Note that while [4] provides sufficient conditions for the localization of SSQWs that can be interpreted as “very weak shift effects”, the model presented in this paper deviates significantly from it.

2.1 Definition of two-dimensional split-step quantum walks

In what follows, we use ${{\mathcal {H}}}$ to denote $\ell ^2({\mathbb {Z}}^2;{\mathbb {C}} ^4)$. Let $(\varvec{p}, \varvec{q}) = (p_1, p_2, q_1, q_2)\in {\mathbb {R}}^2\times {\mathbb {C}}^2$ satisfy $p_j^2+|q_j|^2=1\ (j=1,2)$ and use $\{\varvec{e}_j\}_{j=1}^2$ to denote the standard basis of ${\mathbb {Z}}^2$. We set an operator $S_j\ (j=1,2)$ on $\ell ^2({\mathbb {Z}}^2; {\mathbb {C}}^2)$ as $\displaystyle S_j:= \begin{pmatrix} p_j&{}q_jL_j\\ (q_jL_j)^*&{}-p_j \end{pmatrix},$ where $L_j$ is the $\varvec{e}_j$-shift on $\ell ^2({\mathbb {Z}}^2)$ defined by $(L_jf)(\varvec{x})=f(\varvec{x}+\varvec{e}_j)$ for all $f\in \ell ^2({\mathbb {Z}}^2)$, i.e.,

$$\begin{aligned} (S_j\psi )({\varvec{x}})= \begin{pmatrix} p_j\psi _1({\varvec{x}})+q_j\psi _2({\varvec{x}}+{\varvec{e}_j})\\ q_j^*\psi _1({\varvec{x}}-{\varvec{e}}_j)-p_j\psi _2({\varvec{x}}) \end{pmatrix} \quad \text {for all}\ {\varvec{x}}\in {\mathbb {Z}}^2,\ \psi =\begin{pmatrix} \psi _1\\ \psi _2 \end{pmatrix}\in \ell ^2({\mathbb {Z}}^2;{\mathbb {C}}^2). \end{aligned}$$

Identifying ${\mathcal {H}}$ with $\ell ^2({\mathbb {Z}}^2;{\mathbb {C}}^2)\oplus \ell ^2({\mathbb {Z}}^2;{\mathbb {C}}^2)$, we define the shift operator S on ${\mathcal {H}}$ as $S=S_1\oplus S_2$, i.e.,

$$\begin{aligned} (S\Psi )(\varvec{x}) = \begin{pmatrix} (S_1\Psi _1)(\varvec{x})\\ (S_2\Psi _2)(\varvec{x}) \end{pmatrix} \quad \text {for all}\ {\varvec{x}}\in {\mathbb {Z}}^2,\ \Psi = \begin{pmatrix} \Psi _1\\ \Psi _2 \end{pmatrix}\in {\mathcal {H}}, \, \Psi _j \in \ell ^2({\mathbb {Z}}^2;{\mathbb {C}}^2)\ (j=1,2). \end{aligned}$$

Then, S is self-adjoint and unitary on ${\mathcal {H}}$ and satisfies $S^2=1$. Next, we define the coin operator. Let $\Omega ={}^t(\omega _{1,1}, \omega _{1,2}, \omega _{2,1}, \omega _{2,2})$ and $\Phi ={}^t(\phi _{1,1}, \phi _{1,2}, \phi _{2,1}, \phi _{2,2})$ be normalized vectors on ${\mathbb {C}}^4.$ We define the coin operator C on ${\mathcal {H}}$ as a multiplication operator by $C({\varvec{x}} )$, i.e.,

$$\begin{aligned} (C\Psi )(\varvec{x}) = C(\varvec{x})\Psi (\varvec{x})\quad \text {for all} \, \varvec{x}\in {\mathbb {Z}}^2, \ \Psi \in {\mathcal {H}}. \end{aligned}$$

In our one-defect model, $C(\varvec{x})$ is defined by

$$\begin{aligned} C(\varvec{x})&= 2|\chi (\varvec{x})\rangle \langle \chi (\varvec{x})| - 1, \\ \chi (\varvec{x})&= {}^t(\chi _{1,1}(\varvec{x}), \chi _{1,2}(\varvec{x}), \chi _{2,1}(\varvec{x}), \chi _{2,2}(\varvec{x})) ={\left\{ \begin{array}{ll} \Phi , &{}{\varvec{x}}\in {\mathbb {Z}}^2{\setminus } \{\textbf{0}\},\\ \Omega , &{}{\varvec{x}}=\varvec{0}. \end{array}\right. } \end{aligned}$$

Then, C is self-adjoint and unitary on ${\mathcal {H}}$. We define a time evolution operator U on ${\mathcal {H}}$ as $U=SC$. Since both S and C are self-adjoint and unitary on ${\mathcal {H}}$, the spectral mapping theorem (SMT) of quantum walks [24] can be applied to U.

2.2 Related results for localization

To introduce the SMT results, we provide some definitions. The discriminant operator T on $\ell ^2({\mathbb {Z}}^2)$ with respect to U is defined as $T=dSd^*$, where $d : {\mathcal {H}}\rightarrow \ell ^2({\mathbb {Z}}^2)$ is a boundary operator, i.e.,

$$\begin{aligned} (d\Psi )(\varvec{x}) = \langle \chi (\varvec{x}), \Psi (\varvec{x})\rangle _{{\mathbb {C}}^4}\quad \text {for all} \ \varvec{x}\in {\mathbb {Z}}^2, \Psi \in {\mathcal {H}}. \end{aligned}$$

Furthermore, for a boundary operator d, the following holds:

$$\begin{aligned}&(d^*f)(\varvec{x}) = \chi (\varvec{x})f(\varvec{x})\quad \text {for all}\ \varvec{x}\in {\mathbb {Z}}^2, f\in \ell ^2({\mathbb {Z}}^2),\\&d^*d = \bigoplus _{\varvec{x}\in {\mathbb {Z}}^2}|\chi (\varvec{x})\rangle \langle \chi (\varvec{x})|, \quad dd^* = I_{\ell ^2({\mathbb {Z}}^2)},\\&C = 2d^*d - 1. \end{aligned}$$

Definition 1.1

(Localization) Let $\Psi _0\in {{\mathcal {H}}}$ be an initial state with $\Vert \Psi _0\Vert =1$. We say that a pair $(U,\Psi _0)$ causes localization when there exists ${\varvec{x}}\in {\mathbb {Z}}^2$ such that $\limsup _{t\rightarrow \infty }\Vert (U^t\Psi _0)(\varvec{x})\Vert ^2>0$ holds.

In the following, sufficient conditions for localization are described. By [23, Proposition 2.4], $(U, \Psi _0)$ causes localization if and only if the time evolution operator U has eigenvalue $\mu$ and $\Psi _0$ overlap with the eigenspace $\ker (U-\mu )$. Therefore, for localization to occur, U must have an eigenvalue. We briefly present known results on the sufficient conditions for U to have eigenvalues. Among the results obtained by SMT, the following are of particular interest in this paper:

$$\begin{aligned} \{ e^{\pm i\arccos \lambda }\mid \lambda \in \sigma _{\textrm{p}}(T)\}\subset \sigma _{\textrm{p}}(U),\quad \{ e^{\pm i\arccos \lambda }\mid \lambda \in \sigma _{\textrm{ess}}(T)\}= \sigma _{\textrm{ess}}(U). \end{aligned}$$

(2.1)

From this result, sufficient conditions for localization can be obtained by analyzing the eigenvalues of T. The following is a preparation for analyzing the eigenvalues of T. We define the bounded operator $T_0$ on $\ell ^2({\mathbb {Z}}^2)$ as follows:

$$\begin{aligned}&T_0=a_{\Phi }({\varvec{p}})+\sum _{j=1}^2\left( q_j\phi _{j,1}^*\phi _{j,2}L_j+q_j^*\phi _{j,1}\phi _{j,2}^*L_j^* \right) , \nonumber \\&a_{\Phi }({\varvec{p}})=\sum _{j=1}^2p_j\left( |\phi _{j,1}|^2-|\phi _{j,2}|^2\right) , \quad a_{\Omega }({\varvec{p}})=\sum _{j=1}^2p_j\left( |\omega _{j,1}|^2-|\omega _{j,2}|^2\right) . \end{aligned}$$

(2.2)

By [4, Lemma 3.3], we find

$$\begin{aligned} \sigma _{\textrm{ess}}(T)=\sigma (T_0)= \left[ -2\sum _{j=1}^2|q_j\phi _{j,1}\phi _{j,2}|+a_{\Phi }({\varvec{p}}),2\sum _{j=1}^2|q_j\phi _{j,1}\phi _{j,2}|+a_{\Phi }({\varvec{p}})\right] . \end{aligned}$$

(2.3)

Let $\varvec{1}_A$ be the characteristic function of a set A. Assuming $[-1, 1]{\setminus } \sigma (T_0)\ne \emptyset$, we can define the function f on $[-1, 1]{\setminus } \sigma (T_0)$ as

$$\begin{aligned} f(\lambda ) = \lambda +\langle \varphi _{\varvec{q}},\psi _{\lambda }\rangle _{\ell ^2({\mathbb {Z}}^2)}, \end{aligned}$$

where

$$\begin{aligned} \varphi _{\varvec{q}}=\sum _{j=1}^2\left( q_j\omega _{j,2}\phi _{j,1}^*{\varvec{1}}_{\{-{\varvec{e}}_j\}}+q_j^*\omega _{j,1}\phi _{j,2}^*{\varvec{1}}_{\{ {\varvec{e}}_j\}} \right) ,\quad \psi _{\lambda }=\left( T_0-\lambda \right) ^{-1}\varphi _{\varvec{q}}. \end{aligned}$$

(2.4)

By the method of the paper [4], we obtain the following facts for f and the eigenvalues of T, U:

$$\begin{aligned} f(\lambda )=0 \ \Longrightarrow \ \lambda \in \sigma _{\textrm{p}}(T) \ \Longrightarrow \ e^{\pm i\arccos \lambda }\in \sigma _{\textrm{p}}(U). \end{aligned}$$

(2.5)

Note that the eigenvalues $e^{\pm i\arccos \lambda }$ of U are discrete. The next theorem gives a sufficient condition for U to have eigenvalues.

Theorem 1.2

([4], $d=2$) Assume the following conditions:

$$\begin{aligned}&\forall j\in \{1,2\}, \ \phi _{j,1}\omega _{j,2}+\phi _{j,2}\omega _{j,1}=0, \end{aligned}$$

(2.6)

$$\begin{aligned}&\exists l\in \{1,2\}, \ \phi _{l,1}\omega _{l,2}\ne 0, \end{aligned}$$

(2.7)

$$\begin{aligned}&\exists \varvec{p}_0\in \{-1,1\}^2, \ a_{\Phi }(\varvec{p}_0) \ne a_{\Omega }(\varvec{p}_0). \end{aligned}$$

(2.8)

Then, there exists $\delta > 0$ such that if $(\varvec{p}, \varvec{q})$ satisfies $p_lq_l\ne 0$ and $\Vert (\varvec{p}, \varvec{q}) - (\varvec{p}_0, \varvec{0})\Vert _{{\mathbb {R}}^2\times {\mathbb {C}}^2}<\delta$, then f have a zero, i.e., there exist eigenvalues of U.

Note that Theorem 2.2 is applicable only when the shift is sufficiently weak (i.e., $\varvec{q}$ is sufficiently close to $\varvec{0}$). In the next subsection, we define SSQWs with a strong shift (i.e., $\varvec{p} = \varvec{0}$).

2.3 SSQWs with strong shift

We consider SSQWs in which the shift is so strong that Theorem 2.2 cannot be applied. Therefore, we introduce the following conditions.

• (C.1) $\omega _{1,2}=\phi _{1,2}=0$ and $\omega _{2,1}\phi _{2,2}+\omega _{2,2}\phi _{2,1}=0$,

• (C.2) $\omega _{i,j}, \phi _{i,j} \in {\mathbb {R}}{\setminus }\{0\}$ for $(i,j)\in \{(1,1),(2,1),(2,2)\}$,

• (C.3) $p_1=p_2=0$ and $q_1=q_2=1$.

Remark 2.3

Condition (C.1) is a special case of condition (2.6) in Theorem 2.2. Condition (C.2) is not essential, and the rest of the paper’s argument can be developed similarly even if the parameters are not real values. Note that condition (C.3) is a “strong shift” condition, and the assumption of Theorem 2.2 is not satisfied in the model with this condition imposed.

Hereafter, conditions (C.1)–(C.3) is assumed unless otherwise noted. By conditions (C.1)–(C.3), $a_{\Phi }(\varvec{p}) = a_{\Omega }(\varvec{p}) = 0$ holds and the previously defined $T_0$ and $\varphi _{\varvec{q}}$ can be rewritten as

$$\begin{aligned} T_0=\phi _{2,1}\phi _{2,2}(L_2+L_2^*), \quad \varphi _{{\varvec{q}}}=\phi _{2,1}\omega _{2,2}{\varvec{1}}_{\{-{\varvec{e}}_2\}}+\phi _{2,2}\omega _{2,1}{\varvec{1}}_{\{{\varvec{e}}_2\}}. \end{aligned}$$

Defining $\Lambda := 2|\phi _{2,1}\phi _{2,2}|$, $[-1,1]{\setminus } \sigma (T_0)\ne \emptyset$ holds because $\sigma (T_0) = [-\Lambda , \Lambda ]$ and

$$\begin{aligned} \Lambda = 2|\phi _{2,1}\phi _{2,2}| \le |\phi _{2,1}|^2 + |\phi _{2,2}|^2 < 1. \end{aligned}$$

We set ${{\mathbb {T}}}_-:=[-1,-\Lambda )$ and ${{\mathbb {T}}}_+:=(\Lambda ,1]$, then the domain of f is ${{\mathbb {T}}}_-\cup {\mathbb {T}}_+$.

3 Main results

In this section, we discuss the main results. The main results consist of two parts: one on localization and one on resonance. The models treated in this section are SSQWs with the strong shift defined in the previous section.

3.1 Sufficient conditions for localization of strong shift

In the next theorem, we present the necessary and sufficient conditions for f to have zero points.

Theorem 1.4

Under the conditions (C.1)–(C.3), the following conditions (1) and (2) are equivalent.

• (1) f has zero points on ${{\mathbb {T}}}_-\cup {{\mathbb {T}}}_+$.

• (2) $|\phi _{2,2}|<|\omega _{2,2}|$.

In particular, when either (i.e., both) of the above conditions holds, the zeros of f consist of $\lambda _0^+$ and $\lambda _0^-$ given by

$$\begin{aligned} \lambda _0^{\pm }=\pm \frac{2\omega _{2,1}\omega _{2,2}}{\sqrt{2\omega _{2,2}^2/\phi _{2,2}^2-1}}. \end{aligned}$$

Note that $\lambda _0^{+}\ne \lambda _0^{-}$ by condition (C.2). Furthermore, (2.5) of the eigenvalue correspondence indicates that U has at least four eigenvalues $e^{i\arccos \lambda _0^{\pm }}$ and $e^{-i\arccos \lambda _0^{\pm }}$.

Proof

First, we calculate $f(\lambda )$. By the Fourier transform from $\ell ^2({\mathbb {Z}}^2)$ to $L^2([0,2\pi )^2, d\varvec{k}/(2\pi )^2)$ and condition (C.1), we obtain $\hat{\varphi } _{\varvec{q}}({\varvec{k}})=2i\phi _{2,1}\omega _{2,2}\sin k_2$ and $\hat{T}_0({\varvec{k}})=2\phi _{2,1}\phi _{2,2}\cos k_2$ for ${\varvec{k}}=(k_1,k_2)\in [0,2\pi )^2$. We set $a:=\lambda /2\phi _{2,1}\phi _{2,2}$ for $\lambda \in {\mathbb {T}}_-\cup {\mathbb {T}}_+$. Note that $|a|>1$. Then, we have the following equation:

$$\begin{aligned} \frac{f(\lambda )}{2\phi _{2,1}\phi _{2,2}}&=a+\frac{1}{2\phi _{2,1}\phi _{2,2}}\langle \hat{\varphi } ,(\hat{T}_0(\cdot )-\lambda )^{-1}\hat{\varphi } \rangle \\&=a+\frac{1}{4\phi ^2_{2,1}\phi ^2_{2,2}}\int _{[0,2\pi )^2}\frac{d{\varvec{k}}}{(2\pi )^2}\frac{4\phi _{2,1}^2\omega _{2,2}^2\sin ^2k_2}{\cos k_2-a }\\&=a+\frac{\omega _{2,2}^2}{\phi ^2_{2,2}}\int _{[0,2\pi )}\frac{dk_2}{(2\pi )}\frac{\sin ^2k_2}{\cos k_2-a }\\&=a+\frac{\omega _{2,2}^2}{\phi _{2,2}^2}(-a+\textrm{sgn}(a)\sqrt{a^2-1}). \end{aligned}$$

Thus, if there exist zero points $\lambda _0\in {\mathbb {T}}_-\cup {\mathbb {T}}_+$ of f, i.e., $f(\lambda _0)=0$, then

$$\begin{aligned}&\Lambda = 2|\phi _{2,1}\phi _{2,2}| < |\lambda _0| \le 1, \end{aligned}$$

(3.1)

$$\begin{aligned}&\left( \frac{\omega _{2,2}^2}{\phi _{2,2}^2}-1\right) |a| = \frac{\omega _{2,2}^2}{\phi _{2,2}^2}\sqrt{a^2-1} \ (>0). \end{aligned}$$

(3.2)

Note that condition (3.1) and (3.2) leads to conditions

$$\begin{aligned}&|\phi _{2,2}| \ne |\omega _{2,2}|, \quad 2|\omega _{2,1}\omega _{2,2}|\le \sqrt{2\omega _{2,2}^2/\phi _{2,2}^2-1} \end{aligned}$$

(3.3)

$$\begin{aligned}&\text {and} \quad |\phi _{2,2}| < |\omega _{2,2}|, \end{aligned}$$

(3.4)

respectively, but (3.3) is fully subsumed by (3.4). From the above discussion, we obtain the following:

$$\begin{aligned} \exists \lambda _0\in {\mathbb {T}}_-\cup {\mathbb {T}}_+, \ f(\lambda _0) = 0\ \Longrightarrow \ |\phi _{2,2}| < |\omega _{2,2}|. \end{aligned}$$

(3.5)

The reverse direction of (3.5) can be easily seen. When the above conditions (1) or (and) (2) hold, solving equation (3.2) yields

$$\begin{aligned} \lambda _0=\pm \frac{2\omega _{2,1}\omega _{2,2}}{\sqrt{2\omega _{2,2}^2/\phi _{2,2}^2-1}}. \end{aligned}$$

$\square$

Theorem 1.5

Assume (2) in Theorem 3.1 under conditions (C.1)–(C.3). Let $\mu _0$ be one of the eigenvalues of U corresponding to the zero point $\lambda _0$ of f, i.e., $\mu _0 \in \{e^{\pm i\arccos \lambda _0}\}, \ f(\lambda _0)=0$. Then, the eigenvector $\Psi _{\mu _0}$ of U corresponding to the eigenvalue $\mu _0$ satisfies for all ${\varvec{x}} = (x_1, x_2)\in {\mathbb {Z}}^2$,

$$\begin{aligned} \Psi _{\mu _0}({\varvec{x}})=\begin{pmatrix} \chi _{1,1}({\varvec{x}})\left( \psi _{\lambda _0}({\varvec{x}})-{\varvec{1}}_{\{0\}}({\varvec{x}})\right) \\ -\mu _0\chi _{1,1}({\varvec{x}}-{\varvec{e}}_1)\left( \psi _{\lambda _0}({\varvec{x}}-{\varvec{e}}_1)-{\varvec{1}}_{\{0\}}({\varvec{x}}-{\varvec{e}}_1)\right) \\ \chi _{2,1}({\varvec{x}})\left( \psi _{\lambda _0}({\varvec{x}})-{\varvec{1}}_{\{0\}}({\varvec{x}})\right) -\mu _0\chi _{2,2}({\varvec{x}}+{\varvec{e}}_2)\left( \psi _{\lambda _0}({\varvec{x}}+{\varvec{e}}_2)-{\varvec{1}}_{\{0\}}({\varvec{x}}+{\varvec{e}}_2)\right) \\ \chi _{2,2}({\varvec{x}})\left( \psi _{\lambda _0}({\varvec{x}})-{\varvec{1}}_{\{0\}}({\varvec{x}})\right) -\mu _0\chi _{2,1}({\varvec{x}}-{\varvec{e}}_2)\left( \psi _{\lambda _0}({\varvec{x}}-{\varvec{e}}_2)-{\varvec{1}}_{\{0\}}({\varvec{x}}-{\varvec{e}}_2)\right) \\ \end{pmatrix}, \end{aligned}$$

(3.6)

where

$$\begin{aligned}&\psi _{\lambda _0}({\varvec{x}})=[(T_0-\lambda _0)^{-1}\varphi _{\varvec{q}}]({\varvec{x}})={\varvec{1}}_{\{x_1=0\}}({\varvec{x}})\int _{[0,2\pi )}\frac{2i\omega _{2,2}\phi _{2,1}\sin k_2}{2\phi _{2,1}\phi _{2,2}\cos k_2-\lambda _0}e^{ik_2x_2}\frac{dk_2}{2\pi }. \end{aligned}$$

(3.7)

Proof

We set $P:=I-|{\varvec{1}}_{\{ \textbf{0 }\}}\rangle \langle {\varvec{1}}_{\{ \textbf{0 }\}}|$ and $\psi _{\lambda _0}:=(T_0-\lambda _0)\varphi _{\varvec{q}}$. By conditions (C.1) and (C.2), $\psi _{\lambda _0}\in \textrm{Ran}P\backslash \{0\}$ holds. Recall the discussion in [4] using the Feshbach map $F(\lambda )$. Since $\lambda _0$ is a zero of f, $\lambda _0$ is also an eigenvalue of T, and $F(\lambda _0)\psi _{\lambda _0}=0$ holds. Furthermore, by a theorem on Feshbach maps in the paper [2, Theorem II.1], the eigenvector of T corresponding to the eigenvalue $\lambda _0$ can be written as follows:

$$\begin{aligned}{}[P-(P^{\perp }(T-\lambda _0)P^{\perp })^{-1}P^{\perp }TP]\psi _{\lambda _0}. \end{aligned}$$

(3.8)

By [4, Lemma 4.1 and Proposition 4.3], we have $(P^{\perp }(T-\lambda _0)P^{\perp })^{-1}=-\lambda _0^{-1} P^{\perp }$ and $P^{\perp }TP=|{\varvec{1}}_{\{0\}}\rangle \langle \varphi _{{\varvec{q}}}|.$ Thus,

$$\begin{aligned}{}[P-(P^{\perp }(T-\lambda _0)P^{\perp })^{-1}P^{\perp }TP]\psi _{\lambda _0}=\psi _{\lambda _0}+\frac{\langle \varphi _{\varvec{q}},\psi _{\lambda _0}\rangle }{\lambda _0}{\varvec{1}}_{\{ 0 \}}. \end{aligned}$$

Since $\lambda _0$ is a zero point of f, $f(\lambda _0)=\lambda _0+\langle \varphi _{\varvec{q}}, \psi _{\lambda _0}\rangle =0$ holds. Thus we have $[P-(P^{\perp }(T-\lambda )P^{\perp })^{-1}P^{\perp }TP]\psi _{\lambda _0}=\psi _{\lambda _0}-{\varvec{1}}_{\{0\}}.$ Next, by [24], the following vector is the eigenvector of U corresponding to the eigenvalue $\mu _0 \in \{e^{\pm i\arccos \lambda _0}\}$:

$$\begin{aligned} \Psi _{\mu _0}=(1-\mu _0S)d^*(\psi _{\lambda _0}-{\varvec{1}}_{\{\textbf{0}\}}). \end{aligned}$$

By direct calculation, we obtain (3.6) and (3.7). Note that (3.7) is obtained by Fourier transform and inverse Fourier transform. $\square$

Remark 3.3

Theorem 3.1 shows that SSQWs with the strong shift are localized as in [4]. Moreover, we see that the localized location ${\varvec{x}}=(x_1,x_2)$ always satisfies $x_1=0$ or $x_1 = 1$ by Theorem 3.2. The cause is condition $\omega _{1,2}=\phi _{1,2}=0$ in (C.1). If this condition is removed, localization is expected to occur even for $x_1 \ne 0$ and $x_1 \ne 1$.

3.2 Threshold resonances and generalized eigenfunction

In this subsection, we consider the case of $|\phi _{2,2}|=|\omega _{2,2}|$. When $|\phi _{2,2}|=|\omega _{2,2}|$ holds, we obtain a quantity called the threshold resonance of U as the quantity corresponding to the eigenvalue $\mu _0$ of U obtained in the case $|\phi _{2,2}|<|\omega _{2,2}|$ in Theorem 3.2. First, we define resonance and threshold resonance.

Definition 1.7

(Threshold resonances) We can naturally extend U to $U_{\infty }$ acting on ${{\mathcal {H}}}_{\infty }:=\ell ^{\infty }({\mathbb {Z}}^2;{\mathbb {C}} ^4)$. Let $\Psi \in {\mathcal {H}}_{\infty }$ be a solution of $U_{\infty }\Psi =m \Psi$, where m is a complex number. If $\Psi \in {{\mathcal {H}}}_{\infty }\backslash {{\mathcal {H}}}$, then we say that m is a resonance of U and $\Psi$ is a generalized eigenfunction of U corresponding to m. If m is a resonance of U and $m\in \partial \sigma _{\textrm{ess}}(U)$ holds, then m is called a threshold resonance.

Remark 3.5

Note that $\ell ^2({{\mathbb {Z}}}^2;{{\mathbb {C}}}^4)\subsetneq \ell ^{\infty }({{\mathbb {Z}}}^2;{{\mathbb {C}}}^4)$ and $\Vert \Psi \Vert _{\infty }\le \Vert \Psi \Vert _{2}$ for any $\Psi \in \ell ^2({{\mathbb {Z}}}^2;{{\mathbb {C}}}^4)$.

We set $m_{\pm }:=e^{i\arccos (\pm \Lambda )}$. Under the conditions (C.1)–(C.3), $\sigma _{\textrm{ess}}(T)=[-\Lambda ,\Lambda ]$ holds by (2.3). Thus $m_{\pm }^\#\in \partial \sigma _{\textrm{ess}}(U)$ by (2.1), where $m_{\pm }^\#$ is $m_{\pm }$ or $m_{\pm }^*.$ We use (3.6) to construct a candidate for the generalized eigenfunction of U corresponding to $m_{\pm }^\#$. First, We set $\psi _{\pm \Lambda }({\varvec{x}} ):=\lim _{\lambda \rightarrow \pm \Lambda \pm 0}\psi _{\lambda } ({\varvec{x}} )$ for each ${\varvec{x}} \in {\mathbb {Z}}^2$ and

$$\begin{aligned} \Sigma _{\pm }:=\{ e^{i\arccos \lambda } \mid \lambda \in {{\mathbb {T}}}_{\pm }\},\quad \Sigma _{\pm }^*:=\{ e^{-i\arccos \lambda } \mid \lambda \in {{\mathbb {T}}}_{\pm }\}. \end{aligned}$$

Then, $\bigcup _{\star =\pm }\left( \Sigma _{\star }\cup \Sigma _{\star }^*\right) ={{\mathbb {S}}}^1\backslash \sigma _{\textrm{ess}}(U)$ holds, where ${\mathbb {S}}^1$ is the unit circle on ${\mathbb {C}}$. Also, we define a vector

$$\begin{aligned} \Psi _{\mu }:=(1-\mu S)d^*(\psi _{\lambda } -{\varvec{1}}_{\{\varvec{0}\}})\in {\mathcal {H}}=\ell ^2({\mathbb {Z}}^2;{\mathbb {C}}^4) \end{aligned}$$

(3.9)

for $\mu \in {{\mathbb {S}}}^1\backslash \sigma _{\textrm{ess}}(U)$ and $\lambda =\textrm{Re}\, \mu \in {{\mathbb {T}}}_-\cup {{\mathbb {T}}}_+.$ Then, the following equations are satisfied in the same way as in the proof of Theorem 3.2:

$$\begin{aligned}&\psi _{\lambda }({\varvec{x}})={\varvec{1}}_{\{x_1=0\}}({\varvec{x}})\int _{[0,2\pi )}\frac{2i\omega _{2,2}\phi _{2,1}\sin k_2}{2\phi _{2,1}\phi _{2,2}\cos k_2-\lambda }e^{ik_2x_2}\frac{dk_2}{2\pi }, \end{aligned}$$

(3.10)

$$\begin{aligned}&\Psi _{\mu }({\varvec{x}})=\begin{pmatrix} \chi _{1,1}({\varvec{x}})\left( \psi _{\lambda }({\varvec{x}})-{\varvec{1}}_{\{0\}}({\varvec{x}})\right) \\ -\mu \chi _{1,1}({\varvec{x}}-{\varvec{e}}_1)\left( \psi _{\lambda }({\varvec{x}}-{\varvec{e}}_1)-{\varvec{1}}_{\{0\}}({\varvec{x}}-{\varvec{e}}_1)\right) \\ \chi _{2,1}({\varvec{x}})\left( \psi _{\lambda }({\varvec{x}})-{\varvec{1}}_{\{0\}}({\varvec{x}})\right) -\mu \chi _{2,2}({\varvec{x}}+{\varvec{e}}_2)\left( \psi _{\lambda }({\varvec{x}}+{\varvec{e}}_2)-{\varvec{1}}_{\{0\}}({\varvec{x}}+{\varvec{e}}_2)\right) \\ \chi _{2,2}({\varvec{x}})\left( \psi _{\lambda }({\varvec{x}})-{\varvec{1}}_{\{0\}}({\varvec{x}})\right) -\mu \chi _{2,1}({\varvec{x}}-{\varvec{e}}_2)\left( \psi _{\lambda }({\varvec{x}}-{\varvec{e}}_2)-{\varvec{1}}_{\{0\}}({\varvec{x}}-{\varvec{e}}_2)\right) \\ \end{pmatrix}, \end{aligned}$$

(3.11)

where $\mu \in {{\mathbb {S}}}^1\backslash \sigma _{\textrm{ess}}(U)$ and $\lambda =\textrm{Re}\, \mu \in {{\mathbb {T}}}_-\cup {{\mathbb {T}}}_+.$

Also, we define

$$\begin{aligned} \Psi _{m _{\pm }^\#}({\varvec{x}} ):=\lim _{{\mathop {\mu \in \Sigma _{\pm }^\#}\limits ^{\mu \rightarrow m_{\pm }^\#}}}\Psi _{\mu }({\varvec{x}} ),\quad {\varvec{x}}\in {{\mathbb {Z}}}^2. \end{aligned}$$

The following are the main results of this subsection.

Theorem 1.9

Under the conditions (C.1)–(C.3), we assume $|\phi _{2,2}|=|\omega _{2,2}|$. Then, $m_{\pm }^\#$ are threshold resonances of U, i.e.,

$$\begin{aligned} \Psi _{m_{\pm }^\#}\in {{\mathcal {H}}}_{\infty }\backslash {{\mathcal {H}}}, \ U_{\infty }\Psi _{m_{\pm }^\#}=m_{\pm }^\#\Psi _{m_{\pm }^\#}. \end{aligned}$$

Remark 3.7

In this paper, $p_j=0$ and $q_j=1$ are fixed. No results are known for intermediate states $p_j$ and $q_j$. However, if we could evaluate various integrals as they appear in this paper, we might obtain results similar to Theorems 3.1 and  3.6 for such $p_j$ and $q_j$. The following points are important in this paper. First, we proved that localization occurs for parameters $p_j$ and $q_j$, which at a glance do not localize quantum walks, and then we explicitly wrote down eigenvalues, eigenvectors, resonances, and generalized eigenfunctions. Let us emphasize that this is the result achieved by fixing $p_j=0, q_j=1$. In particular, the specific presentation of eigenvalues and eigenvectors not only presents where localization occurs but also leads to the discovery of resonances and generalized eigenfunctions. This paper describes this method. Now that resonances and generalized eigenfunctions have been explicitly shown, we also know that when eigenvectors converge to generalized eigenfunctions, the meaning of that convergence is not $\ell ^2$, but $\ell ^{\infty }$.

Proposition 1.11

Under the conditions (C.1)–(C.3), we assume $|\phi _{2,2}|=|\omega _{2,2}|$. Then, the following equations hold for each $\varvec{x}\in {\mathbb {Z}}^2$:

$$\begin{aligned} \psi _{\pm \Lambda }({\varvec{x}})=(\pm \textrm{sgn}(\phi _{2,1}\phi _{2,2}))^{x_2}\textrm{sgn}(x_2)\frac{\omega _{2,2}}{\phi _{2,2}}\varvec{1}_{\{x_1=0\}\cup \{x_2\not = 0\}}({\varvec{x}}). \end{aligned}$$

(3.12)

In particular, the above equations (3.12) show that $\Psi _{m_{\pm }^\#}\in {{\mathcal {H}}}_{\infty }\backslash {{\mathcal {H}}}$ holds.

Proof

For $\lambda \in {\mathbb {T}}_-\cup {\mathbb {T}}_+$ and $x\in {\mathbb {Z}}$, we set

$$\begin{aligned} I_{\pm }(x)&:=\lim _{\lambda \rightarrow \pm \Lambda \pm 0}\int _{-\pi }^{\pi }\frac{-i\sin k}{2\phi _{2,1}\phi _{2,2}\cos k-\lambda }e^{ikx}\frac{dk}{2\pi } \\&= 2\lim _{\lambda \rightarrow \pm \Lambda \pm 0}\int _0^{\pi }\frac{\sin k\sin xk}{2\phi _{2,1}\phi _{2,2}\cos k-\lambda }\frac{dk}{2\pi }. \end{aligned}$$

Then, $\psi _{\pm \Lambda }({\varvec{x}})=\lim _{\lambda \rightarrow \pm \Lambda \pm 0}\psi _{\lambda }(\varvec{x}) = -2\omega _{2,2}\phi _{2,1}{\varvec{1}}_{\{x_1=0\}}({\varvec{x}})I_{\pm }(x_2)$ by (3.10).

First, we consider the case where $\phi _{2,1}\phi _{2,2}>0$ holds. In this case, since $\Lambda =2\phi _{2,1}\phi _{2,2}<\lambda$, we have

$$\begin{aligned} \left| \frac{\sin k\sin xk}{2\phi _{2,1}\phi _{2,2}\cos k-\lambda }\right| \le \frac{1}{\Lambda }\frac{|\sin k\sin xk|}{1-\cos k} ,\quad k\in [0,\pi ],\ x\in {\mathbb {Z}}. \end{aligned}$$

(3.13)

The right-hand side of (3.13) is found to be bounded with respect to k on $(0,\pi ]$ by continuity and the following calculation:

$$\begin{aligned} \lim _{k\rightarrow 0+0}\frac{|\sin k\sin xk|}{1-\cos k}&=\lim _{k\rightarrow 0+0}\frac{\sin xk}{\sin k}\frac{k}{xk}(1+\cos k)x=2x,\quad x\in {{\mathbb {Z}}}\backslash \{0\}. \end{aligned}$$

Thus, the following inequality holds for all $x\in {{\mathbb {Z}}}$:

$$\begin{aligned} \int _0^{\pi }\frac{1}{\Lambda }\frac{|\sin k\sin xk|}{1-\cos k}\frac{dk}{\pi }<\infty . \end{aligned}$$

Therefore, by the dominated convergence theorem, we obtain

$$\begin{aligned} I_{+}(x)=-\frac{1}{\Lambda \pi }\int _0^{\pi }\frac{\sin k\sin xk}{1-\cos k}dk \end{aligned}$$

in the case of $\phi _{2,1}\phi _{2,2}>0$. We set

$$\begin{aligned} J_{\pm }(x):=\int _0^{\pi }\frac{\sin k\sin xk}{1\pm \cos k}dk,\quad x\in {\mathbb {Z}}, \end{aligned}$$

then $I_+(x)=-(1/\Lambda \pi )J_-(x)$ for $x\in {{\mathbb {Z}}}$ in the case of $\phi _{2,1}\phi _{2,2}>0$.

In the other cases, we have the above table for $I_{\pm }$ in the same way.

	$\phi _{2,1}\phi _{2,2}>0$	$\phi _{2,1}\phi _{2,2}<0$
$I_+(x)$	$-\frac{1}{\Lambda \pi }J_-(x)$	$-\frac{1}{\Lambda \pi }J_+(x)$
$I_-(x)$	$\frac{1}{\Lambda \pi }J_+(x)$	$\frac{1}{\Lambda \pi }J_-(x)$

For $x\ge 4$, we have

$$\begin{aligned} J_{\pm }(x)&=\int _0^{\pi }\frac{\sin k\left( \sin k \cos (x-1)k+\cos k\sin (x-1)k\right) }{1\pm \cos k}dk\\&=\int _0^{\pi }\left( 1\mp \cos k\right) \cos (x-1)k\ dk\\&\quad +\int _0^{\pi }\frac{\sin k\cos k}{1\pm \cos k}\left( \sin k\cos (x-2)k+\cos k\sin (x-2)k\right) dk\\&=\int _0^{\pi }\frac{\left( \sin k-\sin ^3k\right) \sin (x-2)k}{1\pm \cos k}dk\\&=J_{\pm }(x-2)\\&={\left\{ \begin{array}{ll} J_{\pm }(2),\quad x:\text {even}, \\ J_{\pm }(3),\quad x:\text {odd}. \end{array}\right. } \end{aligned}$$

Note that $J_{\pm }(-x) = - J_{\pm }(x)$ for all $x\in {\mathbb {Z}}$. A direct calculation using these relational equations for $J_{\pm }(x)$ yields the following equations:

$$\begin{aligned} J_{\pm }(x)= {\left\{ \begin{array}{ll} \textrm{sgn}(x)(\mp 1)^{x+1}\pi , &{} x \ne 0,\\ 0, &{} x = 0. \end{array}\right. } \end{aligned}$$

Hence by (3.10), we obtain (3.12). In particular, because $|\psi _{\pm \Lambda }({\varvec{x}})|=\varvec{1}_{\{x_1=0\}\cup \{x_2\not = 0\}}({\varvec{x}})$ and (3.11), we get $\Psi _{m_{\pm }^\#}\in {{\mathcal {H}}}_{\infty }\backslash {{\mathcal {H}}}.$ $\square$

We set $\varepsilon (\lambda ):=-\langle \varphi _{\varvec{q}},\psi _{\lambda }\rangle /\lambda$ for all $\lambda \in {\mathbb {T}}_-\cup {\mathbb {T}}_+.$

Lemma 1.12

We assume (C.1)–(C.3) and $|\phi _{2,2}|=|\omega _{2,2}|$. Then, $\lim _{\lambda \rightarrow \pm \Lambda \pm 0}\varepsilon (\lambda )=1$ holds.

Proof

By the assumption $|\phi _{2,2}|=|\omega _{2,2}|$ and the proof of Theorem 3.1, we have the following equation:

$$\begin{aligned} \frac{f(\lambda )}{2\phi _{2,1}\phi _{2,2}}=\textrm{sgn}(a)\sqrt{a^2-1},\quad \text { for } \lambda \in {\mathbb {T}}_-\cup {\mathbb {T}}_+, \end{aligned}$$

where $a=\lambda /2\phi _{2,1}\phi _{2,2}$. This means that $\lim _{\lambda \rightarrow \pm \Lambda \pm 0}f(\lambda )=0$. Since $\varepsilon (\lambda )=-f(\lambda )/\lambda +1$ and $\pm \Lambda \not =0,$ we have $\lim _{\lambda \rightarrow \pm \Lambda \pm 0}\varepsilon (\lambda )=1.$ $\square$

Proposition 1.13

Under the conditions (C.1)–(C.3), we assume $|\phi _{2,2}|=|\omega _{2,2}|$. For all $\mu \in {{\mathbb {S}}}^1\backslash \sigma _{\textrm{ess}}(U)$, we set $\lambda :=\textrm{Re}\, \mu \in {\mathbb {T}}_-\cup {\mathbb {T}}_+$. Then, the following equation holds:

$$\begin{aligned} (U-\mu )\Psi _{\mu }=(\varepsilon (\lambda )-1)\{ 2\mu Sd^*\varphi _{\varvec{q}}+(U-\mu )(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}\}. \end{aligned}$$

In particular,

$$\begin{aligned} \lim _{{\mathop {\mu \in \Sigma _{\pm }^\#}\limits ^{\mu \rightarrow m_{\pm }^\#}}}\Vert (U-\mu )\Psi _{\mu }\Vert _2=0. \end{aligned}$$

Before proving Proposition 3.10, we prepare some related lemmas. Following Eq. (3.8), we define $h_{\lambda }$ as follows:

$$\begin{aligned} h_{\lambda }:=[P-(P^{\perp }(T-\lambda )P^{\perp })^{-1}P^{\perp }TP]\psi _{\lambda }. \end{aligned}$$

Lemma 1.14

$h_{\lambda }=\psi _{\lambda }-\varepsilon (\lambda ){\varvec{1}}_{\{\textbf{0}\}}$.

Proof

By the same argument as the proof of Theorem 3.2, we have $(P^{\perp }(T-\lambda )P^{\perp })^{-1}=-\lambda ^{-1}P^{\perp }$ and $P^{\perp }TP=|{\varvec{1}}_{\{\textbf{0}\}}\rangle \langle \varphi _{\varvec{q}}|.$ Moreover, $\psi _{\lambda }\in \textrm{Ran}P$ holds. Thus, we obtain

$$\begin{aligned} h_{\lambda }=P\psi _{\lambda }-\lambda ^{-1}P^{\perp }TP \psi _{\lambda }=\psi _{\lambda }-\varepsilon (\lambda ){\varvec{1}}_{ \{\textbf{0}\}}. \end{aligned}$$

$\square$

Lemma 1.15

$(T-\lambda )h_{\lambda }=(1-\varepsilon (\lambda ))\varphi _{\varvec{q}}.$

Proof

By [4, Lemma 3.2], the discriminant operator T can be written as follows:

$$\begin{aligned} T=\chi _{2,1}L_2\chi _{2,2}+\chi _{2,2}L_2^*\chi _{2,1}, \end{aligned}$$

(3.14)

where $\chi _{2,1}$ and $\chi _{2,2}$ are multiplication operators by $\chi _{2,1}({\varvec{x}})$ and $\chi _{2,2}({\varvec{x}})$, respectively. Using Eqs. (2.2) and (3.14), we obtain

$$\begin{aligned}&(T_0h_{\lambda })({\varvec{x}})=\phi _{2,1}\phi _{2,2}((L_2h_{\lambda }) ({\varvec{x}})+(L_2^*h_{\lambda } )({\varvec{x}})),\\&(Th_{\lambda })({\varvec{x}})=\chi _{2,1}({\varvec{x}})\chi _{2,2}({\varvec{x}}+{\varvec{e}}_2)(L_2h_{\lambda })({\varvec{x}})+\chi _{2,2}({\varvec{x}})\chi _{2,1}({\varvec{x}}-{\varvec{e}}_2)(L_2^*h_{\lambda })({\varvec{x}}). \end{aligned}$$

We define an operator W and a function $\beta$ as $W:=T-T_0$ and $\beta ({\varvec{x}}):=\chi _{2,1}({\varvec{x}})\chi _{2,2}({\varvec{x}}+{\varvec{e}}_2)-\phi _{2,1}\phi _{2,2}$. Here, noting $\psi _{\lambda }\in \textrm{Ran}P$, we have $\psi _{\lambda }(0)=0$. With this fact and Lemma 3.11, we obtain

$$\begin{aligned} (Wh_{\lambda })({\varvec{x}})&=\left( \beta (\textbf{0}){\varvec{1}}_{\{\textbf{0}\}}({\varvec{x}})+\beta (-{\varvec{e}}_2){\varvec{1}}_{\{-{\varvec{e}}_2\}}({\varvec{x}}) \right) (L_2h_{\lambda })({\varvec{x}})\nonumber \\&\quad +\left( \beta (-{\varvec{e}}_2){\varvec{1}}_{\{\textbf{0}\}}({\varvec{x}})+\beta (\textbf{0}){\varvec{1}}_{\{{\varvec{e}}_2\}}({\varvec{x}}) \right) (L_2^*h_{\lambda })({\varvec{x}})\nonumber \\&=\beta (\textbf{0})\psi _{\lambda }({\varvec{e}}_2){\varvec{1}}_{\{\textbf{0}\}}({\varvec{x}})-\varepsilon (\lambda )\beta (-{\varvec{e}}_2){\varvec{1}}_{\{-{\varvec{e}}_2 \}}({\varvec{x}})\nonumber \\&\quad +\beta (-{\varvec{e}}_2)\psi _{\lambda }(-{\varvec{e}}_2){\varvec{1}}_{\{\textbf{0}\}}({\varvec{x}})-\varepsilon (\lambda )\beta (\textbf{0}){\varvec{1}}_{\{ {\varvec{e}}_2 \}}({\varvec{x}}). \end{aligned}$$

(3.15)

On the other hand, we get the following equation by (2.4) and Lemma 3.11:

$$\begin{aligned} (T_0-\lambda )h_{\lambda }=\varphi _{\varvec{q}}-\varepsilon (\lambda )\phi _{2,1}\phi _{2,2}\left( {\varvec{1}}_{ \{ -{\varvec{e}}_2 \}}+ {\varvec{1}}_{ \{ {\varvec{e}}_2 \}}\right) +\lambda \varepsilon (\lambda ){\varvec{1}}_{\{\textbf{0}\}}. \end{aligned}$$

(3.16)

Combining (3.15) and (3.16), we have

$$\begin{aligned} (T-\lambda )h_{\lambda }&=(T_0-\lambda )h_{\lambda }+Wh_{\lambda }\\&=\varphi _{\varvec{q}}+ A_1 {\varvec{1}}_{\{\textbf{0}\}} -\varepsilon (\lambda ) A_2 {\varvec{1}}_{\{ -{\varvec{e}}_2 \}} -\varepsilon (\lambda ) A_3 {\varvec{1}}_{\{ {\varvec{e}}_2 \}}, \end{aligned}$$

where

$$\begin{aligned}&A_1:=\beta (\textbf{0})\psi _{\lambda }({\varvec{e}}_2)+\beta (-{\varvec{e}}_2)\psi _{\lambda }(-{\varvec{e}}_2)+\lambda \varepsilon (\lambda ),\\&A_2:=\beta (-{\varvec{e}}_2)+\phi _{2,1}\phi _{2,2}, \\&A_3:=\beta (\textbf{0})+\phi _{2,1}\phi _{2,2}. \end{aligned}$$

By (2.4),

$$\begin{aligned} -\lambda \varepsilon (\lambda )&= \langle \omega _{2,1}\phi _{2,2}{\varvec{1}}_{\{ {\varvec{e}}_2 \}}+\omega _{2,2}\phi _{2,1}{\varvec{1}}_{ \{ -{\varvec{e}}_2 \} },\psi _{\lambda }\rangle \\&=\omega _{2,1}\phi _{2,2}\psi _{\lambda }({\varvec{e}}_2)+\omega _{2,2}\phi _{2,1}\psi _{\lambda }(-{\varvec{e}}_2)\\&=\beta (\textbf{0})\psi _{\lambda }({\varvec{e}}_2)+\beta (-{\varvec{e}}_2)\psi _{\lambda }(-{\varvec{e}}_2)+\phi _{2,1}\phi _{2,2}\left( \psi _{\lambda }({\varvec{e}}_2)+\psi _{\lambda }(-{\varvec{e}}_2) \right) . \end{aligned}$$

Since $2\cos k\cdot (-2i\omega _{2,1}\phi _{2,2}\sin k)/(2\phi _{2,1}\phi _{2,2}\cos k-\lambda )$ is an odd function, we obtain

$$\begin{aligned} \psi _{\lambda }({\varvec{e}}_2)+\psi _{\lambda }(-{\varvec{e}}_2)=\int _{-\pi }^{\pi } 2\cos k_2\cdot \frac{-2i\omega _{2,1}\phi _{2,2}\sin k_2}{2\phi _{2,1}\phi _{2,2}\cos k_2-\lambda } \frac{dk_2}{2\pi }=0 \end{aligned}$$

by (3.10). Thus we get $A_1=\langle \varphi _{\varvec{q}},\psi _{\lambda }\rangle +\lambda \varepsilon (\lambda )=0$. On the other hand, we obtain $A_2=\omega _{2,2}\phi _{2,1},\ A_3=\omega _{2,1}\phi _{2,2}$ by the definition of $\beta .$ Hence, we have

$$\begin{aligned} (T-\lambda )h_{\lambda }=(1-\varepsilon (\lambda ))\varphi _{\varvec{q}}. \end{aligned}$$

$\square$

Proof of Proposition 3.10

Lemma 3.11 and (3.9) lead to the following equation:

$$\begin{aligned} \Psi _{\mu }=(1-\mu S)d^*h_{\lambda }+(\varepsilon (\lambda )-1)(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}. \end{aligned}$$

Then, we have

$$\begin{aligned} (U-\mu )\Psi _{\mu }&=(SC-\mu )(1-\mu S)d^*h_{\lambda }+(\varepsilon (\lambda )-1)(U-\mu )(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}\\&=SCd^*h_{\lambda }-\mu d^*h_{\lambda }-\mu SCSd^*h_{\lambda }+\mu ^2Sd^*h_{\lambda }+(\varepsilon (\lambda )-1)(U-\mu )(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}\\&= (1-2\mu \lambda +\mu ^2)Sd^*h_{\lambda }+2\mu (\varepsilon (\lambda )-1)Sd^*\varphi _{\varvec{q}}+(\varepsilon (\lambda )-1)(U-\mu )(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}. \end{aligned}$$

The last equality in the above equation is due to the fact that $SCd^*h_{\lambda }=Sd^*h_{\lambda },\ SCSd^*=S(2d^*d-1)Sd^*=2Sd^*T-d^*$ and Lemma 3.12. Also, the equation $1-2\mu \lambda +\mu ^2=0$ holds, since $\lambda =\textrm{Re}\ \mu$ and $|\mu |=1.$ Thus,

$$\begin{aligned} (U-\mu )\Psi _{\mu }=(\varepsilon (\lambda )-1)\{ 2\mu Sd^*\varphi _{\varvec{q}}+(U-\mu )(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}\}. \end{aligned}$$

Therefore, by $|\mu |=1$, Lemma 3.9 and unitarity of U and S, we obtain

$$\begin{aligned} \lim _{{\mathop {\mu \in \Sigma _{\pm }^\#}\limits ^{\mu \rightarrow m_{\pm }^\#}}}\Vert (U-\mu )\Psi _{\mu }\Vert _2&= \lim _{\lambda \rightarrow \pm \Lambda \pm 0}|\varepsilon (\lambda )-1|\Vert 2\mu Sd^*\varphi _{\varvec{q}}+(U-\mu )(1-\mu S)d^*{\varvec{1}}_{\{ \varvec{0}\}}\Vert _2\\&\le \lim _{\lambda \rightarrow \pm \Lambda \pm 0}2|\varepsilon (\lambda )-1|\left( \Vert Sd^*\varphi _{\varvec{q}}\Vert _2+2\Vert d^*{\varvec{1}}_{\{ \varvec{0}\}}\Vert _2 \right) =0. \end{aligned}$$

$\square$

We are now ready to prove our main result on threshold resonances in this subsection, Theorem 3.6.

Proof of Theorem 3.6

In what follows, we show that $m_+$ is a threshold resonance of U. By the triangle inequality, for $\mu \in \Sigma _+$ and ${\varvec{x}}\in {{\mathbb {Z}}}^2,$ we obtain

$$\begin{aligned} \Vert (U_{\infty }\Psi _{m_{+}})({\varvec{x}} )-m_{+}\Psi _{m_{+}}({\varvec{x}} )\Vert _{{\mathbb {C}} ^4}&\le \Vert (U_{\infty }\Psi _{m_{+}})({\varvec{x}} )-(U_{\infty }\Psi _{\mu })({\varvec{x}} )\Vert _{{\mathbb {C}} ^4} \end{aligned}$$

(3.17)

$$\begin{aligned}&\quad +\Vert (U_{\infty }\Psi _{\mu })({\varvec{x}} )-\mu \Psi _{\mu }({\varvec{x}} ) \Vert _{{\mathbb {C}} ^4} \end{aligned}$$

(3.18)

$$\begin{aligned}&\quad +\Vert \mu \Psi _{\mu }({\varvec{x}} ) -m_{+}\Psi _{m_{+}}({\varvec{x}})\Vert _{{\mathbb {C}} ^4}. \end{aligned}$$

(3.19)

By the definition of $\Psi _{m_{+}}$, the term (3.19) converges to 0 as $\mu \rightarrow m_{+}$. Noting Remark 3.5, the term (3.18) can be evaluated from above as follows:

$$\begin{aligned} \Vert (U_{\infty }\Psi _{\mu })({\varvec{x}} )-\mu \Psi _{\mu }({\varvec{x}} ) \Vert _{{\mathbb {C}} ^4}\le \Vert U_{\infty }\Psi _{\mu }-\mu \Psi _{\mu } \Vert _{\infty }\le \Vert U\Psi _{\mu }-\mu \Psi _{\mu } \Vert _{2}. \end{aligned}$$

(3.20)

By Proposition 3.10, the right-hand side of (3.20) converges to 0 as $\mu \rightarrow m_{+}.$ Then, the term (3.18) converges to 0 as $\mu \rightarrow m_{+}.$ Finally, we prove that the term (3.17) converges to 0 as $\mu \rightarrow m_+.$ Let $\Phi _{\mu }:=\Psi _{\mu }-\Psi _{m_+}$ and $C_{\infty }$ be the natural extension of C acting on ${{\mathcal {H}}}_{\infty }$. We have

$$\begin{aligned} (U_{\infty }\Phi _{\mu })({\varvec{x}} )=\begin{pmatrix} (C_{\infty }\Phi _{\mu })_2({\varvec{x}} +{\varvec{e}}_1)\\ (C_{\infty }\Phi _{\mu })_1({\varvec{x}} -{\varvec{e}}_1)\\ (C_{\infty }\Phi _{\mu })_4({\varvec{x}} +{\varvec{e}}_2)\\ (C_{\infty }\Phi _{\mu })_3({\varvec{x}} -{\varvec{e}}_2)\\ \end{pmatrix}, \end{aligned}$$

where $(C_{\infty }\Phi _{\mu })_i({\varvec{x}} )$ is the i-th component of $(C_{\infty }\Phi _{\mu })({\varvec{x}} )$ for each ${\varvec{x}} \in {\mathbb {Z}}^2$, $i=1,2,3,4$. Since $C_{\infty }$ is one-defect, there exists a constant $c>0$ such that $|(C_{\infty }\Phi _{\mu })_i({\varvec{x}} )|\le c|(\Phi _{\mu })_i({\varvec{x}} )|$ for ${\varvec{x}} \in {\mathbb {Z}}^2,\ i=1,2,3,4.$ Thus, we obtain

$$\begin{aligned} \Vert (U_{\infty }\Phi _{\mu })({\varvec{x}} )\Vert _{{\mathbb {C}} ^4}^2&\le c\{ |(\Phi _{\mu })_1({\varvec{x}} -{\varvec{e}}_1)|^2+|(\Phi _{\mu })_2({\varvec{x}} +{\varvec{e}}_1)|^2+|(\Phi _{\mu })_3({\varvec{x}} -{\varvec{e}}_2)|^2+|(\Phi _{\mu })_4({\varvec{x}} +{\varvec{e}}_2)|^2 \} \\&\rightarrow 0 \end{aligned}$$

as $\mu \rightarrow m_{+}.$ This means that the term (3.17) convergences to 0 as $\mu \rightarrow m_+.$ Hence, $\Vert (U_{\infty }\Psi _{m_+})({\varvec{x}} )-m_+\Psi _{m_+}({\varvec{x}} )\Vert _{{\mathbb {C}} ^4}=0$ holds for all $\varvec{x}\in {\mathbb {Z}}^2$, i.e.,

$$\begin{aligned} U_{\infty }\Psi _{m_{+}} =m_{+}\Psi _{m_{+}}. \end{aligned}$$

In addition to the above results, by Proposition 3.8, $m_+$ is a threshold resonance of U. For other threshold resonances, the theorem can be proved in the same way as above. $\square$