**4. Asymptotics of Energy**

If *ω* ∈ *∂B*, then by Corollary 3, it is immediately obtained that

$$\lim\_{M \to \infty} \frac{\mathcal{E}\_M(\omega)}{M} = \frac{2}{3}. \tag{20}$$

Let us consider the case of *ω* ∈ *Bin* ∪ *Bout* as follows. Note that

$$\lambda\_{\pm} = \begin{cases} \text{sgn}(\cos k) e^{\pm \theta} & : \omega \in B\_{\text{out}}, \\ e^{\pm i\theta} & : \omega \in B\_{\text{in}}. \end{cases}$$

where (1/|*a*|) cos *k* = cosh *θ* (*ω* ∈ *Bout*), while (1/|*a*|) cos *k* = cos *θ* (*ω* ∈ *Bin*) such that sin *θ* > 0 and sinh *θ* > 0. To observe the asymptotics of E*M*(*ω*) for *ω* ∈/ *∂B*, we rewrite E*M*(*ω*) as follows:

$$\mathcal{L}\_M(\omega) = \begin{cases} \frac{1}{|a|^2 \sinh^2 \theta + |b|^2 \sinh^2 M \theta} \left\{ (-|b|^2 + |a|^2 \sinh^2 \theta) M + \frac{|b|^2 \sinh 2M\theta \sinh 2\theta}{4} \right\} & : \omega \in B\_{\text{out}} \\\\ \frac{1}{|a|^2 \sin^2 \theta + |b|^2 \sin^2 M \theta} \left\{ (|b|^2 + |a|^2 \sin^2 \theta) M - \frac{|b|^2}{4} \frac{\sin 2M\theta \sin 2\theta}{\sin^2 \theta} \right\} & : \omega \in B\_{\text{in}} \end{cases} \tag{21}$$

From now on, let us consider the asymptotics of E*M*(*ω*) for large *M*. We summarize our results on the asymptotics of E*M*(*ω*) in Table 1. In the following, we regard E*M*(*ω*) as a function of *θ*, *M*; that is E(*<sup>M</sup>*, *θ*) because *θ* can be expressed by *ω* and consider the asymptotics for large *M*.

$$\begin{aligned} \text{4.1. } \omega \in B\_{\text{out}} \\ \text{Let us see that} \end{aligned}$$

$$\lim\_{M \to \infty} \mathcal{E}\_M(\omega) = \frac{\cosh \theta}{\sinh \theta} = \frac{\left| \frac{\cos k}{a} \right|}{\sqrt{\left| \frac{\cos k}{a} \right|^2 - 1}}. \tag{22}$$

Note that sinh *Mθ* ∼ *eM<sup>θ</sup>*/2 ( *M*. Then by (21), we have

$$\mathcal{E}\_M(\omega) \sim \frac{1}{|b|^2 e^{2M\theta}} \times \frac{|b|^2}{4} \frac{e^{2M\theta} \sinh 2\theta}{\sinh^2 \theta} = \frac{\cosh \theta}{\sinh \theta}.$$

By (22), if *ω* → *ω*∗ ∈ *∂B*, then E*M*(*ω*) ∼ 1/*θ* → ∞. To connect it to the limit for the case of *ω*∗ ∈ *∂B* described by (20) continuously, we consider *M* → ∞ and *θ* → 0 simultaneously, so that *Mθ* ∼ *θ*∗ ∈ (0, <sup>∞</sup>). Let us see that

$$\mathcal{E}\_M(\omega) \sim \frac{1}{\sinh^2 \theta\_\*} \left( -1 + \frac{\sinh 2\theta\_\*}{2\theta\_\*} \right) M \tag{23}$$

Noting that sinh *mθ* = sinh *<sup>m</sup>θ*∗ = 0, for *m* = 1, 2 and sinh *θ* ∼ *θ*∗/*M*, we have

$$\begin{split} \mathcal{E}\_{\mathcal{M}}(\omega) &\sim \frac{1}{|b^{2}|\sinh^{2}\theta\_{\*}} \left\{-|b|^{2}\mathcal{M} + \frac{|b|^{2}}{4} \frac{\sinh 2\theta\_{\*} \times (2\theta\_{\*}/M)}{(\theta\_{\*}/M)^{2}}\right\} \\ &= \frac{1}{\sinh^{2}\theta\_{\*}} \left(-1 + \frac{\sinh 2\theta\_{\*}}{2\theta\_{\*}}\right)M \end{split}$$

Therefore, if we design the parameter *θ*∗ so that

$$\frac{2}{3} = \frac{1}{\sinh^2 \theta\_\*} \left( -1 + \frac{\sinh 2\theta\_\*}{2\theta\_\*} \right) ,\tag{24}$$

then the energy of *Bout* continuously closes to that of *∂B* in the sufficient large system size *M*.

*4.2. ω* ∈ *Bin*

In this paper, since we determine *θ* satisfying sin *θ* > 0, we set *θ* ∈ (0, *<sup>π</sup>*). Remark that <sup>E</sup>*M*(*ω*<sup>−</sup><sup>1</sup>) = E*M*(*ω*) for any *ω* ∈ *Bin* because *ei<sup>θ</sup>* is invariant under this deformation.

By (21), if sin *θ* ) sin *Mθ* ) 1, we have

$$\mathcal{E}\_M(\omega) \sim \left(\frac{|a|^2 \sin^2 \theta + |b|^2}{|a|^2 \sin^2 \theta + |b^2| \sin^2 M\theta}\right) \mathcal{M},\tag{25}$$

for sufficiently large *M*, which implies that

$$M \lessapprox \mathcal{E}\_M(\omega) \lessapprox \left( 1 + \frac{|b|^2}{|a|^2 \sin^2 \theta} \right) M \tag{26}$$

if *θ* ∈ { / 0, *π*} is fixed. Then, we conclude that E*M*(*ω*) = *O*(*M*) if *θ* ∈/ Z*π* is fixed for *ω* ∈ *Bin*. On the other hand, if we design *θ* so that the condition of the perfect transmitting is satisfied; *θ* = *<sup>π</sup>*/*M*, ||∈{1, ... , *M* − 1} (see Corollary 1) and choose which is very close to 0 or *M*, then | sin *θ*| \* 1. Note that if | sin *θ*| → 0, which means *ω* → *ω*∗ ∈ *∂B*, then the coefficient of the upper bound in (26) diverges.

Then, from now on, let us consider the following three cases having a magnitude relation between *θ* and *M*;

$$\text{(i)}\ 1\ \otimes M\ \otimes \ 1/\sin\theta; \text{ (ii)}\ M \simeq 1/\sin\theta; \text{ (iii)}\ 1/\sin\theta \ll M.$$

1. Case (i): 1 \* *M* \* 1/ sin *θ* LetevaluateRHSof

 us start to (21). Since

$$\frac{\sin 2M\theta \sin 2\theta}{4\sin^2 \theta} \sim M \left\{ 1 - \frac{1}{3} (1 + 2M^2) \theta \right\} \prime$$

the "{ }" part in RHS of (21) can be evaluated by 2|*b*|<sup>2</sup>*M*3*θ*2/3. The denominator of (21) is evaluated by 1/(|*b*|<sup>2</sup>*M*2*θ*<sup>2</sup>). Combining them, we have

$$\mathcal{E}\_M(\omega) \sim \frac{2M}{3} \tag{27}$$

This is consistent with (20).

2. Case (ii): *M* ) 1/| sin *θ*|

> Under this condition, the parameter *θ* lives around 0 or *π* if *M* is large. Since we consider *θ* ∈ (0, *<sup>π</sup>*), we can evaluate sin *θ* by sin *θ* ∼ *θ*, or sin *θ* ∼ (*π* − *θ*) for large *M*. We define *θ* = *θ* if 0 < *θ* < *π*/2 and *θ* = *π* − *θ* if *π*/2 ≤ *θ* < *π*. Because *M* sin *θ* ) 1 by the assumption, we have *Mθ* ) 1. Therefore, we put *Mθ* = *θ*∗ + with *θ*∗ ) 1 and || \* 1. Then up to the value *θ*<sup>∗</sup>, let us see

$$\mathcal{E}\_{\mathcal{M}}(\omega) \sim \begin{cases} \frac{1}{\sin^2 \theta\_\*} \left( 1 - \frac{\sin 2\theta\_\*}{2\theta\_\*} \right) M & : \theta\_\* \notin \mathbb{Z}\pi, \\\frac{|b|^2}{|a|^2 \theta\_\*^2} M^3 & : \theta\_\* \in \mathbb{Z}\pi \text{ and } \epsilon M \ll 1 \\\frac{M}{\epsilon^2} & : \theta\_\* \in \mathbb{Z}\pi \text{ and } \epsilon M \gg 1 \end{cases} \tag{28}$$

Note that if *θ*∗ ∈/ Z*π*, then sin *θ* = sin *θ* ∼ *θ*∗/*M* and sin<sup>2</sup> *Mθ* = sin<sup>2</sup> *Mθ* ∼ sin<sup>2</sup> *θ*∗ = 0, sin 2*Mθ* = sin 2*Mθ* ∼ sin 2*θ*∗ and so on. Inserting them into (21), we have

$$\begin{split} \mathcal{E}\_{M}(\omega) &\sim \frac{1}{|a|^{2}\theta\_{\*}^{2}/M^{2}+|b|^{2}\sin^{2}\theta\_{\*}} \left\{ (|a|^{2}\theta\_{\*}^{2}/M^{2}+|b|^{2})M - \frac{|b|^{2}}{4} \frac{\sin 2\theta\_{\*}\cdot 2\theta\_{\*}/M}{\theta\_{\*}^{2}/M^{2}} \right\} \\ &\sim \frac{1}{\sin^{2}\theta\_{\*}} \left(1-\frac{\sin 2\theta\_{\*}}{2\theta\_{\*}}\right)M \end{split}$$

On the other hand, if *θ*∗ ∈ Z*π*, since sin *θ* ∼ *θ*∗/*M* and sin *Mθ*∗ ∼ , by (21), we have

$$\begin{split} \mathcal{E}\_{M}(\omega) &\sim \frac{1}{|a|^{2}\theta^{2} + |b|^{2}\epsilon^{2}} \Bigg{(} |b|^{2}M - \frac{|b|^{2}}{4} \frac{2\epsilon \cdot 2\theta\_{\*}/M}{(\theta\_{\*}/M)^{2}} \Bigg{)} \\ &\sim \frac{|b|^{2}M}{|a|^{2}\theta^{'} + |b|^{2}\epsilon^{2}} \\ &\sim \begin{cases} \frac{|b|^{2}}{|a|^{2}\theta\_{\*}^{2}}M^{3} & : \epsilon \ll \theta\_{\*}/M\\ M/\epsilon^{2} & : \epsilon \gg \theta\_{\*}/M \end{cases} \end{split}$$

3. Case (iii): 1/| sin *θ*| \* *M*

> The "{ }" part in (21) is estimated by (|*b*|<sup>2</sup> + |*a*|<sup>2</sup> sin<sup>2</sup> *θ*)*M* because *Mθ* ( 1. Then, we have

$$\mathcal{E}\_M(\omega) \sim \left(\frac{|a|^2 \sin^2 \theta + |b|^2}{|a|^2 \sin^2 \theta + |b|^2 |\sin^2 M \theta}\right) M\_\prime \tag{29}$$

for sufficiently large *M* which is the same as (25). Let us consider the following case study:


$$\mathcal{E}\_M(\omega) \sim \begin{cases} \frac{1}{\sin^2 M \theta} M & : \sin \theta \ll \sin M\theta \asymp 1\\ (1 + \frac{|b|^2}{|a|^2 \sin^2 \theta}) M & : \sin M\theta \ll \sin \theta \asymp 1 \end{cases} \tag{30}$$

(b) Since | sin *Mθ*| \* 1, we evaluate | sin *Mθ*| by

$$|\sin M\theta| \sim \min\{|M\theta|, \ |\pi - M\theta|, \dots, |M\pi - M\theta|\} =: \delta.$$

Then, there exists a natural number *m* such that |*θ* − *mπ*/*M*| = *δ*/*M*. Note that | sin *θ*| is also sufficiently small. Then, the natural number *m* must be *m*/*M* \* 1 if 0 < *θ* < *π*/2 and (*M* − *m*)/*<sup>M</sup>* \* 1 if *π*/2 ≤ *θ* < *π*. Putting *m* := min{*<sup>m</sup>*, *M* − *<sup>m</sup>*}, we have

$$|\sin\theta| \sim |\frac{m'}{M}\pi \pm \frac{\delta}{M}| \sim \frac{\delta}{M}.$$

Therefore, | sin *θ*|\*| sin *Mθ*| \* 1 holds. Then, (29) implies

$$
\mathcal{E}\_M(\omega) \sim \frac{M}{\delta^2}.
$$

We summarize the above statements in the following theorem by setting *θ* = *<sup>O</sup>*(1/*M*), = 1/*Mα* as a special but natural design of the parameters.

**Theorem 2.** *Let us set ω* ∈ *Bin so that*

$$\theta = \theta(M) = \left(\mathfrak{x}\pi + \frac{1}{M^a}\right)\frac{1}{M}$$

*with the parameters x* ∈ (0, *M*) ⊂ R *and α* ≥ 0*. If x* → 0 *or x* → *M with fixed M, then* E*M*(*ω*) = *<sup>O</sup>*(*M*)*. On the other hand, if we take M* → ∞ *and fix x* = min{*<sup>x</sup>*, *M* − *x*} ) 1*, then we have*

$$\mathcal{E}\_M(\omega) = \begin{cases} O(M^3) & : \mathbf{x'} \text{ is natural number and } \mathfrak{a} \ge 1, \\ O(M^{1+2\mathfrak{a}}) & : \mathbf{x'} \text{ is natural number and } 0 \le \mathfrak{a} < 1, \\ O(M) & : otherwise. \end{cases}$$

**Table 1.** Asymptotics of the energy of E*M*(*ω*): cos *θ* = (*ω* + *<sup>ω</sup>*<sup>−</sup><sup>1</sup>)/(2|*a*|), *Mθ* = *θ*∗ + .

