$\mathrm{i}\,\omega_{*}^{(\mathtt{p})}(\mathtt{h})$ of
${\mathcal{L}}_{\underline{\mu},0}$ in (2.35) for $(\mu,\epsilon)$ close to
$(\underline{\mu},0)$ we apply Kato’s similarity transformation theory for
Hamiltonian operators as developed in [5, 7, 9], that we summarize in the
following lemma. We recall that ${\mathcal{L}}_{\mu,\epsilon}:Y\subset X\to X$
has domain $Y:=H^{1}(\mathbb{T}):=H^{1}(\mathbb{T},\mathbb{C}^{2})$ and range
$X:=L^{2}(\mathbb{T}):=L^{2}(\mathbb{T},\mathbb{C}^{2})$.

######  Lemma 4.1.

(Kato theory for separated eigenvalues of Hamiltonian operators) Fix
$\mathtt{p}\in\mathbb{N}$, $\mathtt{p}\geq 2$. Let $\Gamma$ be a closed
counter-clocked wise oriented curve winding once around the double eigenvalue
$\mathrm{i}\,\omega_{*}^{(\mathtt{p})}(\mathtt{h})$ in (2.36) of
${\mathcal{L}}_{\underline{\mu},0}$, separating
$\sigma_{\mathtt{p}}^{\prime}({\mathcal{L}}_{\underline{\mu},0})$ from the
other part of the spectrum
$\sigma_{\mathtt{p}}^{\prime\prime}({\mathcal{L}}_{\underline{\mu},0})$. Then
there exist $\epsilon_{0},\delta_{0}>0$ such that for any $(\mu,\epsilon)\in
B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{0}}(0)$ the following hold.
1\. The curve $\Gamma$ belongs to the resolvent set of the operator
${\mathcal{L}}_{\mu,\epsilon}:Y\subset X\to X$ defined in (2.14). The
operators

$P(\mu,\epsilon):=-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}({\mathcal{L}}_{\mu,\epsilon}-\lambda)^{-1}\mathrm{d}\lambda:X\to
Y$ (4.1)

are projectors commuting with ${\mathcal{L}}_{\mu,\epsilon}$ on $Y$, i.e.
$P(\mu,\epsilon)^{2}=P(\mu,\epsilon)$ and

$P(\mu,\epsilon){\mathcal{L}}_{\mu,\epsilon}[y]={\mathcal{L}}_{\mu,\epsilon}P(\mu,\epsilon)[y]\,,\quad\forall
y\in Y\,.$ (4.2)

The map $(\mu,\epsilon)\mapsto P(\mu,\epsilon)$ is analytic from
$B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{0}}(0)$ to
${\mathcal{L}}(X,Y)$. The projectors $P(\mu,\epsilon)$ are skew-Hamiltonian
and reversibility preserving, i.e.

$\mathcal{J}P(\mu,\epsilon)=P(\mu,\epsilon)^{*}\mathcal{J}\,,\quad\overline{\rho}P(\mu,\epsilon)=P(\mu,\epsilon)\overline{\rho}\,.$
(4.3)

2\. The domain $Y$ of ${\mathcal{L}}_{\mu,\epsilon}$ decomposes as the direct
sum $Y=\mathcal{V}_{\mu,\epsilon}\oplus\mathcal{K}_{\mu,\epsilon}$ of the
closed subspaces $\mathcal{V}_{\mu,\epsilon}:=\text{Rg}(P(\mu,\epsilon))$,
$\mathcal{K}_{\mu,\epsilon}:=\text{Ker}(P(\mu,\epsilon))\cap Y$, which are
invariant under ${\mathcal{L}}_{\mu,\epsilon}$, and

$\displaystyle\sigma({\mathcal{L}}_{\mu,\epsilon})\cap\\{z\in\mathbb{C}\mbox{
inside
}\Gamma\\}=\sigma({\mathcal{L}}_{\mu,\epsilon}|_{{\mathcal{V}}_{\mu,\epsilon}})=\sigma^{\prime}({\mathcal{L}}_{\mu,\epsilon})\,.$
(4.4)

3\. The projectors $P(\mu,\epsilon)$ are similar one to each other: the
transformation operators

$U_{\mu,\epsilon}:=\big{(}\mathrm{Id}-(P(\mu,\epsilon)-P(\underline{\mu},0))^{2}\big{)}^{-1/2}\big{[}P(\mu,\epsilon)P(\underline{\mu},0)+(\mathrm{Id}-P(\mu,\epsilon))(\mathrm{Id}-P(\underline{\mu},0))\big{]}$
(4.5)

are bounded and invertible in both $Y$ and $X$, and
$U_{\mu,\epsilon}P(\underline{\mu},0)U_{\mu,\epsilon}^{-1}=P(\mu,\epsilon)$.
The map $(\mu,\epsilon)\mapsto U_{\mu,\epsilon}$ is analytic from
$B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{0}}(0)$ to
${\mathcal{L}}\big{(}Y\big{)}$. The transformation operators
$U_{\mu,\epsilon}$ are symplectic and reversibility preserving, namely

$U_{\mu,\epsilon}^{*}\mathcal{J}U_{\mu,\epsilon}=\mathcal{J}\,,\qquad\overline{\rho}U_{\mu,\epsilon}=U_{\mu,\epsilon}\overline{\rho}\,.$
(4.6)

Finally
$\mathcal{V}_{\mu,\epsilon}=U_{\mu,\epsilon}\mathcal{V}_{\underline{\mu},0}$
and $\dim\mathcal{V}_{\mu,\epsilon}=\dim\mathcal{V}_{\underline{\mu},0}=2$ for
any $(\mu,\epsilon)\in B_{\delta_{0}}(\underline{\mu})\times
B_{\epsilon_{0}}(0)$.

We consider the basis of the subspace
${\mathcal{V}}_{\mu,\epsilon}=\mathrm{Rg}(P(\mu,\epsilon))$,

${\mathcal{F}}:=\\{f^{+}(\mu,\epsilon),f^{-}(\mu,\epsilon)\\}\,,\quad
f^{+}(\mu,\epsilon):=U_{\mu,\epsilon}f_{\mathtt{p}}^{+},\ \
f^{-}(\mu,\epsilon):=U_{\mu,\epsilon}f_{0}^{-}\,,$ (4.7)

obtained applying the transformation operators $U_{\mu,\epsilon}$ in (4.5) to
the eigenvectors $f_{0}^{-}:=f_{0}^{-}(\underline{\mu},\mathtt{h})$ and
$f_{\mathtt{p}}^{+}:=f_{\mathtt{p}}^{+}(\underline{\mu},\mathtt{h})$ of
${\mathcal{L}}_{\underline{\mu},0}$ defined in (2.34), which are a symplectic
and reversible basis of
$\mathcal{V}_{\underline{\mu},0}=\mathrm{Rg}(P(\underline{\mu},0))$. Thanks to
(4.6) and (2.25) the basis ${\mathcal{F}}$ is symplectic and reversible as
well, namely

$\displaystyle{\mathcal{W}}_{c}\big{(}f^{+}(\mu,\epsilon),f^{+}(\mu,\epsilon)\big{)}=-\mathrm{i}\,\,,\quad{\mathcal{W}}_{c}\big{(}f^{-}(\mu,\epsilon),f^{-}(\mu,\epsilon)\big{)}=\mathrm{i}\,\,,\quad{\mathcal{W}}_{c}\big{(}f^{+}(\mu,\epsilon),f^{-}(\mu,\epsilon)\big{)}=0\,,$

and $\overline{\rho}f^{\sigma}(\mu,\epsilon)=\sigma f^{+}(\mu,\epsilon)$ for
any $\sigma=\pm$. This directly implies the following lemma, cfr. [9, Lemma
2.3].

######  Lemma 4.2.

(Matrix representation of ${\mathcal{L}}_{\mu,\epsilon}$ on
$\mathcal{V}_{\mu,\epsilon}$) For any integer $\mathtt{p}\geq 2$, the operator
${\mathcal{L}}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$
in (2.14) is represented on the basis ${\mathcal{F}}$ in (4.7) by the $2\times
2$ Hamiltonian and reversible matrix

$\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)=\mathtt{J}\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon),\
\mathtt{J}:=\begin{pmatrix}-\mathrm{i}\,&0\\\
0&\mathrm{i}\,\end{pmatrix}\,,\quad\forall(\mu,\epsilon)\in
B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{0}}(0)\,,$ (4.8)

where

$\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon)=\begin{pmatrix}(\mathfrak{B}(\mu,\epsilon)f^{+}_{\mathtt{p}},f^{+}_{\mathtt{p}})&(\mathfrak{B}(\mu,\epsilon)f^{-}_{0},f^{+}_{\mathtt{p}})\\\
(\mathfrak{B}(\mu,\epsilon)f^{+}_{\mathtt{p}},f^{-}_{0})&(\mathfrak{B}(\mu,\epsilon)f^{-}_{0},f^{-}_{0})\end{pmatrix}=:\begin{pmatrix}\alpha^{(\mathtt{p})}(\mu,\epsilon)&\mathrm{i}\,\beta^{(\mathtt{p})}(\mu,\epsilon)\\\
-\mathrm{i}\,\beta^{(\mathtt{p})}(\mu,\epsilon)&\gamma^{(\mathtt{p})}(\mu,\epsilon)\end{pmatrix}\,,$
(4.9)

with

$\mathfrak{B}(\mu,\epsilon):=[P_{\underline{\mu},0}]^{*}\,[U_{\mu,\epsilon}]^{*}\,{\mathcal{B}}(\mu,\epsilon)\,U_{\mu,\epsilon}\,P_{\underline{\mu},0}\,.$
(4.10)

It results

$\mathtt{L}^{(\mathtt{p})}(\mu,0)=\begin{pmatrix}\mathrm{i}\,\omega_{\mathtt{p}}^{+}(\mu)&0\\\
0&\mathrm{i}\,\omega_{0}^{-}(\mu)\end{pmatrix},\quad\text{i.e.}\quad\mathtt{B}^{(\mathtt{p})}(\mu,0)=\begin{pmatrix}-\omega_{\mathtt{p}}^{+}(\mu)&0\\\
0&\omega_{0}^{-}(\mu)\end{pmatrix}\,,$ (4.11)

where $\omega_{j}^{\sigma}(\mu):=\omega^{\sigma}(j+\mu,\mathtt{h})$ in (2.20)
and
$\omega_{\mathtt{p}}^{+}(\underline{\mu})=\omega_{0}^{-}(\underline{\mu})=\omega_{*}^{(\mathtt{p})}$
in (2.35). Finally the functions
$\alpha^{(\mathtt{p})},\beta^{(\mathtt{p})},\gamma^{(\mathtt{p})}$ are
analytic w.r.t. $(\mu,\epsilon)\in B_{\delta_{0}}(\underline{\mu})\times
B_{\epsilon_{0}}(0)$.

######  Remark 4.3.

The functions
$\alpha^{(\mathtt{p})}(\mu,\epsilon),\beta^{(\mathtt{p})}(\mu,\epsilon)$ and
$\gamma^{(\mathtt{p})}(\mu,\epsilon)$ are actually analytic also w.r. to
$\mathtt{h}>\mathtt{h}_{*}$ for any $\mathtt{h}_{*}>0$ provided $\epsilon$ is
small depending on $\mathtt{h}_{*}$. This follows because the projectors
$P(\mu,\epsilon)$ and the similarity operators $U_{\mu,\epsilon}$ of Lemma 4.1
are actually analytic also in $\mathtt{h}>0$ as
${\mathcal{L}}_{\mu,\epsilon}(\mathtt{h})$, see Remark 2.6. However, we shall
not rely on this property, and the analiticity of
$\mathtt{h}\mapsto\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ will be deduced a
posteriori in view of its explicit expression (5.7)-(5.8).

The eigenvalues of the matrix $\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)$ in
(4.8) are

$\lambda^{\pm}(\mu,\epsilon)=\tfrac{\mathrm{i}\,}{2}S^{(\mathtt{p})}(\mu,\epsilon)\pm\tfrac{1}{2}\sqrt{D^{(\mathtt{p})}(\mu,\epsilon)}$
(4.12)

where

$\displaystyle
S^{(\mathtt{p})}(\mu,\epsilon):=-\mathrm{i}\,\text{Tr}\,\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)=\gamma^{(\mathtt{p})}(\mu,\epsilon)-\alpha^{(\mathtt{p})}(\mu,\epsilon)\,,$
(4.13) $\displaystyle
D^{(\mathtt{p})}(\mu,\epsilon):=\text{Tr}^{2}\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)-4\det\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)=4(\beta^{(\mathtt{p})})^{2}(\mu,\epsilon)-(T^{(\mathtt{p})})^{2}(\mu,\epsilon)\,,$
(4.14)

with

$T^{(\mathtt{p})}(\mu,\epsilon):=\text{Tr}\,\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon)=\alpha^{(\mathtt{p})}(\mu,\epsilon)+\gamma^{(\mathtt{p})}(\mu,\epsilon)\,.$

Note that $S^{(\mathtt{p})}(\mu,\epsilon)$ and
$D^{(\mathtt{p})}(\mu,\epsilon)$ are spectral invariants of
${\mathcal{L}}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$,
i.e. they are independent of the choice of the basis.

In the next sections we prove, for $(\mu,\epsilon)$ close to
$(\underline{\mu},0)$, the existence of eigenvalues of the matrix
$\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)$ with nonzero real part. We shall
apply the following abstract results.

#### Abstract instability criterion.

Consider a $2\times 2$ Hamiltonian and reversible matrix

$\mathtt{L}(\mu,\epsilon):=\mathtt{J}\mathtt{B}(\mu,\epsilon),\quad\mathtt{B}(\mu,\epsilon):=\begin{pmatrix}\alpha(\mu,\epsilon)&\mathrm{i}\,\beta(\mu,\epsilon)\\\
-\mathrm{i}\,\beta(\mu,\epsilon)&\gamma(\mu,\epsilon)\end{pmatrix}\,,$ (4.15)

where $\alpha(\mu,\epsilon),\beta(\mu,\epsilon),\gamma(\mu,\epsilon)$ satisfy
the following

######  Assumption 4.4.

The functions $\alpha(\mu,\epsilon)$, $\beta(\mu,\epsilon)$,
$\gamma(\mu,\epsilon)$ are real analytic in the domain
$B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{0}}(0)$ and admit the
expansions, for some integer $\mathtt{p}\geq 2$, and $\beta_{1}\neq 0$,

$\displaystyle\alpha(\underline{\mu}+\delta,\epsilon)$
$\displaystyle=-\alpha_{0}(\delta)+\alpha_{2}\epsilon^{2}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,$
(4.16a) $\displaystyle\beta(\underline{\mu}+\delta,\epsilon)$
$\displaystyle=\beta_{1}\epsilon^{\mathtt{p}}+\beta_{2}\delta\epsilon^{\mathtt{p}}+r(\epsilon^{\mathtt{p}+1},\delta^{2}\epsilon^{\mathtt{p}-1},\ldots,\delta^{\mathtt{p}}\epsilon)\,,$
(4.16b) $\displaystyle\gamma(\underline{\mu}+\delta,\epsilon)$
$\displaystyle=\gamma_{0}(\delta)+\gamma_{2}\epsilon^{2}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,$
(4.16c)

where

$\alpha_{0}(\delta)=\omega_{*}^{(\mathtt{p})}-\alpha_{1}\delta+r(\delta^{2})\,,\qquad\gamma_{0}(\delta)=\omega_{*}^{(\mathtt{p})}+\gamma_{1}\delta+r(\delta^{2})\,,$
(4.17)

and

$\alpha_{1}+\gamma_{1}>0\,.$ (4.18)

We shall prove in Section 5 that for any $\mathtt{p}\geq 2$ the matrix
$\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)$ in (4.8) satisfies Assumption 4.4.

In view of Assumption 4.4 the trace
$T(\mu,\epsilon):=\text{Tr}\,\mathtt{B}(\mu,\epsilon)=\alpha(\mu,\epsilon)+\gamma(\mu,\epsilon)$
admits the expansion

$\displaystyle
T(\underline{\mu}+\delta,\epsilon)=T(\underline{\mu}+\delta,0)+T_{2}\epsilon^{2}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,\qquad
T_{2}:=\alpha_{2}+\gamma_{2}\,,$ (4.19) $\displaystyle
T(\underline{\mu}+\delta,0)=T_{1}\delta+r(\delta^{2})\,,\qquad
T_{1}:=\alpha_{1}+\gamma_{1}>0\,.$

By (4.19) and the analytic implicit function theorem there exists
$\epsilon_{1}\in(0,\epsilon_{0})$ such that the zero set of the trace
$T(\mu,\epsilon)$ and of the functions

$d_{\pm}(\mu,\epsilon):=T(\mu,\epsilon)\pm 2\beta(\mu,\epsilon)$ (4.20)

are, in the set $B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{1}}(0)$,
graphs of analytic functions

$\mu_{0}\,,\mu_{\pm}:(-\epsilon_{1},\epsilon_{1})\mapsto\mu_{0}(\epsilon)\,,\
\mu_{\pm}(\epsilon)\,,\ \text{i.e.}\ T(\mu_{0}(\epsilon),\epsilon)\equiv 0\
\text{ and }\ d_{\pm}(\mu_{\pm}(\epsilon),\epsilon)\equiv 0\,,$ (4.21)

admitting, by (4.19), the Taylor expansions

$\mu_{0}(\epsilon)=\underline{\mu}-\frac{T_{2}}{T_{1}}\epsilon^{2}+r(\epsilon^{3})\,,\quad\mu_{\pm}(0)=\underline{\mu}\,,\
\mu_{\pm}^{\prime}(0)=0\,.$ (4.22)

We shall prove in Lemma 4.6 that the functions $\mu_{\pm}(\epsilon)$ are
actually $\mathcal{O}(\epsilon^{\mathtt{p}})$-close to $\mu_{0}(\epsilon)$.

Since $T_{1}>0$ (cfr. (4.19)), the functions $d_{\pm}(\mu,\epsilon)$ are
strictly positive (resp. negative) for $\mu>\mu_{\pm}(\epsilon)$ (resp.
$\mu<\mu_{\pm}(\epsilon)$). In addition, since
$d_{\pm}(\mu_{0}(\epsilon),\epsilon)=\pm 2\beta(\mu_{0}(\epsilon),\epsilon)$
we deduce that for any $\epsilon\in(-\epsilon_{1},\epsilon_{1})$

$\begin{cases}\mbox{if }\beta(\mu_{0}(\epsilon),\epsilon)>0&\mbox{then }\
\mu_{+}(\epsilon)<\mu_{0}(\epsilon)<\mu_{-}(\epsilon)\,,\\\ \mbox{if
}\beta(\mu_{0}(\epsilon),\epsilon)<0&\mbox{then }\
\mu_{-}(\epsilon)<\mu_{0}(\epsilon)<\mu_{+}(\epsilon)\,,\\\ \mbox{if
}\beta(\mu_{0}(\epsilon),\epsilon)=0&\mbox{then }\
\mu_{0}(\epsilon)=\mu_{+}(\epsilon)=\mu_{-}(\epsilon)\,.\end{cases}$ (4.23)

The graphs of these functions are represented in Figure 2. By (4.16b), the
fact that $\beta_{1}\neq 0$, and the expansion of $\mu_{0}(\epsilon)$ in
(4.22), we deduce that

$\beta(\mu_{0}(\epsilon),\epsilon)=\beta_{1}\epsilon^{\mathtt{p}}+r(\epsilon^{\mathtt{p}+1})$

is not identically zero, actually it is positive, resp. negative, if
$\beta_{1}>0$, resp. $\beta_{1}<0$. Thus only the first two cases in (4.23)
occur and, for $|\epsilon|$ small enough,
$\mu_{\wedge}(\epsilon)<\mu_{0}(\epsilon)<\mu_{\vee}(\epsilon)$ where

$\mu_{\wedge}(\epsilon):=\min\big{\\{}\mu_{+}(\epsilon),\mu_{-}(\epsilon)\big{\\}}\quad\text{and}\quad\mu_{\vee}(\epsilon):=\max\big{\\{}\mu_{+}(\epsilon),\mu_{-}(\epsilon)\big{\\}}\,.$
(4.24)

######  Theorem 4.5.

Assume that the $2\times 2$ Hamiltonian and reversible matrix
$\mathtt{L}(\mu,\epsilon)=\mathtt{J}\mathtt{B}(\mu,\epsilon)$ in (4.15)
satisfies Assumption 4.4. Then the spectrum of $\,\mathtt{L}(\mu,\epsilon)$,
for $(\mu,\epsilon)$ close to $(\underline{\mu},0)$, consists of two
eigenvalues $\lambda^{\pm}(\mu,\epsilon)$ with opposite non-zero real part if
and only if $(\mu,\epsilon)$ lies inside the open region

$\displaystyle R:$ $\displaystyle=\big{\\{}(\mu,\epsilon)\in
B_{\delta_{0}}(\underline{\mu})\times B_{\epsilon_{1}}(0)\ :\
\mu_{\wedge}(\epsilon)<\mu<\mu_{\vee}(\epsilon)\big{\\}}\,,$ (4.25)

whereas, for $(\mu,\epsilon)\notin R$, the eigenvalues
$\lambda^{\pm}(\mu,\epsilon)$ are purely imaginary, see Figure 2.

###### Proof.

The eigenvalues of $\mathtt{L}(\mu,\epsilon)$ have the form (4.12) and have
nonzero real part if and only if the discriminant

$D(\mu,\epsilon)=4\beta^{2}(\mu,\epsilon)-T^{2}(\mu,\epsilon)=-d_{+}(\mu,\epsilon)d_{-}(\mu,\epsilon)$
(4.26)

is positive, which by construction happens if and only if $(\mu,\epsilon)\in
R$. ∎

In order to describe the unstable eigenvalues, it is convenient to translate
the Floquet exponent $\mu$ around the value $\mu_{0}(\epsilon)$ where
$T(\mu_{0}(\epsilon),\epsilon)\equiv 0$, cfr. (4.21), namely we introduce the
new parameter $\nu$ such that

$\mu=\mu_{0}(\epsilon)+\nu\,,\quad\text{i.e.}\quad\nu:=\delta+\underline{\mu}-\mu_{0}(\epsilon)\,.$
(4.27)

Accordingly we write the functions $\mu_{\pm}(\epsilon)$ in (4.22) as
$\mu_{\pm}(\epsilon)=\mu_{0}(\epsilon)+\nu_{\pm}(\epsilon)$.

######  Lemma 4.6.

The analytic functions
$\nu_{\pm}(\epsilon):=\mu_{\pm}(\epsilon)-\mu_{0}(\epsilon)$ admit the
expansion

$\nu_{\pm}(\epsilon)=\mp\frac{2\beta_{1}}{T_{1}}\epsilon^{\mathtt{p}}+r(\epsilon^{\mathtt{p}+1})\,.$
(4.28)

###### Proof.

The function $\nu_{+}(\epsilon)$ solves

$d_{+}(\mu_{0}(\epsilon)+\nu_{+}(\epsilon),\epsilon)=T(\mu_{0}(\epsilon)+\nu_{+}(\epsilon),\epsilon)+2\beta(\mu_{0}(\epsilon)+\nu_{+}(\epsilon),\epsilon)=0\,.$

Expanding this identity at $\mu=\mu_{0}(\epsilon)$ we get

$\nu_{+}(\epsilon)=-\frac{2\beta(\mu_{0}(\epsilon),\epsilon)}{(\partial_{\mu}T)(\mu_{0}(\epsilon))-2(\partial_{\mu}\beta)(\mu_{0}(\epsilon))}+r(\nu_{+}^{2})(1+r(\epsilon))\,,$

and since
$(\partial_{\mu}T)(\mu_{0}(\epsilon))-2(\partial_{\mu}\beta)(\mu_{0}(\epsilon))=T_{1}(1+r(\epsilon))$
(cfr. (4.19), (4.16b)) we deduce (4.28). ∎

We define

$\nu_{\wedge}(\epsilon):=\min\\{\nu_{+}(\epsilon),\nu_{-}(\epsilon)\\}<0\,,\quad\nu_{\vee}(\epsilon):=\max\\{\nu_{+}(\epsilon),\nu_{-}(\epsilon)\\}>0\,.$
(4.29)

Note that $\nu_{\wedge}(\epsilon)$ and $\nu_{\vee}(\epsilon)$ are the points
where the discriminant $D(\mu_{0}(\epsilon)+\nu,\epsilon)$ in (4.26) of the
matrix $\mathtt{L}(\mu_{0}(\epsilon)+\nu,\epsilon)$ vanishes, whereas for
$\nu_{\wedge}(\epsilon)<\nu<\nu_{\vee}(\epsilon)$ the discriminant
$D(\mu_{0}(\epsilon)+\nu,\epsilon)$ is positive.

######  Theorem 4.7 (Unstable eigenvalues).

Under Assumption 4.4 the matrix $\mathtt{L}(\mu,\epsilon)$ in (4.15) has the
two eigenvalues

$\lambda^{\pm}(\mu_{0}(\epsilon)+\nu,\epsilon)=\begin{cases}\frac{\mathrm{i}\,}{2}S(\mu_{0}(\epsilon)+\nu,\epsilon)\pm\frac{\mathrm{i}\,}{2}\sqrt{|D(\mu_{0}(\epsilon)+\nu,\epsilon)|}&\text{if
}\nu\leq\nu_{\wedge}(\epsilon)\textup{ or
}\nu\geq\nu_{\vee}(\epsilon)\,,\\\\[5.69054pt]
\;\frac{\mathrm{i}\,}{2}S(\mu_{0}(\epsilon)+\nu,\epsilon)\pm\frac{1}{2}\sqrt{D(\mu_{0}(\epsilon)+\nu,\epsilon)}&\text{if
}\nu_{\wedge}(\epsilon)<\nu<\nu_{\vee}(\epsilon)\,,\end{cases}$ (4.30)

where $\mu_{0}(\epsilon)$ is defined in (4.21) and

$\displaystyle D(\mu_{0}(\epsilon)+\nu,\epsilon)$
$\displaystyle=4\beta_{1}^{2}\epsilon^{2\mathtt{p}}-T_{1}^{2}\nu^{2}+8\beta_{1}\beta_{2}\epsilon^{2\mathtt{p}}\nu+r(\epsilon^{{2\mathtt{p}+1}},\nu^{2}\epsilon,\nu^{3})$
(4.31) $\displaystyle S(\mu_{0}(\epsilon)+\nu,\epsilon)$
$\displaystyle=2\omega_{*}+(\gamma_{1}-\alpha_{1})\nu+\Big{(}\gamma_{2}-\alpha_{2}-(\gamma_{1}-\alpha_{1})\frac{T_{2}}{T_{1}}\Big{)}\epsilon^{2}+r(\epsilon^{3},\nu\epsilon^{2},\nu^{2})\,.$
(4.32)

The maximal absolute value of the real part of the unstable eigenvalues in
(4.30) is

$\max\mathfrak{Re}\,\lambda^{\pm}(\mu_{0}(\epsilon)+\nu,\epsilon)=|\beta_{1}|\epsilon^{\mathtt{p}}(1+r(\epsilon))\,.$
(4.33)

_(Isola)_. Assume in addition that the coefficients in (4.16a)-(4.16c) satisfy
also $\alpha_{1}\neq\gamma_{1}$. Then, for any $\epsilon$ small enough, the
pair of unstable eigenvalues $\lambda^{\pm}(\mu_{0}(\epsilon)+\nu,\epsilon)$
depicts in the complex $\lambda$-plane, as $\nu$ varies in the interval
$(\nu_{\wedge}(\epsilon),\nu_{\vee}(\epsilon))$, an ellipse-like curve, i.e. a
closed analytic curve that intersects orthogonally the imaginary axis and
encircles a convex region.

###### Proof.

(Unstable eigenvalues). In view of (4.16b), (4.22) and (4.27),

$\beta(\mu_{0}(\epsilon)+\nu,\epsilon)=\beta_{1}\epsilon^{\mathtt{p}}+\beta_{2}\nu\epsilon^{\mathtt{p}}+r(\epsilon^{\mathtt{p}+1},\nu^{2}\epsilon^{\mathtt{p}-1},\ldots,,\nu^{\mathtt{p}}\epsilon,\nu^{\mathtt{p}+1})\,.$
(4.34)

We now prove the expansion (4.31) of the discriminant

$D(\mu_{0}(\epsilon)+\nu,\epsilon)=4\beta^{2}(\mu_{0}(\epsilon)+\nu,\epsilon)-T^{2}(\mu_{0}(\epsilon)+\nu,\epsilon)\,.$
(4.35)

By (4.21) and (4.19) we get that

$T(\mu_{0}(\epsilon)+\nu,\epsilon)=(\partial_{\mu}T)(\mu_{0}(\epsilon),\epsilon)\nu+r(\nu^{2})=T_{1}\nu+r(\epsilon\nu,\nu^{2})$
(4.36)

and the expansion (4.31) follows by inserting (4.36) and (4.34) into (4.35).
By (4.12) and (4.27), the eigenvalues of $\mathtt{L}(\mu,\epsilon)$ have the
form (4.30). The expansion of $S(\mu_{0}(\epsilon)+\nu,\epsilon)$ in (4.32)
follows from (4.16a)-(4.16c), (4.19) and (4.22). The (opposite) extremal real
parts of the eigenvalues are attained at $\nu=\nu_{\mathfrak{Re}}$, with
$\nu_{\mathfrak{Re}}$ such that
$(\partial_{\mu}D)(\mu_{0}(\epsilon)+\nu_{\mathfrak{Re}},\epsilon)=0$. By
(4.31) we have the expansion

$\nu_{\mathfrak{Re}}(\epsilon)=4\frac{\beta_{1}\beta_{2}}{T_{1}^{2}}\epsilon^{2\mathtt{p}}+r(\epsilon^{2\mathtt{p}+1})\,.$
(4.37)

By plugging (4.37) into (4.31)-(4.30) one obtains the expansion (4.33).

(Isola). For any fixed $\epsilon$ small enough the unstable eigenvalues, in
view of (4.30), branch off from the imaginary axis at
$\nu=\nu_{\wedge}(\epsilon)$, evolve specularly as $\nu$ increases and rejoin
at $\nu=\nu_{\vee}(\epsilon)$ thus forming a closed curve. With the additional
hypothesis $\alpha_{1}\neq\gamma_{1}$, the imaginary part of the eigenvalues

$I(\nu,\epsilon):=\mathfrak{Im}\,\lambda^{\pm}(\mu_{0}(\epsilon)+\nu,\epsilon)$
(4.38)

is monotone w.r.t.
$\nu\in\big{(}\nu_{\wedge}(\epsilon),\nu_{\vee}(\epsilon)\big{)}$ for small
$\epsilon$, since its derivative fulfills

$\partial_{\nu}I(\nu,\epsilon)\stackrel{{\scriptstyle\eqref{eigs}}}{{=}}\frac{\gamma_{1}-\alpha_{1}}{2}+r(\epsilon^{2},\nu)\neq
0\,,\qquad\nu_{\wedge}(\epsilon)\leq\nu\leq\nu_{\vee}(\epsilon)\,.$ (4.39)

Thus the map $\nu\mapsto I(\nu,\epsilon)$ is a diffeomorphism between the
intervals $\big{(}\nu_{\wedge}(\epsilon),\nu_{\vee}(\epsilon)\big{)}$ and
$\big{(}y_{\wedge}(\epsilon),y_{\vee}(\epsilon)\big{)}$ where

$y_{\wedge}(\epsilon):=\min\big{\\{}I(\nu_{\wedge}(\epsilon),\epsilon)\,,I(\nu_{\vee}(\epsilon),\epsilon)\big{\\}}\,,\quad
y_{\vee}(\epsilon):=\max\big{\\{}I(\nu_{\wedge}(\epsilon),\epsilon)\,,I(\nu_{\vee}(\epsilon),\epsilon)\big{\\}}\,.$

Let us denote $\nu(y,\epsilon)$ the inverse of $y=I(\nu,\epsilon)$, with $y$
varying between $y_{\wedge}(\epsilon)<y<y_{\vee}(\epsilon)$. The curve
depicted by the traces of the two unstable eigenvalues in the complex plane is
hence covered by two specular graphs on the imaginary axis, given by

$\Gamma_{r}:=\big{\\{}\big{(}X(y,\epsilon),y\big{)}\;:\;y_{\wedge}(\epsilon)<y<y_{\vee}(\epsilon)\big{\\}}\,,\quad\Gamma_{l}:=\big{\\{}\big{(}-X(y,\epsilon),y\big{)}\;:\;y_{\wedge}(\epsilon)<y<y_{\vee}(\epsilon)\big{\\}}\,,$
(4.40)

where

$X(y,\epsilon):=\frac{1}{2}\sqrt{D\big{(}\mu(y,\epsilon),\epsilon\big{)}}\,,\quad\mu(y,\epsilon):=\mu_{0}(\epsilon)+\nu(y,\epsilon)\,.$
(4.41)

We claim that at the bottom and top of the isola, i.e. at
$y=y_{\wedge}(\epsilon)$ and $y=y_{\vee}(\epsilon)$, the real parts $\pm
X(y,\epsilon)$ of the unstable eigenvalues vanish with derivative that tends
to infinity. Indeed

$\partial_{y}X(y,\epsilon)\stackrel{{\scriptstyle\eqref{Xdepy}}}{{=}}\partial_{y}\frac{1}{2}\sqrt{D(\mu(y,\epsilon),\epsilon)}=\dfrac{(\partial_{\mu}D)(\mu(y,\epsilon),\epsilon)}{4\sqrt{D(\mu(y,\epsilon),\epsilon)}}(\partial_{y}\mu)(y,\epsilon)\,,$
(4.42)

where, by (4.31), (4.28) and (4.41), we have

$\begin{aligned} &\lim_{y\to
y_{\wedge}\,,y_{\vee}}(\partial_{\mu}D)(\mu(y,\epsilon),\epsilon)=(\partial_{\mu}D)(\mu_{0}(\epsilon)+\nu_{\pm}(\epsilon),\epsilon)=\pm
4\beta_{1}T_{1}\epsilon^{\mathtt{p}}+r(\epsilon^{\mathtt{p}+1})\neq 0\,,\\\
&\lim_{y\to
y_{\wedge}\,,y_{\vee}}(\partial_{y}\mu)(y,\epsilon)=\frac{1}{(\partial_{\nu}I)(\mu_{\pm}(\epsilon),\epsilon)}\stackrel{{\scriptstyle\eqref{derivatanonvanishing}}}{{\neq}}0\,,\end{aligned}\quad\text{for
}0<\epsilon\ll 1\,,$

so that the term $|\partial_{y}X(y,\epsilon)|$ tends to $+\infty$ as $y\to
y_{\wedge}(\epsilon)\,,\;y_{\vee}(\epsilon)$. Thus the two branches of the
isola smoothly join together and the isola can be parameterized by a regular
curve having tangent vector parallel to the real axis when the real component
of the curve vanishes.

Finally, we claim that the region encircled by the two graphs is convex.
Indeed, by (4.41),

$\displaystyle\partial_{yy}X(y,\epsilon)=\frac{(\partial_{\mu\mu}D)(\mu(y,\epsilon),\epsilon)\big{(}(\partial_{y}\mu)(y,\epsilon)\big{)}^{2}+(\partial_{\mu}D)(\mu(y,\epsilon),\epsilon)(\partial_{yy}\mu)(y,\epsilon)}{2\big{(}D(\mu(y,\epsilon),\epsilon)\big{)}^{\frac{1}{2}}}$
(4.43)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\frac{\big{(}(\partial_{\mu}D)(\mu(y,\epsilon),\epsilon)(\partial_{y}\mu)(y,\epsilon)\big{)}^{2}}{4\big{(}D(\mu(y,\epsilon),\epsilon)\big{)}^{\frac{3}{2}}}\,.$

In view of (4.31) and (4.28) we have, as
$y_{\wedge}(\epsilon)<y<y_{\wedge}(\epsilon)$,

$\displaystyle(\partial_{\mu\mu}D)(\mu(y,\epsilon),\epsilon)\leq-
T_{1}^{2}\,,\quad|(\partial_{\mu}D)(\mu(y,\epsilon),\epsilon)|\leq
4|T_{1}|\,|\beta_{1}|\epsilon^{\mathtt{p}}(1+r(\epsilon))\,.$ (4.44)

Moreover, since by (4.38), (4.30) and (4.32) one has, for some $c>0$,
$|\partial_{\nu}I(\nu,\epsilon)|\geq c|\gamma_{1}-\alpha_{1}|$ and
$|\partial_{\nu\nu}I(\nu,\epsilon)|\leq c$ for any
$\nu_{\wedge}(\epsilon)<\nu<\nu_{\vee}(\epsilon)$, we have the following
estimates: for some $C_{1}>0$,

$|\partial_{y}\mu(y,\epsilon)|=|\partial_{y}\nu(y,\epsilon)|\leq\frac{C_{1}}{|\gamma_{1}-\alpha_{1}|}\,,\quad|\partial_{yy}\mu(y,\epsilon)|=|\partial_{yy}\nu(y,\epsilon)|\leq
C_{1}\,,\quad\text{for }y_{\wedge}(\epsilon)<y<y_{\vee}(\epsilon)\,.$ (4.45)

By (4.45) and (4.44), provided $\epsilon$ is sufficiently small, we have, for
some $\tilde{C}>0$

$(\partial_{\mu\mu}D)(\mu(y,\epsilon),\epsilon)(\mu_{y}(y,\epsilon))^{2}+(\partial_{\mu})(\mu(y,\epsilon),\epsilon)(\partial_{yy}\mu)(y,\epsilon)\leq-\frac{\tilde{C}T_{1}^{2}}{(\gamma_{1}-\alpha_{1})^{2}}+\tilde{C}\epsilon^{\mathtt{p}}<0\,.$
(4.46)

By (4.43) and (4.46), the function $y\mapsto X(y,\epsilon)$ is concave. ∎

An approximation of the eigenvalues
$\lambda^{\pm}(\mu_{0}(\epsilon)+\nu,\epsilon)$ in Theorem 4.7, obtained
neglecting the remainders $r(\nu^{3})$ of $D(\mu_{0}(\epsilon)+\nu,\epsilon)$
in (4.31) and $r(\nu^{2})$ of $S(\mu_{0}(\epsilon)+\nu,\epsilon)$ in (4.32),
is given by the complex numbers $\widetilde{\lambda}^{\pm}(\nu,\epsilon)$ with

$\begin{cases}x:=\mathfrak{Re}\
\widetilde{\lambda}^{\pm}(\nu,\epsilon)=\pm\frac{1}{2}\sqrt{4\beta_{1}^{2}\epsilon^{2\mathtt{p}}(1+r(\epsilon))-T_{1}^{2}\nu^{2}(1+r(\epsilon))+8\beta_{1}\beta_{2}\nu\epsilon^{2\mathtt{p}}(1+r(\epsilon))}\,,\\\
y:=\mathfrak{Im}\
\widetilde{\lambda}^{\pm}(\nu,\epsilon)=\omega_{*}+\Big{(}\dfrac{\gamma_{2}-\alpha_{2}}{2}-\dfrac{T_{2}(\gamma_{1}-\alpha_{1})}{2T_{1}}\Big{)}\epsilon^{2}(1+r(\epsilon))+\dfrac{\gamma_{1}-\alpha_{1}}{2}\nu(1+r(\epsilon^{2}))\,,\end{cases}$
(4.47)

The functions $\widetilde{\lambda}^{\pm}(\nu,\epsilon)$ are defined for $\nu$
in the interval
$\widetilde{\nu}_{\wedge}(\epsilon)\leq\nu\leq\widetilde{\nu}_{\vee}(\epsilon)$
where the argument of the square root in (4.47) is non-negative. These
approximated eigenvalues describe an ellipse in the $(x,y)$-plane.

######  Lemma 4.8.

(Approximating ellipse) Suppose $\gamma_{1}-\alpha_{1}\neq 0$, with
$\alpha_{1}$ and $\gamma_{1}$ as in (4.16a)-(4.16c). As $\nu$ varies between
$\widetilde{\nu}_{\wedge}(\epsilon)$ and $\widetilde{\nu}_{\vee}(\epsilon)$
the approximated eigenvalues $\widetilde{\lambda}^{\pm}(\nu,\epsilon)$ in
(4.47) span an ellipse of equation

$x^{2}+\dfrac{T_{1}^{2}(1+r(\epsilon^{2}))}{(\gamma_{1}-\alpha_{1})^{2}}(y-y_{0}(\epsilon))^{2}=\beta_{1}^{2}\epsilon^{2\mathtt{p}}(1+r(\epsilon))\,,$
(4.48)

centered at $(0,y_{0}(\epsilon))$ where

$y_{0}(\epsilon)=\omega_{*}+\Big{(}\dfrac{\gamma_{2}-\alpha_{2}}{2}-\dfrac{T_{2}(\gamma_{1}-\alpha_{1})}{2T_{1}}\Big{)}\epsilon^{2}+r(\epsilon^{3})\,.$
(4.49)

###### Proof.

We invert the second equation in (4.47) and obtain the function

$\nu(y,\epsilon)=\dfrac{2}{\gamma_{1}-\alpha_{1}}(y-\tilde{y}(\epsilon))(1+r(\epsilon^{2}))\,,$
(4.50)

where

$\widetilde{y}(\epsilon):=\mathfrak{Im}\
\widetilde{\lambda}^{\pm}(0,\epsilon)=\omega_{*}+\Big{(}\dfrac{\gamma_{2}-\alpha_{2}}{2}-\dfrac{T_{2}(\gamma_{1}-\alpha_{1})}{2T_{1}}\Big{)}\epsilon^{2}(1+r(\epsilon^{2}))\,.$

By plugging the expansion (4.50) for $\nu=\nu(y,\epsilon)$ in the equation for
$x^{2}$ obtained by squaring the first line in (4.47), we get the equation of
a conic
$0=e(x,y):=x^{2}-[\mathfrak{Re}\;\widetilde{\lambda}^{\pm}(\nu(y,\epsilon),\epsilon)]^{2}$,
with

$e(x,y):=x^{2}+\dfrac{T_{1}^{2}}{(\gamma_{1}-\alpha_{1})^{2}}(y-\tilde{y}(\epsilon))^{2}(1+r(\epsilon^{2}))-4\dfrac{\beta_{2}\beta_{1}}{\gamma_{1}-\alpha_{1}}(y-\widetilde{y}(\epsilon))\epsilon^{2\mathtt{p}}(1+r(\epsilon))-\beta_{1}^{2}\epsilon^{2\mathtt{p}}(1+r(\epsilon))\,.$

Then the canonical form (4.48)-(4.49) follows. ∎

##  5 Taylor expansion of $\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)$

In this section we prove that, for any integer $\mathtt{p}\geq 2$, the entries
of the matrix $\mathtt{L}^{(\mathtt{p})}(\mu,\epsilon)$ in (4.8) satisfy
Assumption 4.4, see Theorem 5.3.

We first need to introduce some notation and definitions.

######  Notation 5.1.

Given a family of operators $A=A(\mu,\epsilon;x)$, analytic in
$(\mu,\epsilon)$ and $2\pi$-periodic in $x$, we define, for any
$i,j\in\mathbb{N}_{0}$,

$A_{i,j}:=\frac{1}{i!j!}\big{(}\partial^{i}_{\mu}\partial^{j}_{\epsilon}A\big{)}(\underline{\mu},0;x)\delta^{i}\epsilon^{j}\,,\quad
A_{\ell}:=\sum_{\begin{subarray}{c}i+j=\ell,\ \atop i,j\geq
0\end{subarray}}\\!\\!\\!A_{i,j}=\sum_{\begin{subarray}{c}i+j=\ell,\ \atop
i,j\geq
0,\kappa\in\mathbb{Z}\end{subarray}}A_{i,j}^{[\kappa]}=\sum_{\kappa\in\mathbb{Z}}A_{\ell}^{[\kappa]}\,,$
(5.1)

where $A_{i,j}^{[\kappa]}$ denotes the component of the operator $A_{i,j}$
with Fourier harmonic $\kappa$, i.e.

$A_{i,j}^{[\kappa]}:=\frac{e^{\mathrm{i}\,\kappa
x}}{2\pi}\int_{\mathbb{T}}A_{i,j}(\mu,\epsilon;y)e^{-\mathrm{i}\,\kappa
y}\mathrm{d}y\,,\qquad
A_{\ell}^{[\kappa]}:=\\!\\!\sum_{\begin{subarray}{c}i+j=\ell,\atop i,j\geq
0\end{subarray}}\\!\\!A_{i,j}^{[\kappa]}\,,\qquad A^{[\kappa]}:=\sum_{\ell\geq
0}A_{\ell}^{[\kappa]}\,.$ (5.2)

It results

$\big{[}A_{i,j}^{[\kappa]}\big{]}^{*}=(A^{*})_{i,j}^{[-\kappa]}\,.$ (5.3)

######  Definition 5.2 (Entanglement coefficients).

For any $\ell\in\mathbb{N}_{0}$, integers $\kappa,j,j^{\prime}\in\mathbb{Z}$
and signs $\sigma,\sigma^{\prime}=\pm$, we define the entanglement
coefficients

$\begin{matrix}{\mathtt{B}_{\ell}^{[\kappa]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j^{\prime},j}^{\sigma^{\prime},\sigma}\end{matrix}:=\big{(}{\mathcal{B}}_{0,\ell}^{[\kappa]}f_{j}^{\sigma},f_{j^{\prime}}^{\sigma^{\prime}}\big{)}$
(5.4)

where ${\mathcal{B}}(\mu,\epsilon)$ is the self-adjoint operator in (2.14) and
$f_{j}^{\sigma}=f_{j}^{\sigma}(\underline{\mu})$ are the eigenvectors of
${\mathcal{L}}_{\underline{\mu},0}$ in (2.23).

We also denote

$\omega_{j}^{\sigma}:=\omega_{j}^{\sigma}(\mathtt{h}):=\omega^{\sigma}\big{(}j+\underline{\varphi}(\mathtt{p},\mathtt{h}),\mathtt{h}\big{)}\,,\quad
j\in\mathbb{Z}\,,\ \sigma=\pm\,,$ (5.5)

where $\omega^{\sigma}(\varphi,\mathtt{h})$ are defined in (2.21),
$\underline{\varphi}(\mathtt{p},\mathtt{h})$ in Lemma 2.7, and

$\Omega_{j}^{(\mathtt{p})}:=\Omega_{j}^{(\mathtt{p})}(\mathtt{h}):=\sqrt{(j+\underline{\varphi}(\mathtt{p},\mathtt{h}))\tanh(\mathtt{h}(j+\underline{\varphi}(\mathtt{p},\mathtt{h})))}\,,\quad
t_{j}^{(\mathtt{p})}:=t_{j}^{(\mathtt{p})}(\mathtt{h}):=\sqrt{\frac{j+\underline{\varphi}(\mathtt{p},\mathtt{h})}{\tanh(\mathtt{h}(j+\underline{\varphi}(\mathtt{p},\mathtt{h})))}}\,.$
(5.6)

######  Theorem 5.3 (Taylor expansion of $B^{(\mathtt{p})}(\mu,\epsilon)$).

For any $\mathtt{p}\geq 2$, the entries
$\alpha^{(\mathtt{p})}(\mu,\epsilon),\beta^{(\mathtt{p})}(\mu,\epsilon),\gamma^{(\mathtt{p})}(\mu,\epsilon)$
of the $2\times 2$ Hermitian matrix $\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon)$
in (4.9) are analytic functions in $B_{\delta_{0}}(\underline{\mu})\times
B_{\epsilon_{0}}(0)$ satisfying Assumption 4.4:
I) Taylor expansion of the off-diagonal entry
$\mathrm{i}\,\beta^{(\mathtt{p})}(\mu,\epsilon)$. The function
$\beta^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)$ has the expansion
(4.16b) with real coefficient

$\displaystyle\beta_{1}^{(\mathtt{p})}(\mathtt{h})\epsilon^{\mathtt{p}}=b_{0}^{(\mathtt{p})}\epsilon^{\mathtt{p}}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\sum_{\sigma_{1},\dots,\sigma_{q}=\pm}\sigma_{1}\dots\sigma_{q}[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}\epsilon^{\mathtt{p}}\,,$
(5.7) $\displaystyle
b_{0}^{(\mathtt{p})}\epsilon^{\mathtt{p}}:=\frac{1}{\mathrm{i}\,}\,\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}\,,\quad\big{[}b_{q}^{(\mathtt{p})}\big{]}_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}\epsilon^{\mathtt{p}}:=\frac{1}{\mathrm{i}\,}\,\dfrac{\begin{matrix}{\mathtt{B}_{j_{1}}^{[j_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{1},0}^{\sigma_{1},-}\end{matrix}\begin{matrix}{\mathtt{B}_{j_{2}-j_{1}}^{[j_{2}-j_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{2},j_{1}}^{\sigma_{2},\sigma_{1}}\end{matrix}\dots\begin{matrix}{\mathtt{B}_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},j_{q}}^{+,\sigma_{q}}\end{matrix}}{(\omega_{j_{1}}^{\sigma_{1}}-\omega_{*}^{(\mathtt{p})})\dots(\omega_{j_{q}}^{\sigma_{q}}-\omega_{*}^{(\mathtt{p})})}\,,$

where $\omega_{j}^{\sigma}$ are defined in (5.5), $\omega_{*}^{(\mathtt{p})}$
is defined in (2.32) and the entanglement coefficients have the explicit
expression

$\begin{matrix}{\mathtt{B}_{\ell}^{[\ell]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j+\ell,j}^{\sigma^{\prime},\sigma}\end{matrix}=\frac{\epsilon^{\ell}}{4}\sqrt{\sigma}\overline{\sqrt{\sigma^{\prime}}}\sqrt{\Omega_{j}^{(\mathtt{p})}\Omega_{j+\ell}^{(\mathtt{p})}}\big{(}a_{\ell}^{[\ell]}-p_{\ell}^{[\ell]}(\sigma
t_{j}^{(\mathtt{p})}+\sigma^{\prime}t_{j+\ell}^{(\mathtt{p})})\big{)}\,,\quad
j,\ell\in\mathbb{N}_{0}\,,\ \ \sigma,\sigma^{\prime}=\pm\,,$ (5.8)

where $p_{\ell}^{[\ell]}$, $a_{\ell}^{[\ell]}\in\mathbb{R}$ are the Fourier
coefficients of $p_{\ell}(x)$ and $a_{\ell}(x)$ in (2.12) and
$\Omega_{j}^{(\mathtt{p})}$ in $t_{j}^{(\mathtt{p})}$ are defined in (5.6).
The function

$\beta_{1}^{(\mathtt{p})}:(0,+\infty)\to\mathbb{R}\,,\quad\mathtt{h}\mapsto\beta_{1}^{(\mathtt{p})}(\mathtt{h})\,,$

is analytic.
II) Taylor expansion of the diagonal entries
$\alpha^{(\mathtt{p})}(\mu,\epsilon)$ and
$\gamma^{(\mathtt{p})}(\mu,\epsilon)$ . The entries
$\alpha^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)$ and
$\gamma^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)$ of the $2\times 2$
Hermitian matrix $\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon)$ in (4.9) have the
expansions (4.16a), (4.16c),

$\displaystyle\alpha^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)=-\omega_{\mathtt{p}}^{+}(\underline{\mu}+\delta)+\alpha_{2}^{(\mathtt{p})}(\mathtt{h})\epsilon^{2}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,$
(5.9)
$\displaystyle\gamma^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)=\omega_{0}^{-}(\underline{\mu}+\delta)+\gamma_{2}^{(\mathtt{p})}(\mathtt{h})\epsilon^{2}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,$

with

$\omega_{\mathtt{p}}^{+}(\underline{\mu}+\delta)=\omega_{*}^{(\mathtt{p})}-\alpha_{1}^{(\mathtt{p})}(\mathtt{h})\delta+r(\delta^{2})\,,\quad\omega_{0}^{-}(\underline{\mu}+\delta)=\omega_{*}^{(\mathtt{p})}+\gamma_{1}^{(\mathtt{p})}(\mathtt{h})\delta+r(\delta^{2})\,,$
(5.10)

and

$\alpha_{1}^{(\mathtt{p})}(\mathtt{h})+\gamma_{1}^{(\mathtt{p})}(\mathtt{h})>0\,,\quad\gamma_{1}^{(\mathtt{p})}(\mathtt{h})-\alpha_{1}^{(\mathtt{p})}(\mathtt{h})>0\,,\quad\forall\mathtt{h}>0\,.$
(5.11)

The functions $\alpha_{2}^{(\mathtt{p})}(\mathtt{h})$,
$\gamma_{2}^{(\mathtt{p})}(\mathtt{h})$ are analytic in $\mathtt{h}$.

The whole section is devoted to the proof of Theorem 5.3.

The following definition encodes an important structure of the operators
${\mathcal{B}}(\mu,\epsilon)$ in (2.14), $P(\mu,\epsilon)$ in (4.1) and
$\mathfrak{B}(\mu,\epsilon)$ in (4.10). Roughly they are operators with
$2\pi$-periodic coefficients in $x$, as in (5.1), whose jests $A_{i,j}$ have
only Fourier harmonics with modulus not larger than $j$ and with the same
parity of $j$. Such structure is inherited from the analogous one of the
Stokes wave.

######  Definition 5.4 (Space $\mathtt{F}_{\ell}$).

For any $\ell\in\mathbb{N}_{0}$ we define the space $\mathtt{F}_{\ell}$ of
operators $A=A_{\ell}$ such that

$A_{\ell}=\sum_{\kappa\in\mathbb{Z}}A_{\ell}^{[\kappa]}=\sum_{\begin{subarray}{c}i+j=\ell,\
\kappa\in\mathbb{Z}\\\ i,j\geq 0\end{subarray}}A_{i,j}^{[\kappa]}$

with Fourier harmonics $|\kappa|\leq j$ and of the same parity of $j$, namely

$A_{i,j}^{[\kappa]}=0\quad\textup{ if }|\kappa|>j\quad\textup{
or}\quad\kappa{\;\scriptsize\begin{matrix}\cancel{\equiv}\end{matrix}\;}j\
(\textup{mod }2)\,,$ (5.12)

in particular $A_{\ell}^{[\kappa]}=0$ if $|\kappa|>\ell$.

The following properties of the spaces $\mathtt{F}_{\ell}$ will be used
repeatedly.

######  Lemma 5.5.

Let $A\in\mathtt{F}_{\ell}$, $B\in\mathtt{F}_{\ell^{\prime}}$. Then

  * (i)

$A\circ B\in\mathtt{F}_{\ell+\ell^{\prime}}$ with $(A\circ
B)^{[\pm(\ell+\ell^{\prime})]}_{\ell+\ell^{\prime}}=A^{[\pm\ell]}_{\ell}\circ
B^{[\pm\ell^{\prime}]}_{\ell^{\prime}}$;

  * (ii)

$A_{\ell}^{[\pm\ell]}=A_{0,\ell}^{[\pm\ell]}$;

  * (iii)

$A^{*}\in\mathtt{F}_{\ell}$.

###### Proof.

$(i)$ By Definition 5.4 of the space $\mathtt{F}_{\ell}$, we have

$A\circ B=\sum_{{i_{1}+j_{1}=\ell\,,\atop
i_{2}+j_{2}=\ell^{\prime}\,,}\atop\kappa_{1},\kappa_{2}\in\mathbb{Z}}A_{i_{1},j_{1}}^{[\kappa_{1}]}\circ
B_{i_{2},j_{2}}^{[\kappa_{2}]}=\sum_{i+j=\ell+\ell^{\prime},\atop i,j\geq
0,\kappa\in\mathbb{Z}}C_{i,j}^{[\kappa]}\ ,\qquad
C_{i,j}^{[\kappa]}:=\sum_{{i_{1}+i_{2}=i\atop
j_{1}+j_{2}=j}\atop\kappa_{1}+\kappa_{2}=\kappa}A_{i_{1},j_{1}}^{[\kappa_{1}]}\circ
B_{i_{2},j_{2}}^{[\kappa_{2}]}$

where, in view of (5.12), the harmonics $\kappa_{1}$ have modulus not larger
than $j_{1}$ and with the same parity of $j_{1}$, and $\kappa_{2}$ have
modulus not larger than $j_{2}$ and with the same parity of $j_{2}$. Hence
each $\kappa=\kappa_{1}+\kappa_{2}$ has modulus not larger than
$j=j_{1}+j_{2}$ and with the same parity of $j_{1}+j_{2}$. Hence $A\circ
B\in\mathtt{F}_{\ell+\ell^{\prime}}$. Moreover since $A$ has harmonics at most
$\ell$ and $B$ at most $\ell^{\prime}$, to reach harmonics
$\pm(\ell+\ell^{\prime})$ the only possibility is via the terms
$A_{i_{1},j_{1}}^{[\pm\ell]}\circ B_{i_{2},j_{2}}^{[\pm\ell^{\prime}]}$ which,
again by the properties of the class $\mathtt{F}_{\ell}$, are non-trivial only
when $(i_{1},j_{1})=(0,\ell)$, $(i_{2},j_{2})=(0,\ell^{\prime})$. $(ii)$ The
only possibility to have harmonics $[\pm\ell]$ is that $j=\ell$, namely $i=0$.
$(iii)$ follows by (5.3). ∎

Hereafter $\mathcal{O}_{k}$ denotes either a linear operator mapping
$H^{1}(\mathbb{T},\mathbb{C}^{2})$ to $L^{2}(\mathbb{T},\mathbb{C}^{2})$ with
operatorial norm of
$\mathcal{O}(\epsilon^{k},\,\delta\epsilon^{k-1}\,,\dots\,,\delta^{k})$ or a
$2\times 2$ complex matrix with analogous estimates on its norm.

###  5.1 Taylor expansion of ${\mathcal{B}}(\mu,\epsilon)$ and
$P(\mu,\epsilon)$

We first describe the Taylor expansion of the operator
${\mathcal{B}}(\underline{\mu}+\delta,\epsilon)$ defined in (2.14).

We remind that, by (2.12) and recalling Definition 2.1, the coefficients
$p_{\ell}(x)$, $a_{\ell}(x)$ of the functions $p_{\epsilon}(x)$,
$a_{\epsilon}(x)$ have the form

$\displaystyle p_{\ell}(x)$
$\displaystyle=\begin{cases}p_{\ell}^{[0]}+p_{\ell}^{[2]}\cos(2x)+\dots+p_{\ell}^{[\ell]}\cos(\ell
x)&\text{if }\ell\text{ is even}\,,\\\\[5.69054pt]
p_{\ell}^{[1]}\cos(x)+p_{\ell}^{[3]}\cos(3x)+\dots+p_{\ell}^{[\ell]}\cos(\ell
x)&\text{if }\ell\text{ is odd}\,,\end{cases}$ (5.13) $\displaystyle
a_{\ell}(x)$
$\displaystyle=\begin{cases}a_{\ell}^{[0]}+a_{\ell}^{[2]}\cos(2x)+\dots+a_{\ell}^{[\ell]}\cos(\ell
x)&\text{if }\ell\text{ is even}\,,\\\\[5.69054pt]
a_{\ell}^{[1]}\cos(x)+a_{\ell}^{[3]}\cos(3x)+\dots+a_{\ell}^{[\ell]}\cos(\ell
x)&\text{if }\ell\text{ is odd}\,,\end{cases}$

with real coefficients $a_{\ell}^{[i]}$, $p_{\ell}^{[i]}$.

######  Lemma 5.6 (Expansion of ${\mathcal{B}}(\mu,\epsilon$).

The Taylor expansion of the self-adjoint operator
${\mathcal{B}}(\mu,\epsilon)$ in (2.14) at $(\underline{\mu},0)$,

${\mathcal{B}}(\underline{\mu}+\delta,\epsilon)=\sum_{\ell=0}^{\mathtt{p}}{\mathcal{B}}_{\ell}+\mathcal{O}_{\mathtt{p}+1}\
,$ (5.14)

satisfies, for any $\ell\in\mathbb{N}_{0}$,

${\mathcal{B}}_{\ell}\in\mathtt{F}_{\ell}\quad\text{ and
}\quad{\mathcal{B}}_{\ell}^{[\pm\ell]}={\mathcal{B}}_{0,\ell}^{[\pm\ell]}=\epsilon^{\ell}\frac{e^{\pm\mathrm{i}\,\ell
x}}{2}\begin{bmatrix}a_{\ell}^{[\ell]}&-p_{\ell}^{[\ell]}(\partial_{x}+\mathrm{i}\,\underline{\mu})\\\
p_{\ell}^{[\ell]}(\partial_{x}+\mathrm{i}\,\underline{\mu}\pm\mathrm{i}\,\ell)&0\end{bmatrix}\,.$
(5.15)

We have

$\displaystyle{\mathcal{B}}_{0,1}$ $\displaystyle=\epsilon$
$\displaystyle\begin{bmatrix}a_{1}(x)&-p_{1}(x)(\partial_{x}+\mathrm{i}\,\underline{\mu})\\\
(\partial_{x}+\mathrm{i}\,\underline{\mu})\circ p_{1}(x)&0\end{bmatrix}\,,$
(5.16a) $\displaystyle{\mathcal{B}}_{0,2}$ $\displaystyle=\epsilon^{2}$
$\displaystyle\begin{bmatrix}a_{2}(x)&-p_{2}(x)(\partial_{x}+\mathrm{i}\,\underline{\mu})\\\
(\partial_{x}+\mathrm{i}\,\underline{\mu})\circ
p_{2}(x)&{\mathtt{f}}_{2}(D+\underline{\mu})^{2}\big{(}1-\tanh^{2}(\mathtt{h}(D+\underline{\mu}))\big{)}\end{bmatrix}\,.$
(5.16b)

###### Proof.

Recall that the operator ${\mathcal{B}}_{i,j}^{[\kappa]}$ is the
$\kappa$-harmonic of the Taylor coefficient of order $\delta^{i}\epsilon^{j}$
of the operator ${\mathcal{B}}(\underline{\mu}+\delta,\epsilon)$ in (2.14) at
$\delta=\epsilon=0$. The terms of order $\epsilon^{j}$ come either from the
terms of order $\epsilon^{j}$ of $a_{\epsilon}$ and $p_{\epsilon}$ which, by
(5.13) have only harmonics with the same parity of $j$ and not larger than
$j$, or from the $\epsilon^{j}$ Taylor coefficient of the Fourier multiplier
$|D+\mu|\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}$, which,
in view of (2.12), expand only in even powers of $\epsilon$. Then
${\mathcal{B}}_{i,j}^{[\kappa]}$ satisfies (5.12), proving that
${\mathcal{B}}_{\ell}\in\mathtt{F}_{\ell}$. The explicit formula for
${\mathcal{B}}_{0,\ell}^{[\ell]}$ follows by Lemma 5.5 and (5.13). Finally
(5.16) follows computing explicitly the Taylor coefficients of
${\mathcal{B}}(\underline{\mu}+\delta,\epsilon)$ in (2.14) and exploiting
(2.12). ∎

A similar expansion holds for the projector $P(\mu,\epsilon)$ defined in
(4.1).

######  Lemma 5.7 (Expansion of $P(\mu,\epsilon)$).

The Taylor expansion of the projector $P(\mu,\epsilon)$ in (4.1) at
$(\underline{\mu},0)$ is

$P(\underline{\mu}+\delta,\epsilon)=\sum_{\ell=0}^{\mathtt{p}}P_{\ell}+\mathcal{O}_{\mathtt{p}+1}$
(5.17)

with

$P_{0}=P(\underline{\mu},0)\,,\qquad
P_{1}=\mathcal{P}\big{[}\textup{${\mathcal{B}}_{1}$}\big{]}\,,\qquad
P_{\ell}=\sum_{q=1}^{\ell}\sum_{\begin{subarray}{c}\ell_{1}+\dots+\ell_{q}=\ell\\\
\ell_{1},\ldots,\ell_{q}\in\mathbb{N}\end{subarray}}\mathcal{P}[{\mathcal{B}}_{\ell_{q}},\dots,{\mathcal{B}}_{\ell_{1}}]\,,$
(5.18)

where, for any $q\in\mathbb{N}$,

$\mathcal{P}\big{[}A_{1},\dots,A_{q}\big{]}:=\frac{(-1)^{q+1}}{2\pi\mathrm{i}\,}\oint_{\Gamma}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}\mathcal{J}A_{1}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}\dots\mathcal{J}A_{q}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}\mathrm{d}\lambda\,.$
(5.19)

It results

$P_{\ell}\in\mathtt{F}_{\ell}\quad\forall\ell\in\mathbb{N}_{0}\,.$ (5.20)

Furthermore, recalling (5.2),

$\mathcal{P}\big{[}A_{1},\dots,A_{q}\big{]}_{i,j}^{[\kappa]}=\sum_{\begin{subarray}{c}i_{1}+\dots+i_{q}=i\\\
j_{1}+\dots+j_{q}=j\\\ \kappa_{1}+\dots+\kappa_{q}=\kappa\\\
\kappa_{1},\dots,\kappa_{q}\in\mathbb{Z}\end{subarray}}\mathcal{P}\big{[}[A_{1}]_{i_{1},j_{1}}^{[\kappa_{1}]},\dots,[A_{q}]_{i_{q},j_{q}}^{[\kappa_{q}]}\big{]}\,.$
(5.21)

###### Proof.

For any $\lambda$ in the resolvent set of ${\mathcal{L}}_{\underline{\mu},0}$,
we Taylor expand the resolvent
$({\mathcal{L}}_{\underline{\mu}+\delta,\epsilon}-\lambda)^{-1}$ at
$\delta=\epsilon=0$ via Neumann series, getting by (5.14)

$\displaystyle({\mathcal{L}}_{\underline{\mu}+\delta,\epsilon}-\lambda)^{-1}=\left(\mathrm{Id}+({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}\sum_{\ell=1}^{\mathtt{p}}\mathcal{J}{\mathcal{B}}_{\ell}+\mathcal{O}_{\mathtt{p}+1}\right)^{-1}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}$
$\displaystyle=({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}+\sum_{\ell=1}^{\mathtt{p}}\sum_{q=1}^{\ell}(-1)^{q}\sum_{\ell_{1}+\ldots+\ell_{q}=\ell\atop\ell_{1},\ldots,\ell_{q}\geq
1}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}\mathcal{J}{\mathcal{B}}_{\ell_{1}}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}\dots\mathcal{J}{\mathcal{B}}_{\ell_{q}}({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}+\mathcal{O}_{\mathtt{p}+1}\
.$

Inserting such expansion in (4.1) yields (5.17)–(5.19). Property (5.20)
descends from the composition rule of Lemma 5.5 together with the fact that
the operator $\mathcal{J}{\mathcal{B}}_{\ell^{\prime}}$ belongs to
$\mathtt{F}_{\ell^{\prime}}$ (cfr. (5.15)) and
${\mathcal{L}}_{\underline{\mu},0}$ is a Fourier multiplier in
$\mathtt{F}_{0}$. Identity (5.21) descends from (5.19) exploiting that
$({\mathcal{L}}_{\underline{\mu},0}-\lambda)^{-1}$ and $\mathcal{J}$ are
Fourier multipliers independent of $\delta,\epsilon$. ∎

###  5.2 Taylor expansion of $\mathfrak{B}(\mu,\epsilon)$

We now prove an efficient formula to compute the Taylor expansion of the
operator $\mathfrak{B}(\mu,\epsilon)$ in (4.10) at any order.

######  Proposition 5.8 (Expansion of $\mathfrak{B}(\mu,\epsilon)$).

The Taylor expansion of the operator $\mathfrak{B}(\mu,\epsilon)$ in (4.10) at
$(\underline{\mu},0)$ is

$\mathfrak{B}(\underline{\mu}+\delta,\epsilon)=\sum_{\ell=0}^{\mathtt{p}}\mathfrak{B}_{\ell}+\mathcal{O}_{\mathtt{p}+1}$
(5.22)

where

$\mathfrak{B}_{\ell}=P_{0}^{*}\big{(}{\mathcal{B}}_{\ell}+{\mathcal{B}}_{\ell-1}P_{1}+\dots+{\mathcal{B}}_{1}P_{\ell-1}\big{)}P_{0}+P_{0}^{*}{\mathcal{N}}_{\ell}P_{0}\,,\quad\mathfrak{B}_{\ell}\in\mathtt{F}_{\ell}\,,$
(5.23)

the operators ${\mathcal{B}}_{1},\ldots,{\mathcal{B}}_{\ell}$ and
$P_{1},\ldots,P_{\ell-1}$ are the jets of the operators
${\mathcal{B}}(\mu,\epsilon)$ in (2.14) and $P(\mu,\epsilon)$ in (4.1) at
$(\underline{\mu},0)$ respectively, and

${\mathcal{N}}_{\ell}\in\mathtt{F}_{\ell}\qquad\textup{satisfy}\qquad{\mathcal{N}}_{0}={\mathcal{N}}_{1}={\mathcal{N}}_{2}=0\quad\textup{
and
}\quad{\mathcal{N}}_{\ell}^{[\pm\ell]}=0\,,\quad\forall\ell=3,\dots,\mathtt{p}\,.$
(5.24)

The rest of section is devoted to the proof of Proposition 5.8.
We will use that the operators ${\mathcal{B}}(\mu,\epsilon)$ and
$P(\mu,\epsilon)$ satisfy

$\displaystyle({\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon))^{*}=P(\mu,\epsilon)^{*}{\mathcal{B}}(\mu,\epsilon)={\mathcal{B}}_{\mu,\epsilon}P(\mu,\epsilon)\,,$
(5.25) $\displaystyle
Q(P(\mu,\epsilon)-P_{0})^{2}=(P(\mu,\epsilon)-P_{0})^{2}Q\quad\mbox{ with
}\quad Q\in\\{P(\mu,\epsilon),P_{0}\\}\,,$ (5.26)

where $P_{0}=P(\underline{\mu},0)$. Property (5.25) follows by (4.2), (4.3),
$\mathcal{J}^{2}=-\text{Id}$ and since ${\mathcal{B}}(\mu,\epsilon)$ is self-
adjoint. Property (5.26) is satisfied by any pair of projectors.

For any operator $A$ we set
$\textup{{Sym}}[A]:=\frac{1}{2}A+\frac{1}{2}A^{*}$.

######  Lemma 5.9 (Splitting of $\mathfrak{B}(\mu,\epsilon)$).

The operator $\mathfrak{B}(\mu,\epsilon)$ in (4.10) can be written as

$\textup{$\mathfrak{B}$}(\mu,\epsilon)=P_{0}^{*}\textup{$\mathbf{Sym}$}\big{[}{\mathcal{B}}(\mu,\epsilon)+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)+{\mathcal{R}}(\mu,\epsilon)\big{]}P_{0}$
(5.27)

with
$\mathring{{\mathcal{B}}}(\mu,\epsilon):={\mathcal{B}}(\mu,\epsilon)-{\mathcal{B}}_{0}$,
$\mathring{P}(\mu,\epsilon):=P(\mu,\epsilon)-P_{0}$ and

${\mathcal{R}}_{\ell}\in\mathtt{F}_{\ell}\qquad\textup{satisfy}\qquad{\mathcal{R}}_{0}={\mathcal{R}}_{1}={\mathcal{R}}_{2}=0\,,\quad{\mathcal{R}}_{\ell}^{[\pm\ell]}=0\,,\quad\forall\ell=3,\dots,\mathtt{p}\,.$
(5.28)

###### Proof.

By (4.5) and denoting $g(x):=(1-x)^{-\frac{1}{2}}-1$, we first write

$U_{\mu,\epsilon}P_{0}=\big{(}\mathrm{Id}+g(\mathring{P}(\mu,\epsilon)^{2})\big{)}P(\mu,\epsilon)P_{0}\,.$

Then, by (4.10) and using also (5.26) with $Q=P(\mu,\epsilon)$, we get

$\displaystyle{\textup{$\mathfrak{B}$}}(\mu,\epsilon)$
$\displaystyle=P_{0}^{*}U_{\mu,\epsilon}^{*}{\mathcal{B}}(\mu,\epsilon)U_{\mu,\epsilon}P_{0}$
(5.29)
$\displaystyle=P_{0}^{*}\big{(}\mathrm{Id}+g^{*}(\mathring{P}(\mu,\epsilon)^{2})\big{)}P(\mu,\epsilon)^{*}{\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)\big{(}\mathrm{Id}+g(\mathring{P}(\mu,\epsilon)^{2})\big{)}P_{0}\,.$

Next using (5.25) we obtain

$\displaystyle{\textup{$\mathfrak{B}$}}(\mu,\epsilon)$
$\displaystyle=P_{0}^{*}\big{(}\mathrm{Id}+g^{*}(\mathring{P}(\mu,\epsilon)^{2})\big{)}{\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)\big{(}\mathrm{Id}+g(\mathring{P}(\mu,\epsilon)^{2})\big{)}P_{0}$
(5.30)
$\displaystyle=P_{0}^{*}\textup{$\mathbf{Sym}$}\big{[}{\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)+2{\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)g(\mathring{P}(\mu,\epsilon)^{2})+g^{*}(\mathring{P}(\mu,\epsilon)^{2}){\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)g(\mathring{P}(\mu,\epsilon)^{2})\big{]}P_{0}\,.$

Finally, since
$P(\mu,\epsilon)P_{0}=(\mathrm{Id}+\mathring{P}(\mu,\epsilon))P_{0}$ we write

${\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)P_{0}={\mathcal{B}}(\mu,\epsilon)P_{0}+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)P_{0}+{\mathcal{B}}_{0}\mathring{P}(\mu,\epsilon)P_{0}$

and, using that $g(\mathring{P}(\mu,\epsilon)^{2})$ commutes with $P_{0}$ by
(5.26) with $Q=P_{0}$, we conclude that

$\textup{$\mathfrak{B}$}(\mu,\epsilon)=P_{0}^{*}\textup{$\mathbf{Sym}$}\big{[}{\mathcal{B}}(\mu,\epsilon)+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)+\mathcal{R}(\mu,\epsilon)\big{]}P_{0}$
(5.31)

where

$\displaystyle\mathcal{R}(\mu,\epsilon)$
$\displaystyle:=P_{0}^{*}\big{[}{{\mathcal{B}}_{0}\mathring{P}(\mu,\epsilon)+2{\mathcal{B}}_{0}g(\mathring{P}(\mu,\epsilon)^{2})}+2\mathring{{\mathcal{B}}}(\mu,\epsilon)g(\mathring{P}(\mu,\epsilon)^{2})$
(5.32) $\displaystyle\ \
+2{\mathcal{B}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)g(\mathring{P}(\mu,\epsilon)^{2})+g^{*}(\mathring{P}(\mu,\epsilon)^{2}){\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)g(\mathring{P}(\mu,\epsilon)^{2})\big{]}P_{0}\,.$

We now show that $\mathcal{R}(\mu,\epsilon)$ satisfies the properties claimed
in (5.28).
${\mathcal{R}}_{\ell}\in\mathtt{F}_{\ell}$. It is a consequence of Lemma 5.5,
(5.15) and the fact that
$\big{[}g(\mathring{P}(\mu,\epsilon)^{2})\big{]}_{\ell}\in\mathtt{F}_{\ell}$,
which, writing
$\big{[}g(\mathring{P}(\mu,\epsilon)^{2})\big{]}_{\ell}=\sum_{k\geq
1}c_{k}\sum_{\ell_{1}+\ldots+\ell_{2k}=\ell\atop\ell_{1},\ldots,\ell_{2k}\geq
1}P_{\ell_{1}}\circ\cdots\circ P_{\ell_{2k}}$ (the $c_{k}$’s are the Taylor
coefficients of $g(x)$ at $x=0$), follows by (5.20) and the composition
property of Lemma 5.5.
$\mathcal{R}_{0}=\mathcal{R}_{1}=\mathcal{R}_{2}=0$. We first exhibit a
cancellation in the first two terms in (5.32). Precisely we claim that

$P_{0}^{*}{\mathcal{B}}_{0}\mathring{P}(\mu,\epsilon)P_{0}+2P_{0}^{*}{\mathcal{B}}_{0}g(\mathring{P}(\mu,\epsilon)^{2})P_{0}=2P_{0}^{*}{\mathcal{B}}_{0}{\mathcal{Q}}_{\mu,\epsilon}\mathring{P}(\mu,\epsilon)^{4}P_{0}\,,\quad{\mathcal{Q}}_{\mu,\epsilon}:=\widetilde{g}(\mathring{P}(\mu,\epsilon)^{2})\,,$
(5.33)

where $\widetilde{g}(x)$ is the analytic function such that
$g(x)=\frac{1}{2}x+\widetilde{g}(x)x^{2}$. Since the operator
${\mathcal{Q}}_{\mu,\epsilon}$ in (5.33) is $\mathcal{O}_{4}$ and all the
other terms in (5.32) are at least $\mathcal{O}_{3}$, we deduce that
$\mathcal{R}_{0}=\mathcal{R}_{1}=\mathcal{R}_{2}=0$. Moreover as above one has
${\mathcal{Q}}_{\ell}\in\mathtt{F}_{\ell}$ (argue as for
${\mathcal{R}}_{\ell}$). Let us prove (5.33). By definition of
$\widetilde{g}(x)$ we have
$g(\mathring{P}(\mu,\epsilon)^{2})=\tfrac{1}{2}\mathring{P}(\mu,\epsilon)^{2}+{\mathcal{Q}}_{\mu,\epsilon}\mathring{P}(\mu,\epsilon)^{4}\,,$
and, using also (5.25),

$P_{0}^{*}{\mathcal{B}}_{0}\mathring{P}(\mu,\epsilon)P_{0}+2P_{0}^{*}{\mathcal{B}}_{0}g(\mathring{P}(\mu,\epsilon)^{2})P_{0}=\underbrace{P_{0}^{*}{\mathcal{B}}_{0}P_{0}\mathring{P}(\mu,\epsilon)P_{0}+P_{0}^{*}{\mathcal{B}}_{0}\mathring{P}(\mu,\epsilon)^{2}P_{0}}_{=0}+2P_{0}^{*}{\mathcal{B}}_{0}{\mathcal{Q}}_{\mu,\epsilon}\mathring{P}(\mu,\epsilon)^{4}P_{0}$
(5.34)

where the underbraced term vanishes because of the identity
$P_{0}\mathring{P}(\mu,\epsilon)P_{0}=-\mathring{P}(\mu,\epsilon)^{2}P_{0}$,
which comes from the projector identity
$P_{0}+\mathring{P}(\mu,\epsilon)=(P_{0}+\mathring{P}(\mu,\epsilon))^{2}$.
This proves (5.33).
${\mathcal{R}}_{\ell}^{[\pm\ell]}=0$ for any $\ell=3,\dots,\mathtt{p}$. We
first note that, by (5.32)-(5.33) and (5.26) with $Q=P_{0}$, the operator
$\mathcal{R}(\mu,\epsilon)$ has the form

$\textup{${\mathcal{R}}(\mu,\epsilon)$}=P_{0}^{*}\textup{$\mathcal{E}$}(\mu,\epsilon)P_{0}\mathring{P}(\mu,\epsilon)^{2}P_{0}$
(5.35)

with

$\displaystyle\mathcal{E}(\mu,\epsilon)$
$\displaystyle:=2{\mathcal{B}}_{0}\mathcal{Q}_{\mu,\epsilon}\mathring{P}(\mu,\epsilon)^{2}+2\mathring{{\mathcal{B}}}(\mu,\epsilon)\,\breve{g}(\mathring{P}(\mu,\epsilon)^{2})+2{\mathcal{B}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)\,\breve{g}(\mathring{P}(\mu,\epsilon)^{2})$
(5.36)
$\displaystyle\quad+g^{*}(\mathring{P}(\mu,\epsilon)^{2}){\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)\,\breve{g}(\mathring{P}(\mu,\epsilon)^{2})\,,\quad\mathcal{E}_{0}=0\,,$

where $\breve{g}(x)$ is the analytic function such that $g(x)=\breve{g}(x)x$.
By Lemma 5.5, (5.14), (5.17) we deduce that

$\mathcal{E}_{\ell}\in\mathtt{F}_{\ell}\,,\qquad\ell\in\mathbb{N}\,.$ (5.37)

We now show that ${\mathcal{R}}_{\ell}^{[\pm\ell]}=0$, for any
$\ell=3,\dots,\mathtt{p}$, or, equivalently,

$\big{(}{\mathcal{R}}_{\ell}^{[\pm\ell]}\phi_{1},\phi_{2}\big{)}\stackrel{{\scriptstyle\eqref{thisR}}}{{=}}\big{(}P_{0}^{*}{\mathcal{R}}_{\ell}^{[\pm\ell]}P_{0}\phi_{1},\phi_{2}\big{)}=\big{(}{\mathcal{R}}_{\ell}^{[\pm\ell]}P_{0}\phi_{1},P_{0}\phi_{2}\big{)}=0\,,\quad\forall\phi_{1},\phi_{2}\in
L^{2}(\mathbb{T})\,,$

which boils down to prove that, for any $\ell=3,\dots,\mathtt{p}$,

$({\mathcal{R}}_{\ell}^{[\pm\ell]}f_{0}^{-},f_{0}^{-})=0\,,\quad({\mathcal{R}}_{\ell}^{[\pm\ell]}f_{\mathtt{p}}^{+},f_{\mathtt{p}}^{+})=0\,,\quad({\mathcal{R}}_{\ell}^{[\pm\ell]}f_{\mathtt{p}}^{+},f_{0}^{-})=0\,,\quad({\mathcal{R}}_{\ell}^{[\pm\ell]}f_{0}^{-},f_{\mathtt{p}}^{+})=0\,.$
(5.38)

Since ${\mathcal{R}}_{\ell}\in\mathtt{F}_{\ell}$, by directly comparing the
harmonics in the left and right vectors of the scalar product, all these
identities all trivial except for
$({\mathcal{R}}_{\mathtt{p}}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})=0$
and
$({\mathcal{R}}_{\mathtt{p}}^{[-\mathtt{p}]}f_{\mathtt{p}}^{+},f_{0}^{-})=0$.
In view of (5.35) and Lemma 5.5 we have

$\big{(}{\mathcal{R}}_{\mathtt{p}}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+}\big{)}=\sum_{\ell_{1}+\ell_{2}+\ell_{3}=\mathtt{p}\atop\ell_{1},\ell_{2},\ell_{3}\geq
1}\big{(}\mathcal{E}_{\ell_{1}}^{[\ell_{1}]}\underbrace{P_{0}P_{\ell_{2}}^{[\ell_{2}]}P_{\ell_{3}}^{[\ell_{3}]}f_{0}^{-}}_{=:\phi},f_{\mathtt{p}}^{+}\big{)}\,.$
(5.39)

Since $\phi=\alpha f_{0}^{-}+\beta
f_{\mathtt{p}}^{+}\in{\mathcal{V}}_{\underline{\mu},0}$ and, by (5.37),
$\mathcal{E}_{\ell_{1}}^{[\ell_{1}]}\in\mathtt{F}_{\ell_{1}}$ with
$1\leq\ell_{1}\leq\mathtt{p}-1$, the scalar product in (5.39) vanishes.
Similarly
$({\mathcal{R}}_{\mathtt{p}}^{[-\mathtt{p}]}f_{\mathtt{p}}^{+},f_{0}^{-})=0$.
Thus (5.38) and consequently the last identity in (5.28) hold. ∎

In the next lemma we simplify the first terms in (5.27).

######  Lemma 5.10.

One has

$P_{0}^{*}\textup{$\mathbf{Sym}$}\big{[}{\mathcal{B}}(\mu,\epsilon)+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)\big{]}P_{0}=P_{0}^{*}\big{(}{\mathcal{B}}(\mu,\epsilon)+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)\big{)}P_{0}\,.$
(5.40)

###### Proof.

We begin by proving the following identities

$P_{0}^{*}\textup{${\mathcal{B}}_{0}$}=-\mathrm{i}\,\omega_{*}^{(\mathtt{p})}P_{0}^{*}\textup{$\mathcal{J}$}\,,\qquad
P_{0}^{*}\textup{${\mathcal{B}}_{0}$}P(\mu,\epsilon)P_{0}=P_{0}^{*}P(\mu,\epsilon)^{*}\textup{${\mathcal{B}}_{0}$}P_{0}\,.$
(5.41)

The first identity descends from the fact that, for any $f,g$ in
$L^{2}(\mathbb{T},\mathbb{C}^{2})$,

$\big{(}P_{0}^{*}\textup{${\mathcal{B}}_{0}$}f,g\big{)}=\big{(}f,\textup{${\mathcal{B}}_{0}$}P_{0}g\big{)}=\big{(}\mathcal{J}f,{\mathcal{L}}_{\underline{\mu},0}P_{0}g\big{)}=-\mathrm{i}\,\omega_{*}^{(\mathtt{p})}\big{(}\mathcal{J}f,P_{0}g\big{)}=-\mathrm{i}\,\omega_{*}^{(\mathtt{p})}\big{(}P_{0}^{*}\mathcal{J}f,g\big{)}\,.$

The second identity descends from the first one, since

$\displaystyle
P_{0}^{*}{\mathcal{B}}_{0}P(\mu,\epsilon)P_{0}=-\mathrm{i}\,\omega_{*}^{(\mathtt{p})}P_{0}^{*}\mathcal{J}P(\mu,\epsilon)P_{0}$
$\displaystyle\stackrel{{\scriptstyle\eqref{propPU}}}{{=}}P_{0}^{*}P(\mu,\epsilon)^{*}(-\mathrm{i}\,\omega_{*}^{(\mathtt{p})}P_{0}^{*}\mathcal{J})$
$\displaystyle=P_{0}^{*}P(\mu,\epsilon)^{*}P_{0}^{*}{\mathcal{B}}_{0}\stackrel{{\scriptstyle\eqref{BP}}}{{=}}P_{0}^{*}P(\mu,\epsilon)^{*}{\mathcal{B}}_{0}P_{0}\,.$

We use (5.41) to deduce that
$P_{0}^{*}\big{(}{\mathcal{B}}(\mu,\epsilon)+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)\big{)}P_{0}$
is selfadjoint. Indeed

$\displaystyle
P_{0}^{*}\big{(}{\mathcal{B}}(\mu,\epsilon)+\mathring{{\mathcal{B}}}(\mu,\epsilon)\mathring{P}(\mu,\epsilon)\big{)}P_{0}$
$\displaystyle=P_{0}^{*}\big{(}{\mathcal{B}}(\mu,\epsilon)+{\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)+{\mathcal{B}}_{0}P_{0}-{\mathcal{B}}_{0}P(\mu,\epsilon)-{\mathcal{B}}(\mu,\epsilon)P_{0}\big{)}P_{0}$
$\displaystyle=P_{0}^{*}\big{(}\overbrace{{\mathcal{B}}(\mu,\epsilon)P(\mu,\epsilon)}^{(\mathrm{I})}+\overbrace{{\mathcal{B}}_{0}P_{0}}^{(\mathrm{II})}-\overbrace{{\mathcal{B}}_{0}P(\mu,\epsilon)}^{(\mathrm{III})}\big{)}P_{0}\,.$

The terms (I) and (II) are self-adjoint by (5.25). Also the operator
$P_{0}^{*}(\mathrm{III})P_{0}$ is self-adjoint by the second identity in
(5.41). This proves (5.40). ∎

Proof of Proposition 5.8. By (5.27) and (5.40) we have

$\mathfrak{B}(\mu,\epsilon)=P_{0}^{*}\big{(}{\mathcal{B}}(\mu,\epsilon)+({\mathcal{B}}(\mu,\epsilon)-{\mathcal{B}}(\underline{\mu},0))(P(\mu,\epsilon)-P(\underline{\mu},0))+{\mathcal{N}}(\mu,\epsilon)\big{)}P_{0}$

where
${\mathcal{N}}(\mu,\epsilon):=\textup{$\mathbf{Sym}$}[{\mathcal{R}}(\mu,\epsilon)]$.
Consequently the jets $\mathfrak{B}_{\ell}$ of $\mathfrak{B}(\mu,\epsilon)$
have the form in (5.23) and, by Lemma 5.5, the fact that
${\mathcal{B}}_{\ell^{\prime}},P_{\ell^{\prime}}\in\mathtt{F}_{\ell^{\prime}}$
for any $1\leq\ell^{\prime}\leq\ell$ (cfr. (5.15), (5.20)), and (5.28) we
conclude that $\mathfrak{B}_{\ell}\in\mathtt{F}_{\ell}$. By (5.28) the
operator ${\mathcal{N}}(\mu,\epsilon)$ fulfills (5.24), using also Lemma 5.5
for the last identity. ∎

###  5.3 Entanglement coefficients

We restrict the following arguments to depths $\mathtt{h}>0$ such that
$\underline{\mu}=\underline{\varphi}(\mathtt{p},\mathtt{h})$ in Lemma 2.7 is
not an integer so that the eigenvectors
$\\{f_{j}^{\sigma}\\}_{j\in\mathbb{Z},\sigma=\pm}$,
$f_{j}^{\sigma}=f_{j}^{\sigma}(\underline{\mu})$ in (2.23) at
$\mu=\underline{\mu}$ form a complex symplectic basis of
$L^{2}(\mathbb{T},\mathbb{C}^{2})$, cfr. Remark 2.10. This is very useful for
computational purposes. Since the function
$\mathtt{h}\mapsto\underline{\varphi}(\mathtt{p},\mathtt{h})$ is bounded and
real-analytic the excluded depths are isolated and form a closed set, thus its
complementary is dense in $\mathtt{h}>0$. In particular, the formula for
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ in (5.7), although proved for
$\mathtt{h}$ in a dense set of $(0,+\infty)$, holds for all $\mathtt{h}>0$ by
continuity (the function $\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ is actually
analytic in $\mathtt{h}>0$).

By (5.1), (5.3) and the self-adjointness of ${\mathcal{B}}_{\mu,\epsilon}$,
the entanglement coefficients in Definition 5.2 fulfill

$\begin{matrix}{\mathtt{B}_{\ell}^{[\kappa]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j^{\prime},j}^{\sigma^{\prime},\sigma}\end{matrix}=0\quad\text{
if }j^{\prime}\neq j+\kappa\,,\quad\text{ and
}\quad\overline{\begin{matrix}{\mathtt{B}_{\ell}^{[\kappa]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j^{\prime},j}^{\sigma^{\prime},\sigma}\end{matrix}}=\
\begin{matrix}{\mathtt{B}_{\ell}^{[-\kappa]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j,j^{\prime}}^{\sigma,\sigma^{\prime}}\end{matrix}\,.$
(5.42)

The next lemma, proved in [9, Lemma 4.1] (Item ($ii$) is formula [9, (4.19)]),
provides effective formulas to compute the action of the operators
${\mathcal{P}}[{\mathcal{B}}_{\ell_{1}},\dots,{\mathcal{B}}_{\ell_{q}}]$ on
the vector basis.

######  Lemma 5.11.

Let
$\begin{matrix}{\mathtt{B}_{\ell}^{[\kappa]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j^{\prime},j}^{\sigma^{\prime},\sigma}\end{matrix}$
denote the entanglement coefficients in (5.4) and
$f_{j}^{\sigma}:=f_{j}^{\sigma}(\underline{\mu})$ in (2.23), with
$\underline{\mu}\in\mathbb{R}\setminus\mathbb{Z}$. Then:
(i) for any $q\in\mathbb{N}$, $\ell_{1},\dots,\ell_{q}\in\mathbb{N}$ integers
$j,j^{\prime},\kappa_{1},\dots,\kappa_{q}\in\mathbb{Z}$ and
$\sigma,\sigma^{\prime}=\pm$, one has

$\displaystyle\big{(}{\mathcal{B}}_{0,\ell_{q+1}}^{[\kappa_{q+1}]}\mathcal{P}[{\mathcal{B}}_{0,\ell_{q}}^{[\kappa_{q}]},\dots,{\mathcal{B}}_{0,\ell_{1}}^{[\kappa_{1}]}]f_{j}^{\sigma},f_{j^{\prime}}^{\sigma^{\prime}}\big{)}$
(5.43)
$\displaystyle\qquad=\sum_{\sigma_{1},\dots,\sigma_{q}=\pm}\sigma_{1}\dots\sigma_{q}\begin{matrix}{\mathtt{B}_{\ell_{1}}^{[\kappa_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{1},j}^{\sigma_{1},\sigma}\end{matrix}\begin{matrix}{\mathtt{B}_{\ell_{2}}^{[\kappa_{2}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{2},j_{1}}^{\sigma_{2},\sigma_{1}}\end{matrix}\dots\begin{matrix}{\mathtt{B}_{\ell_{q}}^{[\kappa_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{q},j_{q-1}}^{\sigma_{q},\sigma_{q-1}}\end{matrix}\begin{matrix}{\mathtt{B}_{\ell_{q+1}}^{[\kappa_{q+1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j^{\prime},j_{q}}^{\sigma^{\prime},\sigma_{q}}\end{matrix}\mathtt{R}{\scriptsize\begin{matrix}\sigma,&\sigma_{1},&\dots,&\sigma_{q}\\\\[1.99168pt]
j,&j_{1},&\dots,&j_{q}\end{matrix}}$

where $j_{1}:=j+\kappa_{1}$, $j_{2}:=j_{1}+\kappa_{2}$, …,
$j_{q}=j_{q-1}+\kappa_{q}$ and

$\mathtt{R}{\scriptsize\begin{matrix}\sigma,&\sigma_{1},&\dots,&\sigma_{q}\\\\[1.99168pt]
j,&j_{1},&\dots,&j_{q}\end{matrix}}:=\dfrac{(-\mathrm{i}\,\\!)^{q}}{2\pi\mathrm{i}\,}\oint_{\Gamma}\dfrac{\mathrm{d}\lambda}{(\lambda-\mathrm{i}\,\omega_{j}^{\sigma})(\lambda-\mathrm{i}\,\omega_{j_{1}}^{\sigma_{1}})\dots(\lambda-\mathrm{i}\,\omega_{j_{q}}^{\sigma_{q}})}\,,$
(5.44)

where $\Gamma$ is a circuit winding once around
$\mathrm{i}\,\omega_{*}^{(\mathtt{p})}$ counterclockwise and
$\omega_{j}^{\sigma}$ in (5.5).
(ii) Let $j_{1},\dots,j_{q}\in\mathbb{Z}$ and
$\sigma_{1},\dots,\sigma_{q}=\pm$, with
$(j_{1},\sigma_{1}),\dots,(j_{q},\sigma_{q})\notin\\{(0,-),(\mathtt{p},+)\\}$.
Then

$\mathtt{R}{\scriptsize\begin{matrix}-,&\sigma_{1},&\dots,&\sigma_{q}\\\\[1.99168pt]
0,&j_{1},&\dots,&j_{q}\end{matrix}}=\mathtt{R}{\scriptsize\begin{matrix}+,&\sigma_{1},&\dots,&\sigma_{q}\\\\[1.99168pt]
p,&j_{1},&\dots,&j_{q}\end{matrix}}=\dfrac{1}{(\omega_{j_{1}}^{\sigma_{1}}-\omega_{*}^{(\mathtt{p})})\ldots(\omega_{j_{q}}^{\sigma_{q}}-\omega_{*}^{(\mathtt{p})})}$
(5.45)

with $\omega_{j}^{\sigma}$ in (5.5).

We also compute the entanglement coefficients in (5.8). We use that
$\Omega_{j}^{(\mathtt{p})}$ and $t_{j}^{(\mathtt{p})}$ in (5.6) satisfy

$\tanh\big{(}\mathtt{h}(j+\underline{\varphi}(\mathtt{p},\mathtt{h}))\big{)}=\tfrac{\Omega_{j}^{(\mathtt{p})}}{t_{j}^{(\mathtt{p})}}\,,\quad
j+\underline{\varphi}(\mathtt{p},\mathtt{h})=\Omega_{j}^{(\mathtt{p})}t_{j}^{(\mathtt{p})}\,,\quad
j\in\mathbb{N}_{0}\,,$ (5.46)

since $j+\underline{\varphi}(\mathtt{p},\mathtt{h})>0$ for any $j\geq 0$.

######  Lemma 5.12.

For any $j\in\mathbb{Z}$, $\sigma,\sigma^{\prime}=\pm 1$,
$\ell=1,\dots,\mathtt{p}$ the coefficient
$\begin{matrix}{\mathtt{B}_{\ell}^{[\ell]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j+\ell,j}^{\sigma^{\prime},\sigma}\end{matrix}$
in (5.4) has the expression in (5.8).

###### Proof.

By (5.15), (2.23) and (5.6) we have

${\mathcal{B}}_{0,\ell}^{[\pm\ell]}f_{j}^{\sigma}=\frac{e^{\mathrm{i}\,(j\pm\ell)x}}{2\sqrt{2\Omega_{j}^{(\mathtt{p})}}}\begin{bmatrix}-a_{\ell}^{[\ell]}\sqrt{\sigma}\Omega_{j}^{(\mathtt{p})}-\mathrm{i}\,\sqrt{-\sigma}p_{\ell}^{[\ell]}(j+\underline{\mu})\\\
-\mathrm{i}\,\sqrt{\sigma}p_{\ell}^{[\ell]}(j\pm\ell+\underline{\mu})\Omega_{j}^{(\mathtt{p})}\end{bmatrix}\epsilon^{\ell}\,,$

whence, by (5.4), (2.23) we obtain

$\begin{matrix}{\mathtt{B}_{\ell}^{[\pm\ell]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j\pm\ell,j}^{\sigma^{\prime},\sigma}\end{matrix}=\frac{\epsilon^{\ell}}{4\sqrt{\Omega_{j}^{(\mathtt{p})}\Omega_{j\pm\ell}^{(\mathtt{p})}}}\big{(}\sqrt{\sigma}\overline{\sqrt{\sigma^{\prime}}}a_{\ell}^{[\ell]}\Omega_{j}^{(\mathtt{p})}\Omega_{j\pm\ell}^{(\mathtt{p})}+\mathrm{i}\,\sqrt{-\sigma}\overline{\sqrt{\sigma^{\prime}}}p_{\ell}^{[\ell]}(j+\underline{\mu})\Omega_{j\pm\ell}^{(\mathtt{p})}-\mathrm{i}\,\sqrt{\sigma}\overline{\sqrt{-\sigma^{\prime}}}p_{\ell}^{[\ell]}(j\pm\ell+\underline{\mu})\Omega_{j}^{(\mathtt{p})}\big{)}\,.$

By exploiting the second identity in (5.46) and using that, for
$\sigma,\sigma^{\prime}=\pm$,

$\displaystyle\mathrm{i}\,\sqrt{-\sigma}\overline{\sqrt{\sigma^{\prime}}}=-\sigma\sqrt{\sigma}\overline{\sqrt{\sigma^{\prime}}},\quad-\mathrm{i}\,\sqrt{\sigma}\overline{\sqrt{-\sigma^{\prime}}}=-\sigma^{\prime}\sqrt{\sigma}\overline{\sqrt{\sigma^{\prime}}},\quad\sqrt{-\sigma}\overline{\sqrt{-\sigma^{\prime}}}=\sigma\sigma^{\prime}\sqrt{\sigma}\overline{\sqrt{\sigma^{\prime}}}\,,$
(5.47)

we obtain formula (5.8). ∎

###  5.4 Taylor expansion of $\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon)$

We now complete the proof of Theorem 5.3.

For any $\mathtt{p}\geq 2$ the matrix
$\mathtt{B}^{(\mathtt{p})}(\mu,\epsilon)$ in (4.9) admits the Taylor expansion

$\mathtt{B}^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)=\sum_{\ell=0}^{\mathtt{p}}\mathtt{B}_{\ell}+\mathcal{O}_{\mathtt{p}+1}\qquad\text{where}\qquad\mathtt{B}_{\ell}:=\begin{pmatrix}\big{(}\mathfrak{B}_{\ell}f_{\mathtt{p}}^{+},f_{\mathtt{p}}^{+}\big{)}&\big{(}\mathfrak{B}_{\ell}f_{0}^{-},f_{\mathtt{p}}^{+}\big{)}\\\
\big{(}\mathfrak{B}_{\ell}f_{\mathtt{p}}^{+},f_{0}^{-}\big{)}&\big{(}\mathfrak{B}_{\ell}f_{0}^{-},f_{0}^{-}\big{)}\end{pmatrix}$
(5.48)

and the operators $\mathfrak{B}_{\ell}$ have the expansion (5.23).

######  Lemma 5.13 (Off-diagonal entries of $\mathtt{B}_{\ell}$).

The entries $(\mathfrak{B}_{\ell}f_{0}^{-},f_{\mathtt{p}}^{+})$ of the
matrices $\mathtt{B}_{\ell}$ in (5.48) are

$(\mathfrak{B}_{\ell}f_{0}^{-},f_{\mathtt{p}}^{+})=0\,,\quad\forall\ell=0,\dots,\mathtt{p}-1\,,\quad\text{and}\qquad(\mathfrak{B}_{\mathtt{p}}f_{0}^{-},f_{\mathtt{p}}^{+})=\mathrm{i}\,\beta_{1}^{(\mathtt{p})}\epsilon^{\mathtt{p}}$

with $\beta_{1}^{(\mathtt{p})}$ in (5.7).

###### Proof.

The scalar product
$(\mathfrak{B}_{\ell}f_{0}^{-},f_{\mathtt{p}}^{+})=(\mathfrak{B}_{\ell}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})$
vanishes for any $\ell=0,\ldots,\mathtt{p}-1$ because
$\mathfrak{B}_{\ell}\in\mathtt{F}_{\ell}$ by (5.23) (recall (5.12)). We now
prove the formula (5.7) for $\beta_{1}^{(\mathtt{p})}$. We have
$(\textup{$\mathfrak{B}$}_{\mathtt{p}}f_{0}^{-},f_{\mathtt{p}}^{+})=(\textup{$\mathfrak{B}$}_{\mathtt{p}}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})$,
and, by (5.23) and (5.24),

$\displaystyle(\textup{$\mathfrak{B}$}_{\mathtt{p}}f_{0}^{-},f_{\mathtt{p}}^{+})$
$\displaystyle\,\;\,=\;\,\,({\mathcal{B}}_{\mathtt{p}}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})+([{\mathcal{B}}_{\mathtt{p}-1}P_{1}]^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})+\dots+([{\mathcal{B}}_{1}P_{\mathtt{p}-1}]^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})$
$\displaystyle\stackrel{{\scriptstyle\text{Lemma}\,\ref{TFjprop}}}{{=}}({\mathcal{B}}_{\mathtt{p}}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})+({\mathcal{B}}_{\mathtt{p}-1}^{[\mathtt{p}-1]}P_{1}^{[1]}f_{0}^{-},f_{\mathtt{p}}^{+})+\dots+({\mathcal{B}}_{1}^{[1]}P_{\mathtt{p}-1}^{[\mathtt{p}-1]}f_{0}^{-},f_{\mathtt{p}}^{+})$
(5.49)
$\displaystyle\stackrel{{\scriptstyle\eqref{cBellell}}}{{=}}({\mathcal{B}}_{0,p}^{[\mathtt{p}]}f_{0}^{-},f_{\mathtt{p}}^{+})+({\mathcal{B}}_{0,\mathtt{p}-1}^{[\mathtt{p}-1]}P_{0,1}^{[1]}f_{0}^{-},f_{\mathtt{p}}^{+})+\dots+({\mathcal{B}}_{0,1}^{[1]}P_{0,\mathtt{p}-1}^{[\mathtt{p}-1]}f_{0}^{-},f_{\mathtt{p}}^{+})\,.$

Thus
$(\textup{$\mathfrak{B}$}_{\mathtt{p}}f_{0}^{-},f_{\mathtt{p}}^{+})=\mathrm{i}\,\beta_{1}^{(\mathtt{p})}\epsilon^{\mathtt{p}}$
with

$\displaystyle\mathrm{i}\,\beta_{1}^{(\mathtt{p})}\epsilon^{\mathtt{p}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{entcoeff}}}{{=}}\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}+\sum_{i=1}^{\mathtt{p}-1}({\mathcal{B}}_{0,\mathtt{p}-i}^{[\mathtt{p}-i]}P_{0,i}^{[i]}f_{0}^{-},f_{\mathtt{p}}^{+})$
$\displaystyle\stackrel{{\scriptstyle\eqref{Psani1}}}{{=}}\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}+\sum_{i=1}^{\mathtt{p}-1}\sum_{q=1}^{i}\sum_{\ell_{1}+\dots+\ell_{q}=i}({\mathcal{B}}_{0,\mathtt{p}-i}^{[\mathtt{p}-i]}\mathcal{P}[{\mathcal{B}}_{\ell_{q}},\dots,{\mathcal{B}}_{\ell_{1}}]_{0,i}^{[i]}f_{0}^{-},f_{\mathtt{p}}^{+})$
$\displaystyle\stackrel{{\scriptstyle\eqref{hPbis},\eqref{TFjprop}}}{{=}}\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}+\sum_{i=1}^{\mathtt{p}-1}\sum_{q=1}^{i}\sum_{\ell_{1}+\dots+\ell_{q}=i}({\mathcal{B}}_{0,\mathtt{p}-i}^{[\mathtt{p}-i]}\mathcal{P}[{\mathcal{B}}_{0,\ell_{q}}^{[\ell_{q}]},\dots,{\mathcal{B}}_{0,\ell_{1}}^{[\ell_{1}]}]f_{0}^{-},f_{\mathtt{p}}^{+})$
$\displaystyle\stackrel{{\scriptstyle\eqref{cBactspar}}}{{=}}\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}+\sum_{i=1}^{\mathtt{p}-1}\sum_{q=1}^{i}\sum_{\begin{subarray}{c}\ell_{1}+\dots+\ell_{q}=i\\\
\sigma_{1},\dots,\sigma_{q}=\pm\end{subarray}}\\!\\!\\!\\!\\!\sigma_{1}\dots\sigma_{q}\mathtt{R}{\scriptsize\begin{matrix}-,\sigma_{1},\dots,\sigma_{q}\\\\[1.99168pt]
0,\ell_{1},\dots,\ell_{q}\end{matrix}}\,\begin{matrix}{\mathtt{B}_{\ell_{1}}^{[\ell_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{1},0}^{\sigma_{1},-}\end{matrix}\begin{matrix}{\mathtt{B}_{\ell_{2}}^{[\ell_{2}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{2},j_{1}}^{\sigma_{2},\sigma_{1}}\end{matrix}\dots\begin{matrix}{\mathtt{B}_{\ell_{q}}^{[\ell_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{q},j_{q-1}}^{\sigma_{q},\sigma_{q-1}}\end{matrix}\begin{matrix}{\mathtt{B}_{\mathtt{p}-i}^{[\mathtt{p}-i]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},j_{q}}^{+,\sigma_{q}}\end{matrix}$

with $j_{1}:=\ell_{1}$, $j_{2}:=j_{1}+\ell_{2}=\ell_{1}+\ell_{2}$, …,
$j_{q}:=j_{q-1}+\ell_{q}=\ell_{1}+\dots+\ell_{q}=i$. Thus $1\leq
j_{1}<\ldots<j_{q}\leq\mathtt{p}-1$ and
$(j_{1},\sigma_{1}),\dots,(j_{q},\sigma_{q})\notin\\{(0,-),(\mathtt{p},+)\\}$
so that, using (5.45), we have

$\mathrm{i}\,\beta_{1}^{(\mathtt{p})}\epsilon^{\mathtt{p}}=\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}+\sum_{i=1}^{\mathtt{p}-1}\sum_{q=1}^{i}\sum_{\begin{subarray}{c}\ell_{1}+\dots+\ell_{q}=i\\\
\sigma_{1},\dots,\sigma_{q}=\pm\end{subarray}}\\!\\!\\!\\!\\!\sigma_{1}\dots\sigma_{q}\dfrac{\begin{matrix}{\mathtt{B}_{\ell_{1}}^{[\ell_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{1},0}^{\sigma_{1},-}\end{matrix}\begin{matrix}{\mathtt{B}_{\ell_{2}}^{[\ell_{2}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{2},j_{1}}^{\sigma_{2},\sigma_{1}}\end{matrix}\dots\begin{matrix}{\mathtt{B}_{\ell_{q}}^{[\ell_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{q},j_{q-1}}^{\sigma_{q},\sigma_{q-1}}\end{matrix}\begin{matrix}{\mathtt{B}_{\mathtt{p}-i}^{[\mathtt{p}-i]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},j_{q}}^{+,\sigma_{q}}\end{matrix}}{(\omega_{j_{1}}^{\sigma_{1}}-\omega_{*}^{(\mathtt{p})})\ldots(\omega_{j_{q}}^{\sigma_{q}}-\omega_{*}^{(\mathtt{p})})}\,.$
(5.50)

We sum differently (5.50) noting that the following map between sets of
natural indices is a bijection

$\displaystyle\phi:\big{\\{}(i,q;\ell_{1},\dots,\ell_{q})\,:\,1\leq
i\leq\mathtt{p}-1\,,\,1\leq q\leq
i\,,\,\ell_{1}+\dots+\ell_{q}=i\big{\\}}\rightarrow$ $\displaystyle\quad\
\big{\\{}(q;j_{1},\dots,j_{q})\,:\,1\leq q\leq\mathtt{p}-1\,,\
0<j_{1}<\dots<j_{q}<\mathtt{p}\big{\\}}$ $\displaystyle\quad\
(i,q;\ell_{1},\dots,\ell_{q})\mapsto(q;j_{1}:=\ell_{1},j_{2}:=j_{1}+\ell_{2},\dots,j_{q}:=j_{q-1}+\ell_{q}=i)$

with inverse

$\phi^{-1}(q,j_{1},\dots,j_{q})=\big{(}j_{q},q,j_{1},j_{2}-j_{1},\dots,j_{q}-j_{q-1}\big{)}\,.$

Thus, by (5.4), identity (5.50) becomes

$\mathrm{i}\,\beta_{1}^{(\mathtt{p})}\epsilon^{\mathtt{p}}=\begin{matrix}{\mathtt{B}_{\mathtt{p}}^{[\mathtt{p}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},0}^{+,-}\end{matrix}+\sum_{q=1}^{\mathtt{p}-1}\
\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\
\sum_{\sigma_{1},\dots,\sigma_{q}=\pm}\sigma_{1}\dots\sigma_{q}\dfrac{\begin{matrix}{\mathtt{B}_{\ell_{1}}^{[\ell_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{1},0}^{\sigma_{1},-}\end{matrix}\begin{matrix}{\mathtt{B}_{\ell_{2}}^{[\ell_{2}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{2},j_{1}}^{\sigma_{2},\sigma_{1}}\end{matrix}\dots\begin{matrix}{\mathtt{B}_{\ell_{q}}^{[\ell_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{q},j_{q-1}}^{\sigma_{q},\sigma_{q-1}}\end{matrix}\begin{matrix}{\mathtt{B}_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},j_{q}}^{+,\sigma_{q}}\end{matrix}}{(\omega_{j_{1}}^{\sigma_{1}}-\omega_{*}^{(\mathtt{p})})\ldots(\omega_{j_{q}}^{\sigma_{q}}-\omega_{*}^{(\mathtt{p})})}$

with $\ell_{1}:=j_{1}$, $\ell_{2}:=j_{2}-j_{1}$, $\dots$,
$\ell_{q}:=j_{q}-j_{q-1}$. This proves formula (5.7).∎

Expansion (5.48) together with Lemma 5.13 prove (5.7). The function
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ is analytic in $\mathtt{h}$ as the
coefficients ${\mathtt{c}}_{\mathtt{h}}p_{\ell}^{[\ell]},a_{\ell}^{[\ell]}$
belong to ${\mathcal{Q}}({\mathtt{c}}_{\mathtt{h}}^{2})$ (cfr. Lemma 2.5) and
$\Omega_{j}^{(\mathtt{p})}$ in $t_{j}^{(\mathtt{p})}$ are analytic in
$\mathtt{h}$, see (5.6). This concludes the proof of the first part of Theorem
5.3.

We now turn our attention to the second part of Theorem 5.3, namely we prove
(LABEL:abc). By first Taylor expanding in $\epsilon$ the functions
$\alpha^{(\mathtt{p})}(\mu,\epsilon)$ and
$\gamma^{(\mathtt{p})}(\mu,\epsilon)$ at a point $(\underline{\mu}+\delta,0)$
and using (4.11) we obtain

$\displaystyle\alpha^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)=\big{(}\mathfrak{B}(\mu,\epsilon)f_{\mathtt{p}}^{+},f_{\mathtt{p}}^{+}\big{)}$
$\displaystyle=-\omega_{\mathtt{p}}^{+}(\underline{\mu}+\delta)+\underbrace{\big{(}\mathfrak{B}_{0,1}f_{\mathtt{p}}^{+},f_{\mathtt{p}}^{+}\big{)}}_{=\epsilon\widetilde{\alpha_{1}}^{(\mathtt{p})}}+\underbrace{\big{(}\mathfrak{B}_{1,1}f_{\mathtt{p}}^{+},f_{\mathtt{p}}^{+}\big{)}}_{=\delta\epsilon\widetilde{\alpha_{2}}^{(\mathtt{p})}}+\underbrace{\big{(}\mathfrak{B}_{0,2}f_{\mathtt{p}}^{+},f_{\mathtt{p}}^{+}\big{)}}_{=\epsilon^{2}\alpha_{2}^{(\mathtt{p})}}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,$

and

$\displaystyle\gamma^{(\mathtt{p})}(\underline{\mu}+\delta,\epsilon)=\big{(}\mathfrak{B}(\mu,\epsilon)f_{0}^{-},f_{0}^{-}\big{)}$
$\displaystyle=\omega_{0}^{-}(\underline{\mu}+\delta)+\underbrace{\big{(}\mathfrak{B}_{0,1}f_{0}^{-},f_{0}^{-}\big{)}}_{=\epsilon\widetilde{\gamma_{1}}^{(\mathtt{p})}}+\underbrace{\big{(}\mathfrak{B}_{1,1}f_{0}^{-},f_{0}^{-}\big{)}}_{=\delta\epsilon\widetilde{\gamma_{2}}^{(\mathtt{p})}}+\underbrace{\big{(}\mathfrak{B}_{0,2}f_{0}^{-},f_{0}^{-}\big{)}}_{=\epsilon^{2}\gamma_{2}^{(\mathtt{p})}}+r(\epsilon^{3},\delta\epsilon^{2},\delta^{2}\epsilon)\,,$

with the operator $\mathfrak{B}(\mu,\epsilon)$ defined in (4.10). The
coefficients $\widetilde{\alpha_{1}}^{(\mathtt{p})}$,
$\widetilde{\alpha_{2}}^{(\mathtt{p})}$,
$\widetilde{{\gamma_{1}}^{(\mathtt{p})}}$,
$\widetilde{{\gamma_{2}}^{(\mathtt{p})}}$ vanish because, by Proposition 5.8,
$\mathfrak{B}_{0,1}\in\mathtt{F}_{1}$ and
$\mathfrak{B}_{1,1}\in\mathtt{F}_{2}$, so that the zero harmonics component
$\mathfrak{B}_{0,1}^{[0]}=\mathfrak{B}_{1,1}^{[0]}=0$ are zero. We remark that
the second order coefficients $\alpha_{2}^{(\mathtt{p})}$ and
$\gamma_{2}^{(\mathtt{p})}$ can be computed explicitly by means of the
entanglement coefficients.

Finally, by (4.11), formula (5.10) holds with
$\alpha_{1}^{(\mathtt{p})}(\mathtt{h})$ and
$\gamma_{1}^{(\mathtt{p})}(\mathtt{h})$ given by

$\displaystyle\alpha_{1}^{(\mathtt{p})}(\mathtt{h}):=-(\partial_{\mu}\omega^{+})(\mathtt{p}+\underline{\mu},\mathtt{h})\stackrel{{\scriptstyle\eqref{omeghino}}}{{=}}-{\mathtt{c}}_{\mathtt{h}}+(\partial_{\varphi}\Omega)(\mathtt{p}+\underline{\mu},\mathtt{h})\,,$
$\displaystyle\gamma_{1}^{(\mathtt{p})}(\mathtt{h}):=(\partial_{\mu}\omega^{-})(\underline{\mu},\mathtt{h})\stackrel{{\scriptstyle\eqref{omeghino}}}{{=}}{\mathtt{c}}_{\mathtt{h}}+(\partial_{\varphi}\Omega)(\underline{\mu},\mathtt{h})\,,$

and, for any $\mathtt{h}>0$, by (2.22)

$\displaystyle\alpha_{1}^{(\mathtt{p})}(\mathtt{h})+\gamma_{1}^{(\mathtt{p})}(\mathtt{h})=(\partial_{\varphi}\Omega)(\underline{\mu}+\mathtt{p},\mathtt{h})+(\partial_{\varphi}\Omega)(\underline{\mu},\mathtt{h})>0$
$\displaystyle\gamma_{1}^{(\mathtt{p})}(\mathtt{h})-\alpha_{1}^{(\mathtt{p})}(\mathtt{h})=2{\mathtt{c}}_{\mathtt{h}}+(\partial_{\varphi}\Omega)(\underline{\mu},\mathtt{h})-(\partial_{\varphi}\Omega)(\underline{\mu}+\mathtt{p},\mathtt{h})>2{\mathtt{c}}_{\mathtt{h}}$

because the function
$x\mapsto(\partial_{\varphi}\Omega)(\underline{\mu}+x,\mathtt{h})$ is strictly
decreasing, by (2.22). Thus also (5.11) is proved.

##  6  The function $\beta_{1}^{(\mathtt{p})}(\mathtt{h})$

In this section we prove that for any $\mathtt{p}\geq 2$ the analytic function
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ in (5.7) satisfies (1.9). In Section
6.1 we prove the shallow water limit $\lim_{\mathtt{h}\to
0^{+}}\beta_{1}^{(\mathtt{p})}(\mathtt{h})=-\infty$. Actually in Theorem 6.1
we compute the exact asymptotic expansion of
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ as $\mathtt{h}\to 0^{+}$. The first
order term of such expansion vanishes, see Lemma 6.5, and we need to
explicitly compute the higher order term of the Laurent expansion. A key
property is the asymptotic expansion of the Stokes waves at any order proved
in Proposition 3.14. In section 6.2 we also prove that
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})\to 0$ as $\mathtt{h}\to+\infty$. Such
degeneracy is ultimately a consequence of the structural property of the
Stokes waves in deep water at any order proved in Proposition 3.16.

###  6.1 Asymptotic behavior of $\beta_{1}^{(\mathtt{p})}$ as $\mathtt{h}\to
0^{+}$

######  Theorem 6.1 (Shallow-water limit of
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$).

For any $\mathtt{p}\in\mathbb{N}$, $\mathtt{p}\geq 2$, the function
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ in (5.7) admits the following
asymptotic expansion, as $\mathtt{h}\to 0^{+}$,

$\beta_{1}^{(\mathtt{p})}(\mathtt{h})=-\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\Big{(}\frac{3}{8}\Big{)}^{\mathtt{p}-1}\frac{\mathtt{p}^{2}(\mathtt{p}+1)^{2}}{24}\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}+O\big{(}\mathtt{h}^{\frac{15}{2}-3\mathtt{p}}\big{)}\,.$
(6.1)

The rest of the section is devoted to the proof of Theorem 6.1.
The following asymptotic expansions will be often used in the sequel.

######  Lemma 6.2.

For any $j\in\mathbb{N}_{0}$ the quantities $\Omega_{j}^{(\mathtt{p})}$ and
$t_{j}^{(\mathtt{p})}$ in (5.6) satisfy, as $\mathtt{h}\to 0^{+}$,

$\displaystyle\Omega_{j}^{(\mathtt{p})}$
$\displaystyle=\mathtt{h}^{\frac{1}{2}}\Big{(}j+\frac{\mathtt{p}^{3}-\mathtt{p}-2j^{3}}{12}\mathtt{h}^{2}+\frac{38j^{5}-30j^{2}\mathtt{p}(\mathtt{p}^{2}-1)-4\mathtt{p}^{5}-15\mathtt{p}^{3}+19\mathtt{p}}{720}\mathtt{h}^{4}+O(\mathtt{h}^{6})\Big{)}\,,$
(6.2) $\displaystyle t_{j}^{(\mathtt{p})}$
$\displaystyle=\mathtt{h}^{-\frac{1}{2}}\Big{(}1+\frac{\mathtt{h}^{2}j^{2}}{6}+O(\mathtt{h}^{4})\Big{)}\,.$

It results

$\sqrt{\Omega_{0}^{(\mathtt{p})}\Omega_{\mathtt{p}}^{(\mathtt{p})}}=\frac{1}{2}\mathtt{h}^{\frac{3}{2}}\mathtt{p}\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\Big{(}1-\frac{3\mathtt{p}^{2}+8}{40}\mathtt{h}^{2}+O(\mathtt{h}^{4})\Big{)}\,.$
(6.3)

Furthermore $\omega_{j}^{\sigma}$ in (5.5) satisfies, for any
$j\in\mathbb{N}$, $\sigma=\pm$, as $\mathtt{h}\to 0^{+}$,

$\displaystyle\omega_{j}^{+}-\omega_{*}^{(\mathtt{p})}$
$\displaystyle=\mathtt{h}^{\frac{5}{2}}\Big{(}\frac{j^{3}-j+\mathtt{p}-\mathtt{p}^{3}}{6}-\Big{(}\frac{19}{60}(j^{2}+1)\frac{j^{3}-j+\mathtt{p}-\mathtt{p}^{3}}{6}-\frac{\mathtt{p}(\mathtt{p}^{2}-1)(\mathtt{p}^{2}-j^{2})}{90}\Big{)}\mathtt{h}^{2}+O(\mathtt{h}^{4})\Big{)}\,,$
$\displaystyle\omega_{j}^{-}-\omega_{*}^{(\mathtt{p})}$
$\displaystyle=2j\mathtt{h}^{\frac{1}{2}}\Big{(}1-\frac{j^{2}+1}{12}\mathtt{h}^{2}+O(\mathtt{h}^{4})\Big{)}\,.$
(6.4)

Furthermore for any $j\in\mathbb{Z}$ and $\ell\in\mathbb{N}$, as
$\mathtt{h}\to 0^{+}$,

$\displaystyle a_{\ell}^{[\ell]}-p_{\ell}^{[\ell]}(\sigma
t_{j}^{(\mathtt{p})}+\sigma^{\prime}t_{j+\ell}^{(\mathtt{p})})$ (6.5)
$\displaystyle=\begin{cases}3\ell\Big{(}\dfrac{3}{8}\Big{)}^{\ell-1}\mathtt{h}^{2-3\ell}\big{(}1+\dfrac{19\ell^{2}+12j\ell-27\ell+12j^{2}+2}{54}\mathtt{h}^{2}+O(\mathtt{h}^{4})\big{)}\,,&\sigma=\sigma^{\prime}=+\,,\\\\[5.69054pt]
-5\ell\Big{(}\dfrac{3}{8}\Big{)}^{\ell-1}\mathtt{h}^{2-3\ell}\big{(}1+O(\mathtt{h}^{2})\big{)}\,,&\sigma=\sigma^{\prime}=-\,,\\\\[5.69054pt]
-\ell\Big{(}\dfrac{3}{8}\Big{)}^{\ell-1}\mathtt{h}^{2-3\ell}\big{(}1+\dfrac{25\ell^{2}-12j\ell-9\ell+2}{18}\mathtt{h}^{2}+O(\mathtt{h}^{4})\big{)}\,,&\sigma\neq\sigma^{\prime}=+\,,\\\\[5.69054pt]
-\ell\Big{(}\dfrac{3}{8}\Big{)}^{\ell-1}\mathtt{h}^{2-3\ell}\big{(}1+O(\mathtt{h}^{2})\big{)}\,,&\sigma\neq\sigma^{\prime}=-\,.\end{cases}$

###### Proof.

By (5.6) and (2.30) we deduce (6.2) and (6.3). Similarly (2.20), (5.6) and
(6.2) imply (6.4). The last expansion (6.5) relies on the previous ones
together with Lemma 3.15. ∎

We first show that many of the addends in (5.7), namely those with at least
two negative signs $\sigma_{i}=\sigma_{j}=-$, $i\neq j$, do not contribute to
the main terms of the asymptotic expansion in (6.9).

######  Lemma 6.3.

For any integer $q=1,\dots,\mathtt{p}-1$ and any signs
$\sigma_{1},\dots,\sigma_{q}\in\\{\pm\\}$ we denote
$\vec{\sigma}:=(\sigma_{1},\dots,\sigma_{q})$ and its cardinality by
$|\vec{\sigma}|:=|\\{\sigma_{j}=-\\}|$. Then, for any
$0<j_{1}<\dots<j_{q}<\mathtt{p}$, the term
$[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}$ in
(5.7) satisfies, as $\mathtt{h}\to 0^{+}$,

$[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}=O\big{(}\mathtt{h}^{\frac{7}{2}-3\mathtt{p}+2|\vec{\sigma}|}\big{)}\,.$
(6.6)

###### Proof.

By (5.8), (6.2) and (6.5), for any $j,\ell>0$ we have
$\epsilon^{-\ell}\begin{matrix}{\mathtt{B}_{\ell}^{[\ell]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j+\ell,j}^{\sigma^{\prime},\sigma}\end{matrix}=O(\mathtt{h}^{\frac{5}{2}-3\ell})$
whereas when $j=0$ using (6.3) we get
$\epsilon^{-\ell}\begin{matrix}{\mathtt{B}_{\ell}^{[\ell]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\ell,0}^{\sigma^{\prime},\sigma}\end{matrix}=O(\mathtt{h}^{\frac{7}{2}-3\ell})$.
Therefore

$\epsilon^{-\mathtt{p}}\begin{matrix}{\mathtt{B}_{j_{1}}^{[j_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{1},0}^{\sigma_{1},-}\end{matrix}\begin{matrix}{\mathtt{B}_{j_{2}-j_{1}}^{[j_{2}-j_{1}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{j_{2},j_{1}}^{\sigma_{2},\sigma_{1}}\end{matrix}\dots\begin{matrix}{\mathtt{B}_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}}&{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{|}^{|}}}_{\mathtt{p},j_{q}}^{+,\sigma_{q}}\end{matrix}=O(\mathtt{h}^{\frac{5}{2}q+\frac{7}{2}-3\mathtt{p}})\,.$
(6.7)

On the other hand, by (6.4) and since $j_{i}\geq 1$ and
$j_{i}^{3}-j_{i}+\mathtt{p}-\mathtt{p}^{3}<0$ for any $i=1,\dots,q$, we deduce
that

$(\omega_{j_{1}}^{\sigma_{1}}-\omega_{*}^{(\mathtt{p})})\dots(\omega_{j_{q}}^{\sigma_{q}}-\omega_{*}^{(\mathtt{p})})\sim
C_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}\mathtt{h}^{\frac{5}{2}(q-|\vec{\sigma}|)}\mathtt{h}^{\frac{1}{2}|\vec{\sigma}|}\qquad\text{for
some}\quad C_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}\neq 0\,.$ (6.8)

The estimate (6.6) follows by (5.7), (6.7) and (6.8). ∎

We now exhibit the explicit asymptotic expansion of the function
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ in (5.7). We provide the expansion up
to the order $\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}$ since, as we prove in
Lemma 6.5 below, the term of order $\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}$
vanishes.

######  Proposition 6.4.

The function $\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ in (5.7) admits the
asymptotic expansion, as $\mathtt{h}\to 0^{+}$,

$\beta_{1}^{(\mathtt{p})}(\mathtt{h})=-\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\frac{3^{\mathtt{p}-1}\mathtt{p}}{8^{\mathtt{p}}}\Big{\\{}A^{(\mathtt{p})}\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}+\Big{[}\Big{(}\frac{299}{1080}\mathtt{p}^{2}-\frac{1}{2}\mathtt{p}-\frac{4}{45}\Big{)}A^{(\mathtt{p})}+C^{(\mathtt{p})}\Big{]}\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}+O(\mathtt{h}^{\frac{15}{2}-3\mathtt{p}})\Big{\\}}$
(6.9)

where

$\displaystyle
A^{(\mathtt{p})}:=\mathtt{p}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})\,,$
(6.10) $\displaystyle
C^{(\mathtt{p})}:=\frac{28}{27}\mathtt{p}^{3}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})\mathtt{g}^{(\mathtt{p})}_{q}(j_{1},\dots,j_{q})\,,$
(6.11)

with

$\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q}):=(-12)^{q}j_{1}\dots
j_{q}\frac{j_{1}(j_{2}-j_{1})\dots(j_{q}-j_{q-1})(\mathtt{p}-j_{q})}{(\mathtt{p}^{3}-\mathtt{p}+j_{1}-j_{1}^{3})\dots(\mathtt{p}^{3}-\mathtt{p}+j_{q}-j_{q}^{3})}$
(6.12)

and

$\displaystyle\mathtt{g}^{(\mathtt{p})}_{q}(j_{1},\dots,j_{q}):=\frac{1}{3}\Big{[}\frac{49}{45}q+\frac{32}{9}j_{1}^{2}-\frac{4}{9}-\frac{4}{9}\frac{\mathtt{p}^{3}-\mathtt{p}}{j_{1}}+\frac{5}{18}(\mathtt{p}^{3}-\mathtt{p})\Big{(}\frac{1}{j_{1}}+\dots+\frac{1}{j_{q}}\Big{)}$
(6.13)
$\displaystyle+\frac{38}{15}\big{(}j_{1}^{2}+\dots+j_{q}^{2}\big{)}+\frac{\mathtt{p}^{3}-\mathtt{p}}{5}\Big{(}\frac{\mathtt{p}+j_{1}}{\mathtt{p}^{2}+j_{1}^{2}+\mathtt{p}j_{1}-1}+\dots+\frac{\mathtt{p}+j_{q}}{\mathtt{p}^{2}+j_{q}^{2}+\mathtt{p}j_{q}-1}\Big{)}-\frac{13}{9}\big{(}j_{1}j_{2}+j_{2}j_{3}+\dots+j_{q}p\big{)}\Big{]}\,.$

###### Proof.

We split the proof in different steps.
Step 1: reduction. Lemma 6.3 implies that the asymptotic expansion of
$\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ as $\mathtt{h}\to 0^{+}$ is given only
by terms with at most one minus sign appearing in (5.7), i.e.

$\displaystyle\beta_{1}^{(\mathtt{p})}(\mathtt{h})$
$\displaystyle=b_{0}^{(\mathtt{p})}(\mathtt{h})+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{+,\dots,+}(\mathtt{h})$
(6.14) $\displaystyle\
-\sum_{q=1}^{\mathtt{p}-1}\sum_{i=1}^{q}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{i-1},j_{i},j_{i+1},\dots,j_{q}}^{+,\dots,\
+\ ,\ -,\
+,\dots,+}(\mathtt{h})+O\big{(}\mathtt{h}^{\frac{15}{2}-3\mathtt{p}}\big{)}\,.$

We compute the expansion at $\mathtt{h}\to 0^{+}$ of these coefficients.
Step $2$: computation of $b_{0}^{(\mathtt{p})}(\mathtt{h})$. By (5.7), (5.8),
(6.3) and (6.5) we obtain

$\displaystyle
b_{0}^{(\mathtt{p})}(\mathtt{h})=-\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\frac{3^{\mathtt{p}-1}\mathtt{p}}{8^{\mathtt{p}}}\Big{[}\mathtt{p}\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}+\mathtt{b}_{0}^{(\mathtt{p})}\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}+O\big{(}\mathtt{h}^{\frac{15}{2}-3\mathtt{p}}\big{)}\Big{]}$
(6.15)

with

$\mathtt{b}_{0}^{(\mathtt{p})}:=\Big{(}\frac{473}{360}\mathtt{p}^{2}-\frac{1}{2}\mathtt{p}-\frac{4}{45}\Big{)}\mathtt{p}\,.$
(6.16)

Step $3$: computation of
$[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{+,\dots,+}(\mathtt{h})$. For any
$q=1,\dots,\mathtt{p}-1$ and any integer $0<j_{1}<\dots<j_{q}<\mathtt{p}$,
using (5.7) with $\sigma_{1}=\dots=\sigma_{q}=+$ together with (5.8) and
(6.2)-(6.5) we get that

$[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{+,\dots,+}(\mathtt{h})=\frac{N^{+}}{D^{+}}$
(6.17)

where, neglecting terms $O(\mathtt{h}^{4})$,

$\displaystyle N^{+}$
$\displaystyle:=\Big{(}\frac{1}{4}\Big{)}^{q+1}\sqrt{\Omega_{0}^{(\mathtt{p})}\Omega_{\mathtt{p}}^{(\mathtt{p})}}\Omega_{j_{1}}^{(\mathtt{p})}\dots\Omega_{j_{q}}^{(\mathtt{p})}\big{(}a_{j_{1}}^{[j_{1}]}-p_{j_{1}}^{[j_{1}]}(-t_{0}^{(\mathtt{p})}+t_{j_{1}}^{(\mathtt{p})})\big{)}$
(6.18) $\displaystyle\ \
\times\big{(}a_{j_{2}-j_{1}}^{[j_{2}-j_{1}]}-p_{j_{2}-j_{1}}^{[j_{2}-j_{1}]}(t_{j_{1}}^{(\mathtt{p})}+t_{j_{2}}^{(\mathtt{p})})\big{)}\dots\big{(}a_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}-p_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}(t_{\mathtt{p}}^{(\mathtt{p})}+t_{j_{q}}^{(\mathtt{p})})\big{)}$
$\displaystyle=\mathtt{h}^{\frac{7}{2}+\frac{5}{2}q-3\mathtt{p}}\frac{3^{\mathtt{p}-1}}{8^{\mathtt{p}}}\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\mathtt{p}2^{q}\Big{(}1-\tfrac{3\mathtt{p}^{2}+8}{40}\mathtt{h}^{2}\Big{)}j_{1}\dots
j_{q}\prod_{i=1}^{q}\Big{(}1+\tfrac{\mathtt{p}^{3}-\mathtt{p}-2j_{i}^{3}}{12j_{i}}\mathtt{h}^{2}\Big{)}$
$\displaystyle\ \
\times(-j_{1})(j_{2}-j_{1})\dots(\mathtt{p}-j_{q})\big{(}1+\tfrac{25j_{1}^{2}-9j_{1}+2}{18}\mathtt{h}^{2}\big{)}\Big{[}\prod_{i=1}^{q-1}\big{(}1+\tfrac{19(j_{i+1}-j_{i})^{2}+12j_{i}(j_{i+1}-j_{i})-27(j_{i+1}-j_{i})+12j_{i}^{2}+2}{54}\mathtt{h}^{2}\big{)}\Big{]}$
$\displaystyle\ \
\times\big{(}1+\tfrac{19(\mathtt{p}-j_{q})^{2}+12j_{q}(\mathtt{p}-j_{q})-27(\mathtt{p}-j_{q})+12j_{q}^{2}+2}{54}\mathtt{h}^{2}\big{)}$

and

$D^{+}:=\mathtt{h}^{\frac{5}{2}q}(-6)^{-q}\underbrace{(\mathtt{p}^{3}-\mathtt{p}-j_{1}^{3}+j_{1})\dots(\mathtt{p}^{3}-\mathtt{p}-j_{q}^{3}+j_{q})}_{=:D_{1}^{+}}\prod_{i=1}^{q}\Big{[}1-\Big{(}\tfrac{19}{60}(j_{i}^{2}+1)+\tfrac{1}{15}\tfrac{\mathtt{p}(\mathtt{p}^{2}-1)(j_{i}+\mathtt{p})}{j_{i}^{2}+j_{i}\mathtt{p}+\mathtt{p}^{2}-1}\Big{)}\mathtt{h}^{2}\Big{]}\,.$
(6.19)

We write the terms $N^{+},D^{+}$ in (6.18) and (6.19) as

$\displaystyle
N^{+}=-\mathtt{h}^{\frac{7}{2}+\frac{5}{2}q-3\mathtt{p}}\frac{3^{\mathtt{p}-1}}{8^{\mathtt{p}}}\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\mathtt{p}2^{q}\,N_{1}^{+}\big{(}1+\mathtt{h}^{2}N_{2}^{+}+O(\mathtt{h}^{4})\big{)}\,,$
(6.20) $\displaystyle
D^{+}=\mathtt{h}^{\frac{5}{2}q}(-6)^{-q}D_{1}^{+}\big{(}1+\mathtt{h}^{2}D_{2}^{+}+O(\mathtt{h}^{4})\big{)}\,,$

where

$N_{1}^{+}:=j_{1}\ldots j_{q}\,j_{1}(j_{2}-j_{1})\ldots(\mathtt{p}-j_{q})$
(6.21)

whereas, setting $j_{q+1}:=\mathtt{p}$,

$\displaystyle N_{2}^{+}$
$\displaystyle:=-\tfrac{3\mathtt{p}^{2}+8}{40}+\sum_{i=1}^{q}\tfrac{\mathtt{p}^{3}-\mathtt{p}-2j_{i}^{3}}{12j_{i}}+\tfrac{25j_{1}^{2}-9j_{1}+2}{18}+\sum_{i=1}^{q}\tfrac{19(j_{i+1}-j_{i})^{2}+12j_{i}(j_{i+1}-j_{i})-27(j_{i+1}-j_{i})+12j_{i}^{2}+2)}{54}$
$\displaystyle=-\tfrac{3\mathtt{p}^{2}+8}{40}+\tfrac{\mathtt{p}^{3}-\mathtt{p}}{12}\Big{(}\tfrac{1}{j_{1}}+\dots+\tfrac{1}{j_{q}}\Big{)}-\tfrac{1}{6}(j_{1}^{2}+\dots+j_{q}^{2})+\tfrac{25j_{1}^{2}-9j_{1}+2}{18}+\sum_{i=1}^{q}\tfrac{19j_{i+1}^{2}+j_{i}^{2}-26j_{i}j_{i+1}-27(j_{i+1}-j_{i})+2}{54}$
$\displaystyle=-\tfrac{3\mathtt{p}^{2}+8}{40}+\tfrac{\mathtt{p}^{3}-\mathtt{p}}{12}\Big{(}\tfrac{1}{j_{1}}+\dots+\tfrac{1}{j_{q}}\Big{)}-\tfrac{1}{6}(j_{1}^{2}+\dots+j_{q}^{2})-\tfrac{1}{2}\mathtt{p}+\tfrac{1}{27}q+\tfrac{1}{9}$
$\displaystyle\quad+\big{(}\tfrac{25}{18}-\tfrac{19}{54}\big{)}j_{1}^{2}+\tfrac{19}{27}\big{(}j_{1}^{2}+\dots+j_{q}^{2}\big{)}+\tfrac{19}{54}\mathtt{p}^{2}-\tfrac{13}{27}\big{(}j_{1}j_{2}+j_{2}j_{3}+\dots+j_{q}\mathtt{p}\big{)}$
$\displaystyle=\tfrac{299}{1080}\mathtt{p}^{2}-\tfrac{1}{2}\mathtt{p}-\tfrac{4}{45}+\tfrac{1}{27}q+\tfrac{\mathtt{p}^{3}-\mathtt{p}}{12}\Big{(}\tfrac{1}{j_{1}}+\dots+\tfrac{1}{j_{q}}\Big{)}$
(6.22)
$\displaystyle\quad+\tfrac{29}{54}(j_{1}^{2}+\dots+j_{q}^{2})+\tfrac{28}{27}j_{1}^{2}-\tfrac{13}{27}\big{(}j_{1}j_{2}+j_{2}j_{3}+\dots+j_{q}\mathtt{p}\big{)}$

and

$\displaystyle D_{2}^{+}:$
$\displaystyle=-\sum_{i=1}^{q}\tfrac{19}{60}(j_{i}^{2}+1)+\tfrac{1}{15}\tfrac{\mathtt{p}(\mathtt{p}^{2}-1)(j_{i}+\mathtt{p})}{j_{i}^{2}+j_{i}\mathtt{p}+\mathtt{p}^{2}-1}$
(6.23)
$\displaystyle=-\tfrac{19}{60}(j_{1}^{2}+\dots+j_{q}^{2})-\tfrac{19}{60}q-\tfrac{\mathtt{p}(\mathtt{p}^{2}-1)}{15}\big{(}\tfrac{j_{1}+\mathtt{p}}{\mathtt{p}^{2}+j_{1}\mathtt{p}+j_{1}^{2}-1}+\dots+\tfrac{j_{q}+\mathtt{p}}{\mathtt{p}^{2}+j_{q}\mathtt{p}+j_{q}^{2}-1}\big{)}\,.$

In conclusion, in view of (6.17) and (6.20), by the explicit expressions of
$N_{1},D_{1},N_{2},D_{2}$ given respectively in (6.21), (6.19), (6.22), (6.23)
we obtain

$\displaystyle[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{q}}^{+,\dots,+}(\mathtt{h})$
$\displaystyle=\frac{N^{+}}{D^{+}}=-\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}\frac{3^{\mathtt{p}-1}}{8^{\mathtt{p}}}\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\,\mathtt{p}\,(-12)^{q}\,\frac{N_{1}^{+}}{D_{1}^{+}}(1+(N_{2}^{+}-D_{2}^{+})\mathtt{h}^{2}+O(\mathtt{h}^{4}))$
(6.24)
$\displaystyle=-\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\frac{3^{\mathtt{p}-1}\mathtt{p}}{8^{\mathtt{p}}}\Big{[}\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}+\mathtt{b}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}+O\big{(}\mathtt{h}^{\frac{15}{2}-3\mathtt{p}}\big{)}\Big{]}$

with
$\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q}):=(-12)^{q}\frac{N_{1}^{+}}{D_{1}^{+}}$
given explicitly in (6.12) and
$\mathtt{b}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q}):=(-12)^{q}\frac{N_{1}^{+}}{D_{1}^{+}}(N_{2}^{+}-D_{2}^{+})$
given explicitly by

$\displaystyle\mathtt{b}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})=\Big{(}\frac{299}{1080}\mathtt{p}^{2}-\frac{1}{2}\mathtt{p}-\frac{4}{45}+\frac{191}{540}q+\frac{28}{27}j_{1}^{2}+\frac{461}{540}\sum_{i=1}^{q}j_{i}^{2}$
(6.25)
$\displaystyle\qquad\qquad\qquad+\frac{\mathtt{p}^{3}-\mathtt{p}}{12}\sum_{i=1}^{q}\frac{1}{j_{i}}+\frac{\mathtt{p}^{3}-\mathtt{p}}{15}\sum_{i=1}^{q}\frac{\mathtt{p}+j_{i}}{\mathtt{p}^{2}+j_{i}^{2}+\mathtt{p}j_{i}-1}-\frac{13}{27}\sum_{i=1}^{q}j_{i}j_{i+1}\Big{)}\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})$

where we set $j_{q+1}:=\mathtt{p}$.

Step $4$: computation of
$[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{i-1},j_{i},j_{i+1},\dots,j_{q}}^{+,\dots,\
+\ ,\ -,\ +,\dots,+}(\mathtt{h})$. For any $q=1,\dots,\mathtt{p}-1$ and any
integer $0<j_{1}<\dots<j_{q}<\mathtt{p}$, using (5.7) with $\sigma_{1}=-$,
$\sigma_{2}=\dots=\sigma_{q}=+$ together with (5.8) and (6.2)-(6.5) we get
that

$[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{i-1},j_{i},j_{i+1},\dots,j_{q}}^{+,\dots,\
+\ ,\ -,\ +,\dots,+}(\mathtt{h})=\frac{N_{i}}{D_{i}}\,,\quad\forall
i=1,\dots,q\,,$ (6.26)

where, neglecting terms $O(\mathtt{h}^{2})$,

$D_{i}:=2(-6)^{1-q}\mathtt{h}^{\frac{5}{2}q-2}j_{i}\prod_{k=1\atop k\neq
i}^{q}(\mathtt{p}^{3}-\mathtt{p}-j_{k}^{3}+j_{k})\,,\vspace{-10mm}$ (6.27)

and

$\displaystyle N_{1}$
$\displaystyle:=\Big{(}\frac{1}{4}\Big{)}^{q+1}\sqrt{\Omega_{0}^{(\mathtt{p})}\Omega_{\mathtt{p}}^{(\mathtt{p})}}\Omega_{j_{1}}^{(\mathtt{p})}\dots\Omega_{j_{q}}^{(\mathtt{p})}\big{(}a_{j_{1}}^{[j_{1}]}-p_{j_{1}}^{[j_{1}]}(-t_{0}^{(\mathtt{p})}-t_{j_{1}}^{(\mathtt{p})})\big{)}\big{(}a_{j_{2}-j_{1}}^{[j_{2}-j_{1}]}-p_{j_{2}-j_{1}}^{[j_{2}-j_{1}]}(-t_{j_{1}}^{(\mathtt{p})}+t_{j_{2}}^{(\mathtt{p})})\big{)}$
(6.28) $\displaystyle\ \
\times\big{(}a_{j_{3}-j_{2}}^{[j_{3}-j_{2}]}-p_{j_{3}-j_{2}}^{[j_{3}-j_{2}]}(t_{j_{2}}^{(\mathtt{p})}+t_{j_{3}}^{(\mathtt{p})})\big{)}\dots\big{(}a_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}-p_{\mathtt{p}-j_{q}}^{[\mathtt{p}-j_{q}]}(t_{\mathtt{p}}^{(\mathtt{p})}+t_{j_{q}}^{(\mathtt{p})})\big{)}$
$\displaystyle=\mathtt{h}^{\frac{7}{2}+\frac{5}{2}q-3\mathtt{p}}\frac{3^{\mathtt{p}-1}}{8^{\mathtt{p}}}\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\frac{\mathtt{p}2^{q}}{3}j_{1}\dots
j_{q}(-5j_{1})\big{(}-(j_{2}-j_{1})\big{)}(j_{3}-j_{2})\dots(\mathtt{p}-j_{q})\,,$

whereas for any $i=2,\ldots,q$, setting $j_{q+1}:=\mathtt{p}$,

$\displaystyle N_{i}:$
$\displaystyle=\Big{(}\frac{1}{4}\Big{)}^{q+1}\sqrt{\Omega_{0}^{(\mathtt{p})}\Omega_{\mathtt{p}}^{(\mathtt{p})}}\Omega_{j_{1}}^{(\mathtt{p})}\dots\Omega_{j_{q}}^{(\mathtt{p})}\big{(}a_{j_{1}}^{[j_{1}]}-p_{j_{1}}^{[j_{1}]}(-t_{0}^{(\mathtt{p})}+t_{j_{1}}^{(\mathtt{p})})\big{)}\big{(}a_{j_{i}-j_{i-1}}^{[{j_{i}-j_{i-1}}]}-p_{j_{i}-j_{i-1}}^{[{j_{i}-j_{i-1}}]}(t_{j_{i-1}}^{(\mathtt{p})}-t_{j_{i}}^{(\mathtt{p})})\big{)}$
(6.29) $\displaystyle\ \
\times\big{(}a_{j_{i+1}-j_{i}}^{[{j_{i+1}-j_{i}}]}-p_{j_{i+1}-j_{i}}^{[{j_{i+1}-j_{i}}]}(-t_{j_{i}}^{(\mathtt{p})}+t_{j_{i+1}}^{(\mathtt{p})})\big{)}\prod_{k=1\atop
k\neq
i-1,i}^{q}\big{(}a_{j_{k+1}-j_{k}}^{[j_{k+1}-j_{k}]}-p_{j_{k+1}-j_{k}}^{[j_{k+1}-j_{k}]}(t_{j_{k}}^{(\mathtt{p})}+t_{j_{k+1}}^{(\mathtt{p})})\big{)}$
$\displaystyle=\mathtt{h}^{\frac{7}{2}+\frac{5}{2}q-3\mathtt{p}}\frac{3^{\mathtt{p}-1}}{8^{\mathtt{p}}}\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\frac{\mathtt{p}2^{q}}{9}j_{1}\dots
j_{q}(-j_{1})(j_{2}-j_{1})\dots(\mathtt{p}-j_{q})=-\frac{N_{1}}{15}\,.$

By (6.26), (6.27), (6.28) and (6.29) we obtain

$\displaystyle[b_{q}^{(\mathtt{p})}]_{j_{1},\dots,j_{i-1},j_{i},j_{i+1},\dots,j_{q}}^{+,\dots,\
+\ ,\ -,\
+,\dots,+}(\mathtt{h})=-\sqrt{\frac{\mathtt{p}^{2}-1}{3}}\frac{3^{\mathtt{p}-1}\mathtt{p}}{8^{\mathtt{p}}}\Big{[}\widehat{\;\mathtt{b}}_{q}^{(\mathtt{p})}\\!(i;j_{1},\dots,j_{q})\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}+O\big{(}\mathtt{h}^{\frac{15}{2}-3\mathtt{p}}\big{)}\Big{]}$
(6.30)

with

$\displaystyle\widehat{\;\mathtt{b}}_{q}^{(\mathtt{p})}\\!(i;j_{1},\dots,j_{q}):=\begin{cases}\dfrac{5(\mathtt{p}^{3}-\mathtt{p}+j_{1}-j_{1}^{3})}{36j_{1}}\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},j_{2},\dots,j_{q})&\textup{if
}i=1\,,\\\\[8.53581pt]
-\dfrac{\mathtt{p}^{3}-\mathtt{p}+j_{i}-j_{i}^{3}}{108j_{i}}\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},j_{2},\dots
j_{q})&\textup{if }i=2,\ldots,q\,.\end{cases}$ (6.31)

Step 5: conclusion. The sum of the coefficients of order
$\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}$ in (6.15) and (6.24) gives
$A^{(\mathtt{p})}$ in (6.10). By summing the coefficients of order
$\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}$ in (6.15)-(6.16), (6.24)-(6.25) and
(6.30)-(6.31) we obtain the coefficient of
$\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}$ in (6.9) as, setting
$\mathtt{n}_{q}^{(\mathtt{p})}:=\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})$
for brevity,

$\displaystyle\mathtt{b}_{0}^{(\mathtt{p})}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\mathtt{b}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})-\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\sum_{i=1}^{q}\widehat{\;\mathtt{b}}_{q}^{(\mathtt{p})}\\!(i;j_{1},\dots,j_{q})$
(6.32)
$\displaystyle=\Big{[}\Big{(}\tfrac{299}{1080}+\tfrac{28}{27}\Big{)}\mathtt{p}^{2}-\tfrac{1}{2}\mathtt{p}-\tfrac{4}{45}\Big{]}\mathtt{p}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\Big{(}\tfrac{299}{1080}\mathtt{p}^{2}-\tfrac{1}{2}\mathtt{p}-\tfrac{4}{45}\Big{)}\mathtt{n}_{q}^{(\mathtt{p})}$
$\displaystyle+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\Big{[}\tfrac{191}{540}q+\tfrac{28}{27}j_{1}^{2}+\tfrac{461}{540}\sum_{i=1}^{q}j_{i}^{2}+\tfrac{\mathtt{p}^{3}-\mathtt{p}}{12}\sum_{i=1}^{q}\tfrac{1}{j_{i}}+\tfrac{\mathtt{p}^{3}-\mathtt{p}}{15}\sum_{i=1}^{q}\tfrac{\mathtt{p}+j_{i}}{\mathtt{p}^{2}+j_{i}^{2}+\mathtt{p}j_{i}-1}-\tfrac{13}{27}\sum_{i=1}^{q}j_{i}j_{i+1}\Big{]}\mathtt{n}_{q}^{(\mathtt{p})}$
$\displaystyle-\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\Big{[}\tfrac{5(\mathtt{p}^{3}-\mathtt{p}+j_{1}-j_{1}^{3})}{36j_{1}}+\tfrac{\mathtt{p}^{3}-\mathtt{p}+j_{1}-j_{1}^{3}}{108j_{1}}\Big{]}\mathtt{n}_{q}^{(\mathtt{p})}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\sum_{i=1}^{q}\Big{[}\tfrac{\mathtt{p}^{3}-\mathtt{p}}{108j_{i}}+\tfrac{1}{108}-\tfrac{j_{i}^{2}}{108}\Big{]}\mathtt{n}_{q}^{(\mathtt{p})}$
$\displaystyle=\Big{(}\tfrac{299}{1080}\mathtt{p}^{2}-\tfrac{1}{2}\mathtt{p}-\tfrac{4}{45}\Big{)}\Big{(}\overbrace{\mathtt{p}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathtt{n}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})}^{=A^{(\mathtt{p})}\text{
by \eqref{Ap}}}\Big{)}+\tfrac{28}{27}\mathtt{p}^{3}$
$\displaystyle+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\Big{[}\big{(}\tfrac{191}{540}+\tfrac{1}{108}\big{)}q+\tfrac{28}{27}j_{1}^{2}+\big{(}\tfrac{461}{540}-\tfrac{1}{108}\big{)}\sum_{i=1}^{q}j_{i}^{2}+(\mathtt{p}^{3}-\mathtt{p})\big{(}\tfrac{1}{12}+\tfrac{1}{108}\big{)}\sum_{i=1}^{q}\tfrac{1}{j_{i}}$
$\displaystyle+\tfrac{\mathtt{p}^{3}-\mathtt{p}}{15}\sum_{i=1}^{q}\tfrac{\mathtt{p}+j_{i}}{\mathtt{p}^{2}+j_{i}^{2}+\mathtt{p}j_{i}-1}-\tfrac{13}{27}\sum_{i=1}^{q}j_{i}j_{i+1}\Big{]}\mathtt{n}_{q}^{(\mathtt{p})}-\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}\tfrac{4}{27}\Big{[}\tfrac{\mathtt{p}^{3}-\mathtt{p}}{j_{1}}+1-j_{1}^{2}\Big{]}\mathtt{n}_{q}^{(\mathtt{p})}$

which, by elementary sums, is equal to
$\Big{(}\frac{299}{1080}\mathtt{p}^{2}-\frac{1}{2}\mathtt{p}-\frac{4}{45}\Big{)}A^{(\mathtt{p})}+C^{(\mathtt{p})}$
appearing in (6.9). ∎

As anticipated, the problem is degenerate as $\mathtt{h}\to 0^{+}$, namely the
term of order $\mathtt{h}^{\frac{7}{2}-3\mathtt{p}}$ of the asymptotic
expansion (6.9) is actually zero. We need to prove that the subsequent term of
order $\mathtt{h}^{\frac{11}{2}-3\mathtt{p}}$ is nonzero.

######  Lemma 6.5 ( Koutschan - Van Hoeij - Zeilberger).

For any $\mathtt{p}\in\mathbb{N}$, $\mathtt{p}\geq 2$, the coefficient
$A^{(\mathtt{p})}$ in (6.10) is

$A^{(\mathtt{p})}=0\,\ ,$ (6.33)

and the coefficient $C^{(\mathtt{p})}$ in (6.11) is

$C^{(\mathtt{p})}=\frac{\mathtt{p}(\mathtt{p}+1)^{2}}{3}\,\ .$ (6.34)

###### Proof.

A proof of the latter result is presented in [43, Formulae (3)-(4)]. It is
computer-assisted. For the reader’s convenience we report their proof in [10].
We remark that the function $g_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})$ in [43,
Formula (2)] is related to the function
$\mathtt{g}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})$ in (6.13) by the identity

$g_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})=3\mathtt{g}_{q}^{(\mathtt{p})}(j_{1},\dots,j_{q})-\frac{28}{9}\mathtt{p}^{2}\,.$

Using [43, Formula (3)] one readily verifies that [43, Formula (4)] is
equivalent to (6.34). ∎

Proof of Theorem 6.1. It suffices to insert the values of $A^{(\mathtt{p})}$
and $C^{(\mathtt{p})}$ of Lemma 6.5 in (6.9). ∎

In the cases $\mathtt{p}=2,3,4$, inserting the explicit values of the Fourier
coefficients $p_{\ell}^{[\ell]}$, $a_{\ell}^{[\ell]}$, $\ell=1,\dots,4$
computed in [8, Proposition A.7] we have explicit formulas for
$\beta_{1}^{(\mathtt{p})}$ as functions of $\mathtt{h}$. They are plotted in
Figures 3-5. The zeros of $\beta_{1}^{(\mathtt{p})}$ coincide, for
$\mathtt{p}=2,3$, with the values given in previous literature, cfr. [17, 27].

###  6.2 Limit of $\beta_{1}^{(\mathtt{p})}$ as $\mathtt{h}\to+\infty$

We now prove that the term $\beta_{1}^{(\mathtt{p})}(\mathtt{h})$ in (5.7)
vanishes in the infinite-depth limit.

######  Proposition 6.6.

For any $\mathtt{p}\geq 2$, it results
$\lim\limits_{\mathtt{h}\to+\infty}\beta_{1}^{(\mathtt{p})}(\mathtt{h})=0$.

The rest of this section is devoted to the proof of Proposition 6.6.

######  Lemma 6.7.

For any $j\geq 0$, the terms in (5.6) and (5.5) satisfy

$\lim_{\mathtt{h}\to+\infty}\Omega_{j}^{(\mathtt{p})}=\lim_{\mathtt{h}\to+\infty}t_{j}^{(\mathtt{p})}=\sqrt{j+\frac{(\mathtt{p}-1)^{2}}{4}}=:\nu_{j}^{(\mathtt{p})}\,,\quad\lim_{\mathtt{h}\to+\infty}\omega_{j}^{\sigma}-\omega_{*}^{(\mathtt{p})}=\frac{2j-\mathtt{p}+1}{2}-\sigma\nu_{j}^{(\mathtt{p})}\,.$
(6.35)

The terms in (5.7)-(5.8) satisfy, for any $1\leq q\leq\mathtt{p}-1$,
$0<j_{1}<\dots<j_{q}<\mathtt{p}$ and $\sigma_{1},\dots,\sigma_{q}=\pm$,

$\lim_{\mathtt{h}\to+\infty}b_{0}^{(\mathtt{p})}=0\,,\qquad\lim_{\mathtt{h}\to\infty}\big{[}b_{q}^{(\mathtt{p})}\big{]}_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}=A_{j_{1},\dots,j_{q}}B_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}\,,$
(6.36)

where, recalling $\varpi_{j}$ in (3.55) and setting
$(j_{q+1},\sigma_{q+1}):=(\mathtt{p},+)$,

$\displaystyle
A_{j_{1},\dots,j_{q}}:=(-1)^{q}\frac{\sqrt{\mathtt{p}^{2}-1}\nu_{j_{1}}^{(\mathtt{p})}\dots\nu_{j_{q}}^{(\mathtt{p})}\varpi_{j_{1}}\varpi_{j_{2}-j_{1}}\dots\varpi_{j_{q}-j_{q-1}}\varpi_{\mathtt{p}-j_{q}}}{8^{q+1}j_{1}(\mathtt{p}-j_{1})\dots
j_{q}(\mathtt{p}-j_{q})}$ (6.37) $\displaystyle
B_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}:=\big{(}1+\nu_{0}^{(\mathtt{p})}-\sigma_{1}\nu_{j_{1}}^{(\mathtt{p})}\big{)}\prod_{i=1}^{q}\big{(}1-\sigma_{i}\nu_{j_{i}}^{(\mathtt{p})}-\sigma_{i+1}\nu_{j_{i+1}}^{(\mathtt{p})}\big{)}\big{(}2j_{i}-\mathtt{p}+1+2\sigma_{i}\nu_{j_{i}}^{(\mathtt{p})}\big{)}\,.$
(6.38)

###### Proof.

The expansion (6.35) follows by (2.31). By (5.7)-(5.8), (6.35) and (3.55) we
obtain

$\lim_{\mathtt{h}\to\infty}b_{0}^{(\mathtt{p})}=\frac{\sqrt{\mathtt{p}^{2}-1}}{8}\varpi_{\mathtt{p}}\Big{(}1-\frac{\mathtt{p}+1}{2}+\frac{\mathtt{p}-1}{2}\Big{)}=0$

which gives the first identity (6.36). Next, by (5.7)-(5.8), (6.35) and (3.55)
we get

$\displaystyle\lim_{\mathtt{h}\to\infty}\big{[}b_{q}^{(\mathtt{p})}\big{]}_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}=\frac{\sqrt{\mathtt{p}^{2}-1}\nu_{j_{1}}^{(\mathtt{p})}\dots\nu_{j_{q}}^{(\mathtt{p})}\varpi_{j_{1}}\varpi_{j_{2}-j_{1}}\dots\varpi_{j_{q}-j_{q-1}}\varpi_{\mathtt{p}-j_{q}}}{2^{q+3}(2j_{1}-\mathtt{p}+1-2\sigma_{1}\nu_{j_{1}}^{(\mathtt{p})})\dots(2j_{q}-\mathtt{p}+1-2\sigma_{q}\nu_{j_{q}}^{(\mathtt{p})})}\cdot$
$\displaystyle\quad\cdot(1+\nu_{0}^{(\mathtt{p})}-\sigma_{1}\nu_{j_{1}}^{(\mathtt{p})})\Bigg{(}\prod_{i=1}^{q-1}(1-\sigma_{i}\nu_{j_{i}}^{(\mathtt{p})}-\sigma_{i+1}\nu_{j_{i+1}}^{(\mathtt{p})})\Bigg{)}(1-\sigma_{q}\nu_{j_{q}}^{(\mathtt{p})}-\nu_{\mathtt{p}}^{(\mathtt{p})})\,,$

which, since
$(2j-\mathtt{p}+1-2\nu_{j}^{(\mathtt{p})})(2j-\mathtt{p}+1+2\nu_{j}^{(\mathtt{p})})=-4j(\mathtt{p}-j)$,
proves the second identity in (6.36) with $A_{j_{1},\dots,j_{q}}$ and
$B_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}$ defined in (6.38). ∎

The key result in the proof of Proposition 6.6 is the following lemma.

######  Lemma 6.8.

For any $q\in\mathbb{N}$ and integers $0<j_{1}<\dots<j_{q}<\mathtt{p}$ the
terms in (6.38) satisfy

$S_{j_{1},\dots,j_{q}}:=\sum_{\sigma_{1},\dots,\sigma_{q}=\pm}\sigma_{1}\dots\sigma_{q}B_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}=0\,.$
(6.39)

###### Proof.

Identity (6.39) is a direct consequence of the following facts

$\displaystyle
S_{j_{1},\dots,j_{q}}=-4\nu_{j_{q}}^{(\mathtt{p})}j_{q}\,S_{j_{1},\dots,j_{q-1}}\,,\
\forall q\geq 2\,,$ (6.40) $\displaystyle S_{j_{1}}=0\,.$ (6.41)

Proof of (6.40). For any $q\geq 2$ we have

$S_{j_{1},\dots,j_{q}}=\\!\\!\\!\\!\\!\sum_{\sigma_{1},\dots,\sigma_{q-1}=\pm}\\!\\!\\!\\!\\!\sigma_{1}\dots\sigma_{q-1}\Big{(}B_{j_{1},\dots,j_{q-1},j_{q}}^{\sigma_{1},\dots,\sigma_{q-1},+}-B_{j_{1},\dots,j_{q-1},j_{q}}^{\sigma_{1},\dots,\sigma_{q-1},-}\Big{)}$
(6.42)

where by (6.38)

$B_{j_{1},\dots,j_{q-1},j_{q}}^{\sigma_{1},\dots,\sigma_{q-1},+}-B_{j_{1},\dots,j_{q-1},j_{q}}^{\sigma_{1},\dots,\sigma_{q-1},-}=\frac{B_{j_{1},\dots,j_{q-1}}^{\sigma_{1},\dots,\sigma_{q-1}}}{1-\sigma_{q-1}\nu_{j_{q-1}}^{(\mathtt{p})}-\nu_{\mathtt{p}}^{(\mathtt{p})}}\Delta_{q}$
(6.43)

with

$\displaystyle\Delta_{q}\,:=(1-\sigma_{q-1}\nu_{j_{q-1}}^{(\mathtt{p})}-\nu_{j_{q}}^{(\mathtt{p})})(1-\nu_{j_{q}}^{(\mathtt{p})}-\nu_{\mathtt{p}}^{(\mathtt{p})})(2j_{q}-\mathtt{p}+1+2\nu_{j_{q}}^{(\mathtt{p})})$
(6.44)
$\displaystyle-(1-\sigma_{q-1}\nu_{j_{q-1}}^{(\mathtt{p})}+\nu_{j_{q}}^{(\mathtt{p})})(1+\nu_{j_{q}}^{(\mathtt{p})}-\nu_{\mathtt{p}}^{(\mathtt{p})})(2j_{q}-\mathtt{p}+1-2\nu_{j_{q}}^{(\mathtt{p})})\,.$

Setting

$a:=1-\sigma_{q-1}\nu_{j_{q-1}}^{(\mathtt{p})}\,,\quad
b:=1-\nu_{\mathtt{p}}^{(\mathtt{p})}\stackrel{{\scriptstyle\eqref{tjpinfinito}}}{{=}}-\frac{\mathtt{p}-1}{2}\,,\quad
c:=2j_{q}-\mathtt{p}+1\,,\quad\nu:=\nu_{j_{q}}^{(\mathtt{p})}\,,$ (6.45)

the term $\Delta_{q}$ in (6.44) is given by

$\displaystyle\Delta_{q}$
$\displaystyle=(a-\nu)(b-\nu)(c+2\nu)-(a+\nu)(b+\nu)(c-2\nu)=\big{(}4ab-2(a+b)c+4\nu^{2}\big{)}\nu$
$\displaystyle\stackrel{{\scriptstyle\eqref{setabcv}}}{{=}}-4\nu_{j_{q}}^{(\mathtt{p})}j_{q}\Big{(}\frac{1-\mathtt{p}}{2}-\sigma_{q-1}\nu_{j_{q-1}}^{(\mathtt{p})}\Big{)}=-4\nu_{j_{q}}^{(\mathtt{p})}j_{q}\Big{(}1-\sigma_{q-1}\nu_{j_{q-1}}^{(\mathtt{p})}-\nu_{\mathtt{p}}^{(\mathtt{p})}\Big{)}$
(6.46)

since $\frac{\mathtt{p}+1}{2}=\nu_{\mathtt{p}}^{(\mathtt{p})}$ by (6.35). By
(6.42), (6.43), (6.46) we conclude that

$S_{j_{1},\dots,j_{q}}=-4\nu_{j_{q}}^{(\mathtt{p})}j_{q}\,S_{j_{1},\dots,j_{q-1}}=(-4)^{q-1}\nu_{j_{2}}^{(\mathtt{p})}j_{2}\dots\nu_{j_{q}}^{(\mathtt{p})}j_{q}\,S_{j_{1}}\,.$

Proof of (6.41). By (6.38),
$S_{j_{1}}=B_{j_{1}}^{+}-B_{j_{1}}^{-}=:\Delta_{1}$ which is still given by
formula (6.44) setting $(j_{0},\sigma_{0}):=(0,-)$. By (6.46) we have

$S_{j_{1}}=\Delta_{1}=-4\nu_{j_{1}}^{(\mathtt{p})}j_{1}\Big{(}1+\nu_{0}^{(\mathtt{p})}-\frac{\mathtt{p}+1}{2}\Big{)}=0$

by (6.35). The proof of the lemma is complete. ∎

Proof of Proposition 6.6. In view of (5.7), (6.36) and Lemma 6.8 we conclude
that

$\lim_{\mathtt{h}\to+\infty}\beta_{1}^{(\mathtt{p})}(\mathtt{h})=\lim_{\mathtt{h}\to+\infty}b_{0}^{(\mathtt{p})}+\sum_{q=1}^{\mathtt{p}-1}\sum_{0<j_{1}<\dots<j_{q}<\mathtt{p}}A_{j_{1},\dots,j_{q}}\sum_{\sigma_{1},\dots,\sigma_{q}=\pm}\sigma_{1}\dots\sigma_{q}B_{j_{1},\dots,j_{q}}^{\sigma_{1},\dots,\sigma_{q}}=0$

as claimed. ∎

######  Remark 6.9.

Figures 3-5 are obtained in the Mathematica sheet beta1.nb by plugging in
formulas (5.7)-(5.8) the explicit coefficients for $p_{\ell}^{[\ell]}$ and
$a_{\ell}^{[\ell]}$, with $\ell=1,\dots,4$, computed in [8, Proposition A.7]
and encoded in the Mathematica library CoeffsLin.m. The Mathematica code
employed can be found at https://git-scm.sissa.it/amaspero/isolas.

##  Appendix A Spectral collisions and multiple eigenvalues of
${\mathcal{L}}_{\mu,0}$

In this Appendix we prove Lemmata 2.7 and 2.9.
Proof of Lemma 2.7. We shall use that, since
$\Omega(\varphi,\mathtt{h})=\sqrt{\varphi\tanh(\mathtt{h}\varphi)}$ is
strictly concave for $\varphi>0$ (cfr. (2.22)) and $\Omega(0,\mathtt{h})=0$,
then it is strictly sub-additive

$\Omega(\varphi+z,\mathtt{h})<\Omega(\varphi,\mathtt{h})+\Omega(z,\mathtt{h})\,,\qquad\forall\varphi,z>0\,,$
(A.1)

and

$\Omega(z,\mathtt{h})<z\Omega(1,\mathtt{h})\,,\quad\forall z>1\,.$ (A.2)

Proof of Item $i$). We first prove that the equations
$\omega^{\sigma}(\varphi,\mathtt{h})=\omega^{\sigma}(\varphi+\mathtt{p},\mathtt{h})$,
for any $\sigma=\pm$, have no solution, see Figure 6(a). Recalling (2.21),

$\omega^{+}(\varphi,\mathtt{h})=\omega^{+}(\varphi+\mathtt{p},\mathtt{h})\qquad\Leftrightarrow\qquad{\mathtt{c}}_{\mathtt{h}}\mathtt{p}-\Omega(\varphi+\mathtt{p},\mathtt{h})=-\Omega(\varphi,\mathtt{h})\,.$
(A.3)

By (A.2) the function
$d(\varphi,\mathtt{h}):={\mathtt{c}}_{\mathtt{h}}\mathtt{p}-\Omega(\varphi+\mathtt{p},\mathtt{h})$
satisfies $d(0,\mathtt{h})>0$ (since $\mathtt{p}\geq 2$) and
$d(\varphi,\mathtt{h})>-\Omega(\varphi+\mathtt{p},\mathtt{h})\geq-\Omega(\varphi,\mathtt{h})$
for any $\varphi\leq-\mathtt{p}$ as well as
$d(\varphi,\mathtt{h})>0>-\Omega(\varphi,\mathtt{h})$ for any
$\varphi\in(-\mathtt{p},0)$, see the right hand side of Figure 6(a). For any
$\varphi>0$ we deduce by (A.1) and (A.2) that
$\Omega(\varphi+\mathtt{p},\mathtt{h})<\Omega(\varphi,\mathtt{h})+{\mathtt{c}}_{\mathtt{h}}\mathtt{p}$
and therefore the equation (A.3) has no solutions.

Since $\omega^{-}(\varphi,\mathtt{h})=-\omega^{+}(-\varphi,\mathtt{h})$ the
equation
$\omega^{-}(\varphi,\mathtt{h})=\omega^{-}(\varphi+\mathtt{p},\mathtt{h})$ has
no solutions as well.
Proof of Item $ii$). We now prove that there exists a unique positive solution
$\underline{\varphi}(\mathtt{p},\mathtt{h})>0$ of

$\omega^{-}(\varphi,\mathtt{h})=\omega^{+}(\varphi+\mathtt{p},\mathtt{h})\qquad\Leftrightarrow\qquad{\mathtt{c}}_{\mathtt{h}}\mathtt{p}-\Omega(\varphi+\mathtt{p},\mathtt{h})=\Omega(\varphi,\mathtt{h})\,.$
(A.4)

Indeed
$d(\varphi,\mathtt{h})={\mathtt{c}}_{\mathtt{h}}\mathtt{p}-\Omega(\varphi+\mathtt{p},\mathtt{h})$
satisfies $d(0,\mathtt{h})>0$ and it is strictly decreasing to $-\infty$,
whereas $\Omega(0,\mathtt{h})=0$ and $\Omega(\cdot,\mathtt{h})$ is strictly
increasing to $+\infty$, see Figure 7. This proves (2.26). By simmetry
$\varphi=-\underline{\varphi}(\mathtt{p},\mathtt{h})-\mathtt{p}$ is the unique
solution of (A.4) in $(-\infty,-\mathtt{p})$ proving (2.27). We claim that for
$-\mathtt{p}\leq\varphi\leq 0$ there are no further intersection points
between the graphs of $d(\cdot,\mathtt{h})$ and $\Omega(\cdot,\mathtt{h})$ if
$\mathtt{p}\geq 3$ and one intersection point
$\varphi=-\frac{\mathtt{p}}{2}=-1$ if $\mathtt{p}=2$, see Figure 7. Indeed
$d(\varphi,\mathtt{h})>\Omega(\varphi,\mathtt{h})$, for
$\varphi\in[-\mathtt{p},0]$, if and only if

$\mathtt{p}{\mathtt{c}}_{\mathtt{h}}>\Omega\Big{(}\frac{\mathtt{p}}{2}-y,\mathtt{h}\Big{)}+\Omega\Big{(}\frac{\mathtt{p}}{2}+y,\mathtt{h}\Big{)}=:g(y)\,,\quad\forall
y\in\Big{[}-\frac{\mathtt{p}}{2},\frac{\mathtt{p}}{2}\Big{]}\,.$

The function $g$ is even and, by (2.22), strictly concave. Hence it attains at
$y=0$ its maximum value

$g(0)=2\Omega\Big{(}\frac{\mathtt{p}}{2},\mathtt{h}\Big{)}\stackrel{{\scriptstyle\eqref{valzero}}}{{<}}2\frac{\mathtt{p}}{2}\Omega\big{(}1,\mathtt{h}\big{)}=\mathtt{p}{\mathtt{c}}_{\mathtt{h}}\,,\quad\forall\mathtt{p}\geq
3\,,$

with equality holding for $\mathtt{p}=2$.
Proof of Item $iii$). It follows because
$\omega^{-}(\varphi,\mathtt{h})=-\omega^{+}(-\varphi,\mathtt{h})$ amounts to
$\Omega(\varphi,\mathtt{h})+\Omega(\varphi+\mathtt{p},\mathtt{h})=-{\mathtt{c}}_{\mathtt{h}}\mathtt{p}$
and the function $\Omega(\varphi,\mathtt{h})$ is non-negative.

We now prove the properties of the positive function
$\underline{\varphi}(\mathtt{p},\mathtt{h})>0$ stated in Lemma 2.7.
Proof of Item 1. For any $\mathtt{h}>0$ we have

$\mathcal{F}_{\mathtt{p}}(\underline{\varphi}(\mathtt{p},\mathtt{h}),\mathtt{h})=0\qquad\text{where}\qquad\mathcal{F}_{\mathtt{p}}(\varphi,\mathtt{h}):=\Omega(\varphi,\mathtt{h})+\Omega(\varphi+\mathtt{p},\mathtt{h})-{\mathtt{c}}_{\mathtt{h}}\mathtt{p}\,.$
(A.5)

Since $\mathcal{F}_{\mathtt{p}}(\varphi,\mathtt{h})$ is analytic and
$\partial_{\varphi}\mathcal{F}_{\mathtt{p}}(\varphi,\mathtt{h})>0$ by (2.22),
we deduce by the analytic implicit function theorem that
$\mathtt{h}\mapsto\underline{\varphi}(\mathtt{p},\mathtt{h})$ is analytic.
Proof of item 2. By (A.5) and recalling that
${\mathtt{c}}_{\mathtt{h}}=\Omega(1,\mathtt{h})$ we have

$\displaystyle\mathcal{F}_{\mathtt{p}+1}(\underline{\varphi}(\mathtt{p},\mathtt{h}),\mathtt{h})$
$\displaystyle=\Omega(\underline{\varphi}(\mathtt{p},\mathtt{h}),\mathtt{h})+\Omega(\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p}+1,\mathtt{h})-{\mathtt{c}}_{\mathtt{h}}(\mathtt{p}+1)$
$\displaystyle=\Omega(\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p}+1,\mathtt{h})-\Omega(\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p},\mathtt{h})-\Omega(1,\mathtt{h})<0$
(A.6)

by the sub-additivity property in (A.1) with
$\varphi=\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p}$ and $z=1$.
Then (A.6) and the fact that
$\partial_{\varphi}\mathcal{F}_{\mathtt{p}+1}(\varphi,\mathtt{h})>0$ imply
that
$\underline{\varphi}(\mathtt{p},\mathtt{h})<\underline{\varphi}(\mathtt{p}+1,\mathtt{h})$,
for any $\mathtt{p}\geq 2$, proving the first part of (2.28). For any
$\mathtt{h}>0$ the monotone sequence
$\mathtt{p}\mapsto\underline{\varphi}(\mathtt{p},\mathtt{h})$ admits a limit
as $\mathtt{p}\to+\infty$. Let us prove that
$\lim_{\mathtt{p}\to+\infty}\underline{\varphi}(\mathtt{p},\mathtt{h})=+\infty$.
By contradiction suppose that $\sup_{\mathtt{p}\geq
2}\underline{\varphi}(\mathtt{p},\mathtt{h})<+\infty$ is finite. Then by (A.5)
we deduce the contradiction
${\mathtt{c}}_{\mathtt{h}}=\lim_{\mathtt{p}\to\infty}\mathtt{p}^{-1}\Omega(\underline{\varphi}(\mathtt{p},\mathtt{h}))+\mathtt{p}^{-1}\Omega(\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p})=0$.
Proof of item 3. Recalling (A.5),
${\mathtt{c}}_{\mathtt{h}}=\Omega(1,\mathtt{h})$, (A.2) and that
$\Omega(\varphi,\mathtt{h})$ is strictly increasing for $\varphi>0$, we have
that

$\mathcal{F}_{2}(0,\mathtt{h})=\Omega(2,\mathtt{h})-2\Omega(1,\mathtt{h})<0\,,\quad\mathcal{F}_{2}(1,\mathtt{h})=\Omega(3,\mathtt{h})-\Omega(1,\mathtt{h})>0\,,\quad\forall\mathtt{h}>0\,,$

and therefore the unique zero $\underline{\varphi}(2,\mathtt{h})>0$ of
$\mathcal{F}_{2}(\varphi,\mathtt{h})=0$ satisfies
$\underline{\varphi}(2,\mathtt{h})\in(0,1)$ for any $\mathtt{h}>0$.

For $\mathtt{p}\geq 3$ we already know by Item 2 that
$\mathcal{F}_{\mathtt{p}}(\underline{\varphi}(\mathtt{p}-1,\mathtt{h}),\mathtt{h})<0$
and, since $\tfrac{1}{4}(\mathtt{p}-1)^{2}\geq 1$ for any $\mathtt{p}\geq 3$,
we have

$\mathcal{F}_{\mathtt{p}}\big{(}\tfrac{1}{4}(\mathtt{p}-1)^{2},\mathtt{h}\big{)}=\frac{\mathtt{p}-1}{2}\underbrace{\sqrt{\tanh\big{(}\tfrac{\mathtt{h}}{4}(\mathtt{p}-1)^{2}\big{)}}}_{\geq\sqrt{\tanh(\mathtt{h})}}+\frac{\mathtt{p}+1}{2}\underbrace{\sqrt{\tanh\big{(}\tfrac{\mathtt{h}}{4}(\mathtt{p}+1)^{2}\big{)}}}_{>\sqrt{\tanh(\mathtt{h})}}-\mathtt{p}\sqrt{\tanh(\mathtt{h})}>0\,.$
(A.7)

Therefore, since $\Omega(\varphi,\mathtt{h})$ is strictly increasing for
$\varphi>0$, we deduce that
$\underline{\varphi}(\mathtt{p},\mathtt{h})\in\big{(}0,\tfrac{1}{4}(\mathtt{p}-1)^{2}\big{)}$
for any $\mathtt{p}\geq 3$. Furthermore, by the asymptotic expansion

$\sqrt{\tanh(z)}=\Big{(}1-2\frac{e^{-2z}}{1+e^{-2z}}\Big{)}^{\frac{1}{2}}=1-e^{-2z}+O(e^{-4z})\qquad\text{as}\qquad
z\to+\infty\,,$ (A.8)

we deduce that

$\mathcal{F}_{2}\Big{(}\frac{1}{4},\mathtt{h}\Big{)}=\frac{1}{2}\big{(}1-e^{-\frac{\mathtt{h}}{2}}+O(e^{-\mathtt{h}})\big{)}+\frac{3}{2}\big{(}1+O(e^{-\mathtt{h}})\big{)}-2\big{(}1+O(e^{-\mathtt{h}})\big{)}<0\,,\quad\text{for
}\mathtt{h}\geq\mathtt{h}_{*}\textup{ large enough}\,,$

which implies that $\underline{\varphi}(2,\mathtt{h})>1/4$ for any
$\mathtt{h}\geq\mathtt{h}_{*}$.
Proof of item 4. For any $\mathtt{p}\geq 2$, by the previous item the function
$\mathtt{h}\mapsto\underline{\varphi}(\mathtt{p},\mathtt{h})$ is bounded and
$\mathtt{h}\underline{\varphi}(\mathtt{p},\mathtt{h})\to 0^{+}$ as
$\mathtt{h}\to 0^{+}$. By (A.5) and the Taylor expansion
$\sqrt{\tanh(z)}=\sqrt{z}\big{(}1-\frac{z^{2}}{6}+\frac{19}{360}z^{4}+O(z^{6})\big{)}$
as $z\to 0^{+}$ we first deduce that
$\underline{\varphi}:=\underline{\varphi}(\mathtt{p},\mathtt{h})$

$\displaystyle\underline{\varphi}\Big{(}1-\tfrac{1}{6}\mathtt{h}^{2}\underline{\varphi}^{2}+\tfrac{19}{360}\mathtt{h}^{4}\underline{\varphi}^{4}+O(\mathtt{h}^{6})\Big{)}+\big{(}\underline{\varphi}+\mathtt{p}\big{)}\Big{(}1-\tfrac{1}{6}\mathtt{h}^{2}\big{(}\underline{\varphi}+\mathtt{p}\big{)}^{2}+\tfrac{19}{360}\mathtt{h}^{4}\big{(}\underline{\varphi}+\mathtt{p}\big{)}^{4}+O(\mathtt{h}^{6})\Big{)}$
(A.9)
$\displaystyle\qquad=p\Big{(}1-\tfrac{1}{6}\mathtt{h}^{2}+\tfrac{19}{360}\mathtt{h}^{4}+O(\mathtt{h}^{6})\Big{)}\,,$

and then

$\underline{\varphi}(\mathtt{p},\mathtt{h})\big{(}1+\mathcal{O}(\mathtt{h}^{2})\big{)}=\frac{\mathtt{p}}{12}\mathtt{h}^{2}\big{[}(\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p})^{2}-1]+O(\mathtt{h}^{4})\,.$

This proves that
$\underline{\varphi}(\mathtt{p},\mathtt{h})=O(\mathtt{h}^{2})$ as
$\mathtt{h}\to 0^{+}$ and then reinserting the result in (A.9) we obtain the
sharper Taylor expansion in (2.30).

In order to prove that
$\lim_{\mathtt{h}\to+\infty}\underline{\varphi}(\mathtt{p},\mathtt{h})=(\mathtt{p}-1)^{2}/4$
it is sufficient to show that for any sequence $\mathtt{h}_{n}\to\ +\infty$
there is a subsequence $\mathtt{h}_{n_{k}}$ such that
$\underline{\varphi}(\mathtt{p},\mathtt{h}_{n_{k}})\to(\mathtt{p}-1)^{2}/4$.
By item 3 we have that
$1/4\leq\underline{\varphi}(\mathtt{p},\mathtt{h}_{n})\leq(\mathtt{p}-1)^{2}/4<+\infty$
for $n\geq n_{0}$ large enough. By compactness, there exists a subsequence
$\mathtt{h}_{n_{k}}\to+\infty$ such that
$\underline{\varphi}(\mathtt{p},\mathtt{h}_{n_{k}})\to\ell\in[1/4,(\mathtt{p}-1)^{2}/4]$.
Thus by (A.5) we deduce that

$\underset{k\to+\infty}{\lim}\mathcal{F}_{\mathtt{p}}(\underline{\varphi}(\mathtt{p},\mathtt{h}_{n_{k}}),\mathtt{h}_{n_{k}})=\sqrt{\ell}+\sqrt{\ell+\mathtt{p}}-\mathtt{p}=0\,,$
(A.10)

namely $\ell=\tfrac{1}{4}(\mathtt{p}-1)^{2}$, being the unique solution of
(A.10). This proves that
$\underline{\varphi}(\mathtt{p},\mathtt{h})\to\tfrac{1}{4}(\mathtt{p}-1)^{2}$
as $\mathtt{h}\to+\infty$.

We now prove the sharper asymptotic expansion in (2.31). By (A.5) and (A.8) we
have, as $\mathtt{h}\to+\infty$,

$0=\mathcal{F}_{\mathtt{p}}(\underline{\varphi}(\mathtt{p},\mathtt{h}),\mathtt{h})=\underbrace{\sqrt{\underline{\varphi}(\mathtt{p},\mathtt{h})}+\sqrt{\underline{\varphi}(\mathtt{p},\mathtt{h})+\mathtt{p}}-\mathtt{p}}_{=\frac{2\mathtt{p}}{\mathtt{p}^{2}-1}y_{\mathtt{p}}(\mathtt{h})+O(y_{\mathtt{p}}^{2}(\mathtt{h}))}-\sqrt{\underline{\varphi}(\mathtt{p},\mathtt{h})}e^{-2\mathtt{h}\underline{\varphi}(\mathtt{p},\mathtt{h})}+\mathtt{p}e^{-2\mathtt{h}}+O(e^{-4\mathtt{h}\underline{\varphi}(\mathtt{p},\mathtt{h})})+O(e^{-4\mathtt{h}})$
(A.11)

writing
$\underline{\varphi}(\mathtt{p},\mathtt{h})=\frac{(\mathtt{p}-1)^{2}}{4}+y_{\mathtt{p}}(\mathtt{h})$
where $y_{\mathtt{p}}(\mathtt{h})\to 0$ as $\mathtt{h}\to+\infty$. We first
consider the case $\mathtt{p}=2$. By Items 2 and 3 we know that
$y_{2}(\mathtt{h})\to 0^{+}$ as $\mathtt{h}\to+\infty$. Then, by (A.11) and
since $e^{-2\mathtt{h}y_{2}(\mathtt{h})}\leq 1$, we deduce that

$\frac{4}{3}y_{2}(\mathtt{h})=\sqrt{\frac{1}{4}+y_{2}(\mathtt{h})}\,e^{-\frac{\mathtt{h}}{2}}e^{-2\mathtt{h}y_{2}(\mathtt{h})}+O(y_{2}^{2}(\mathtt{h}))+O(e^{-\mathtt{h}})\,.$
(A.12)

Therefore $|y_{2}(\mathtt{h})|\leq Ce^{-\frac{\mathtt{h}}{2}}$ and then,
reinserting this bound in (A.12) we deduce that
$y_{2}(\mathtt{h})=\tfrac{3}{8}e^{-\frac{\mathtt{h}}{2}}+o(e^{-\frac{\mathtt{h}}{2}})$
which is the asymptotic expansion (2.31) for $\mathtt{p}=2$. The cases
$\mathtt{p}=3$ and $\mathtt{p}\geq 4$ follow similarly. By (A.11), writing
$\underline{\varphi}(3,\mathtt{h})=1+y_{3}(\mathtt{h})$ where
$y_{3}(\mathtt{h})\to 0^{-}$, we have

$\frac{3}{4}y_{3}(\mathtt{h})=-3e^{-2\mathtt{h}}+\sqrt{1+y_{3}(\mathtt{h})}e^{-2\mathtt{h}}e^{-2\mathtt{h}y_{3}(\mathtt{h})}+O(y_{3}^{2}(\mathtt{h}))+o(e^{-2\mathtt{h}})$

and therefore (2.31) for $\mathtt{p}=3$ follows. Finally for any
$\mathtt{p}\geq 4$ we deduce by (A.11) and since
$\frac{(\mathtt{p}-1)^{2}}{4}>1$ that
$\frac{2\mathtt{p}}{\mathtt{p}^{2}-1}y_{\mathtt{p}}(\mathtt{h})=-\mathtt{p}e^{-2\mathtt{h}}+O(y_{\mathtt{p}}(\mathtt{h})^{2})+o(e^{-2\mathtt{h}})$
and therefore (2.31) for any $\mathtt{p}\geq 4$. ∎
Proof of Lemma 2.9. In view of (2.20) the operator ${\mathcal{L}}_{\mu,0}$ in
(2.19) has a non-simple eigenvalue if and only if the equation
$\lambda_{j}^{\sigma}(\mu,\mathtt{h})=\lambda_{j^{\prime}}^{\sigma^{\prime}}(\mu,\mathtt{h})$
has a solution $(j,\sigma)\neq(j^{\prime},\sigma^{\prime})$ with
$j,j^{\prime}\in\mathbb{Z}$ and $\sigma,\sigma^{\prime}=\pm$. This amounts to
solve

$\omega^{\sigma}(j+\mu,\mathtt{h})=\omega^{\sigma^{\prime}}(j+\mu+\mathtt{p},\mathtt{h})\quad\text{for
some}\quad\sigma,\sigma^{\prime}\in\\{\pm\\}\,,\ j\in\mathbb{Z}\,,\
\mathtt{p}\in\mathbb{N}_{0}\,,$

namely

$\omega^{\sigma}(\varphi,\mathtt{h})=\omega^{\sigma^{\prime}}(\varphi+\mathtt{p},\mathtt{h})\quad\text{for
some}\quad\sigma,\sigma^{\prime}\in\\{\pm\\}\,,\
\mathtt{p}\in\mathbb{N}_{0}\,,\ \varphi-\mu\in\mathbb{Z}\,.$ (A.13)

In view of Lemma 2.7 and Remark 2.8 the equation (A.13) has a solution with
$\omega^{\sigma}(\varphi,\mathtt{h})=\omega^{\sigma^{\prime}}(\varphi+\mathtt{p},\mathtt{h})\neq
0$ if and only if $\sigma=-$, $\sigma^{\prime}=+$, $\mathtt{p}\geq 2$ and
$\underline{\varphi}(\mathtt{p},\mathtt{h})-\mu\in\mathbb{Z}$. Then Item 1
follows. For Item 2, if
$\underline{\varphi}(\mathtt{p},\mathtt{h})-\mu=:k\in\mathbb{Z}$ then, in view
of (2.26), (2.20), (2.23) and since
$\underline{\varphi}(\mathtt{p},\mathtt{h})\neq 0$ then
$\mathrm{i}\,\omega_{*}^{(\mathtt{p})}(\mathtt{h})$ in (2.32) is a double
eigenvalue of ${\mathcal{L}}_{\mu,0}$ with associated eigenvectors (2.34).
Furthermore (2.33) follows by (2.26) and (2.28) because
$\varphi\mapsto{\mathtt{c}}_{\mathtt{h}}\varphi+\Omega(\varphi,\mathtt{h})$ is
increasing for $\varphi>0$. Finally by (2.27), the opposite double eigenvalue
$-\mathrm{i}\,\omega_{*}^{(\mathtt{p})}(\mathtt{h})$ is obtained for
$\varphi=-\underline{\varphi}(\mathtt{p},\mathtt{h})-\mathtt{p}$ and belongs
to the spectrum of ${\mathcal{L}}_{-\mu,0}$. ∎

## References