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\title[Asymptotic formula for large eigenvalues of the 2-photon QRM] {Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model}
\alttitle{Formule asymptotique pour les grandes valeurs propres du mod\`ele quantique de Rabi \`a deux photons}

\author{\firstname{Anne} \lastname{Boutet de Monvel}}
%\address{Institut de Math\'ematiques de Jussieu, Universit\'e Paris Diderot Paris 7, 175 rue du Chevaleret, 75013 Paris, France}
\address{Institut de Mathématiques de Jusieu, Université Paris Cité, 8 place Aurélie Nemours, 75013 Paris, France}
%\email{aboutet@math.jussieu.fr}
\email{anne.boutet-de-monvel@imj-prg.fr}

\author{\firstname{Lech} \lastname{Zielinski}\IsCorresp}
\address{Laboratoire de Math\'ematiques Pures et Appliqu\'ees Joseph Liouville UR 2597, Universit\'e du Littoral C\^ote d'Opale, 62228 Calais, France}
\email{lech.zielinski@lmpa.univ-littoral.fr}

\begin{abstract}
We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences $(E_m^+)_{m=0}^\infty$, $(E_m^-)_{m=0}^\infty$ satisfying a three-term asymptotic formula with the remainder estimate $O(m^{-1}\ln m)$ when $m$ tends to infinity. By analogy to the one-photon quantum Rabi model, the leading three terms of this asymptotic formula, describe a generalized rotating-wave approximation for large eigenvalues of the two-photon quantum Rabi model.
\end{abstract}

\begin{altabstract}
Nous d\'emontrons que le spectre de l'hamiltonien du mod\`ele quantique de Rabi \`a deux photons est constitu\'e de deux suites de valeurs propres $(E_m^+)_{m=0}^\infty$, $(E_m^-)_{m=0}^\infty$ v\'erifiant une formule asymptotique
\`a trois termes avec l'estimation de l'erreur $O(m^{-1}\ln m)$ quand $m$ tend vers l'infini. Par analogie avec le mod\`ele quantique de Rabi \`a un photon, les trois termes dominants dans cette formule asymptotique d\'ecrivent l'approximation de l'onde tournante pour les grandes valeurs propres du mod\`ele quantique de Rabi \`a deux photons.
\end{altabstract}

\begin{document}
\maketitle

\section{Introduction} \label{sec:1}

The quantum Rabi model describes the interactions between a two-level system and a single-mode quantum field. It is considered as a fundamental model in various domains of theoretical physics, e.g. cavity optics, a theory of nanostructured semiconductors, superconducting circuits, trapped ions and quantum information. We refer to~\cite{Q} for an exhaustive overview of theoretical and experimental works in the domain.

The papers Boutet de Monvel, Naboko, Silva~\cite{BNS2, BNS} and Yanovich~\cite{Tur3} (see also~\cite{Tur2}) investigate the behaviour of large eigenvalues of operators related to the quantum Rabi model. The paper~\cite{BZ6} gives the three-term asymptotic formula for large eigenvalues of the one-photon Rabi model (see Definition~\ref{def}$\MK$\eqref{1.1}). The two-term asymptotic formula for the two-photon Rabi model (see Definition~\ref{def}(2)) is given in~\cite{BZ8} and this note gives the three-term asymptotic formula for this model. Concerning the two-photon Rabi model, we refer to~\cite{Emary1, G, Q}.

In Section~\ref{sec:2} we give the definition of the basic quantum Rabi model in the one-photon and two-photon case. Our main result is Theorem~\ref{thm}, stated in Section~\ref{sec:3}. We describe main ingredients of the proof in Sections~\ref{sec:4}, \ref{sec:5} and \ref{sec:6}.

We remark that our result is closely related to the method of the generalized rotating-wave approximation (GRWA), used by a great number of physicists working with the quantum Rabi model. The method takes its name from the famous paper of Irish~\cite{irish}, but the same idea was also used in the paper Feranchuk, Komarov, Ulyanenkov~\cite{FKU} under the name of the zeroth order approximation of the operator method (see also~\cite{Fer}). It appears that in the case of the one-photon quantum Rabi model, the GRWA given by the formula (25) in~\cite{FKU}, coincides with the three-term asymptotic formula proved in~\cite{BZ6}. In Section~6 we describe the relation between the assertion of Theorem~\ref{thm} and the GRWA for large eigenvalues of the two-photon quantum Rabi model.

\section{Definition of the quantum Rabi model} \label{sec:2}

In what follows, $\mathbb Z$ denotes the set of integers,\, $\mathbb N=\{ n\in \mathbb Z :n\ge 0\}$ and $\ell^2 (\mathbb N)$ is the Hilbert space of square summable sequences $x:\mathbb N\to\mathbb{C}$. For $s\ge 0$ we denote
\[
\ell^{2,s}(\mathbb N)=\left\{ x\in \ell^2(\mathbb N):\, \sum_{k\in \mathbb N} \, \, (1+k^2)^s \, |x(k)|^2 <\infty \right\}.
\]
We define the photon annihilation and creation operators $\hat{a}$ and $\hat{a}^{\dagger}$, as linear maps $\ell^{2,1/2}(\mathbb N)\to \ell^{2}(\mathbb N)$ satisfying
\begin{align*}
\hat{a}^{\dagger}e_m&=\sqrt{m+1}\,e_{m+1}\quad \text{for } m=0,\, 1,\, 2,\dots
\\
\hat{a}e_{m}&=\sqrt{m-1}\,e_{m-1}\quad\text{for } m=1,\, 2,\, 3,\dots \text{ and } \hat{a}e_0=0,
\end{align*}
where $\{ e_m\}_{m\in\mathbb{N}}$ is the canonical basis of $\ell^2(\mathbb{N})$.

\begin{defi}\label{def}
We fix two real parameters: the energy spacing of the two-level system $\Delta$ and the coupling constant $g$.
\begin{enumerate}
\item \label{1.1} The Hamiltonian of the one-photon quantum Rabi model is given by the linear map ${\mathbb C}^2 \otimes \ell^{2,1} (\mathbb N)\to {\mathbb C}^2 \otimes \ell^2 (\mathbb N)$ of the form
\begin{equation}\label{R1}
H_1 =
\begin{pmatrix} \frac{\Delta}{2} & 0 \\ 0 & -\frac{\Delta}{2}
\end{pmatrix} \otimes I_{\ell^2(\mathbb N)} \, +\,
I_{{\mathbb C}^2}\otimes \hat{a}^{\dagger} \hat{a} \, +\,
\begin{pmatrix}0 & g \\ g & 0
\end{pmatrix} \otimes
\left({{\hat a}^\dagger +{\hat a} }\right).
\end{equation}
\item \label{1.2} The Hamiltonian of the two-photon quantum Rabi model is given by the linear map ${\mathbb C}^2 \otimes \ell^{2,1} (\mathbb N)\to {\mathbb C}^2 \otimes \ell^2 (\mathbb N)$ of the form
\begin{equation}\label{R2}
H_2 =
\begin{pmatrix} \frac{\Delta}{2} & 0 \\ 0 & -\frac{\Delta}{2}
\end{pmatrix} \otimes I_{\ell^2(\mathbb N)} \, +\,
I_{{\mathbb C}^2}\otimes \hat{a}^{\dagger} \hat{a} \, +\,
\begin{pmatrix}0 & g \\ g & 0
\end{pmatrix} \otimes
\left({({\hat a}^\dagger)^2 +{\hat a}^2}\right).
\end{equation}
\end{enumerate}
\end{defi}


\section{Main result} \label{sec:3}
In what follows, we assume $0<g<1/2$ and introduce
\begin{equation}\label{beta}
\beta :=\sqrt{1-4g^2}.
\end{equation}
Let $H^0_2$ denote the operator given by~\eqref{R2} with $\Delta =0$. If $0<g<1/2$, then the spectrum of $H^0_2$ is explicitly known (see~\cite{Emary1}): it is composed of the sequence of eigenvalues
\begin{equation}\label{E0}
E^0_{m}=m\beta + (\beta -1)/2,\quad m=0,\, 1,\, 2,\dots
\end{equation}
and each eigenvalue $E^0_{m}$ is of multiplicity 2. Thus $0<g <1/2$ ensures the fact that $H^0_2$ is self-adjoint and has compact resolvent. Since $H_2-H^0_2$ is bounded, the operator $H_2$ is self-adjoint and has compact resolvent if $0<g<1/2$. The explicit values of eigenvalues of $H_2$ are not known when $\Delta \ne 0$, but we can described their asymptotic behavior in

\begin{theo}\label{thm}
Assume that $0<g <1/2$. Then one can find $\{ v_m^+\}_{m\in \mathbb N} \cup \{ v_m^-\}_{m\in \mathbb N}$, an orthonormal basis of ${\mathbb C}^2 \otimes \ell^2 (\mathbb N)$, such that
\[
H_2 v_m^\pm =E_m^\pm v_m^\pm,\quad m=0,\, 1,\, 2,\dots
\]
and the eigenvalue sequences $(E_m^+)_{m\in \mathbb N}$, $(E_m^-)_{m\in \mathbb N}$, satisfy the large $m$ estimates
\begin{equation}\label{AF}
E_m^\pm =m\beta + (\beta -1)/2
\pm {r}_m + O(m^{-1}\ln m)
\end{equation}
with $r_m$ given by the formula
\begin{equation}\label{AF3}
r_m =
\begin{dcases}
\frac{\Delta}{2} \,\sqrt{\frac{\beta}{\pi g m}} \, \, \cos ((2m+1)\alpha) &\text{if\/ $m$ is even}\\
\frac{\Delta}{2} \,\sqrt{\frac{\beta}{\pi g m}} \, \, \sin ((2m+1)\alpha) &\text{if\/ $m$ is odd}
\end{dcases}
\end{equation}
where we have denoted
\begin{equation}\label{alpha}
\alpha :=\arctan \left({ \sqrt{ \frac{1-2g}{1+2g} } \, }\right).
\end{equation}
\end{theo}

\begin{remas*}\leavevmode
\begin{enumerate}\alphenumi
\item \label{ra} One has $E_m^\pm -E_m^0=O(m^{-1/2})$ in spite of the fact that $H_2-H^0_2$ is not compact. A similar fact was established in~\cite{Tur3} for the one-photon Rabi model.
\item \label{rb} Following~\cite{Sah}, one can prove that the spectrum of $H_2$ is not discrete if $g\ge 1/2$.
\end{enumerate}
\end{remas*}


\section{Initial reformulations} \label{sec:4}

It is easy to check that the operator $H_2$ has four closed invariant subspaces:
\begin{align*}
\mathcal H^-_0& \text{ spanned by } \{ (1,0)\otimes e_{4k}: k\in \mathbb N \} \cup
\{ (0,1)\otimes e_{4k+2}: k\in \mathbb N \}
\\
\mathcal H^+_0& \text{ spanned by } \{ (0,1)\otimes e_{4k}: k\in \mathbb N \} \cup
\{ (1,0)\otimes e_{4k+2}: k\in \mathbb N \}
\\
\mathcal H^-_1& \text{ spanned by } \{ (1,0)\otimes e_{4k+1}: k\in \mathbb N \} \cup
\{ (0,1)\otimes e_{4k+3}: k\in \mathbb N \}
\\
\mathcal H^+_1& \text{ spanned by } \{ (0,1)\otimes e_{4k+1}: k\in \mathbb N \} \cup
\{ (1,0)\otimes e_{4k+3}: k\in \mathbb N \}
\end{align*}
and the matrix of $H_2$ in a suitable basis of ${\mathcal H}_\mu^\pm$\, $\mu =$0, 1,\, can be written in the form
\begin{equation}\label{not2}
J_\mu^\pm = \left(
\begin{matrix}
d_\mu^\pm (0) & {b_\mu (0)} & 0 & 0 &\cdots\\
{b_\mu (0)} & d_\mu^\pm (1) & {b_\mu (1)} & 0 &\cdots\\
0 & {b_\mu (1)} & d_\mu^\pm (2) & {b_\mu (2)} &\cdots\\
0 & 0 & {b_\mu (2)} & d_\mu^\pm (3) & \cdots\\
\vdots&\vdots &\vdots&\vdots& \ddots
\end{matrix} \right)
\end{equation}
where
\begin{align*}
d^\pm_{\mu }(m)&:=2m+\mu \pm (-1)^m \Delta /2,
\\
b_\mu (m) &:=g\sqrt{(2m+1+\mu)(2m+2+\mu)}.
\end{align*}
It therefore remains to investigate the asymptotic behavior of eigenvalues of operators defined by $J^-_0$, $J^+_0$, $J^-_1$ and $J^+_1$. For this purpose, we remark that
\begin{equation}\label{23d}
b_{\mu} (m)= 2g(m+\gamma) +O(m^{-1}) \text{ holds with } \gamma := \frac{\mu}{2} + \frac{3}{4}.
\end{equation}
Using the result of Rozenblum stated in Theorem~1.1 of~\cite{Roz}, we find that modulo $O(m^{-1})$, the asymptotic behaviour of the $m$-th eigenvalue remains the same if the entries $\{ b_{\mu} (m)\}_{m\in \mathbb N}$ are replaced by $\{ 2g(m+\gamma) \}_{m\in \mathbb N}$ with $\gamma := \frac{\mu}{2} + \frac{3}{4}$. This fact allows us to deduce the assertion of Theorem~\ref{thm} from

\begin{theo}\label{thm'}
Assume that $0<g<1/2$. Let $\delta$, $\gamma$ be some real numbers and let $\hat{J}^{\delta}_{\gamma}$ be the linear map $\ell^{2,1}(\mathbb N)\to \ell^{2}(\mathbb N)$ defined by the formula
\begin{equation}\label{tildej}
\hat J^{\delta}_\gamma e_m =\bigl(m+(-1)^m \delta \bigr) e_m +g(m+\gamma)e_{m+1}+g(m-1+\gamma) e_{m-1},\quad m\in \mathbb N,
\end{equation}
where $\{ e_m\}_{m\in \mathbb N}$ is the canonical basis in $\ell^2(\mathbb N)$ and by convention $g(m-1+\gamma) e_{m-1}=0$ if $m=0$. Then $\hat{J}^{\delta}_{\gamma}$ has discrete spectrum and its non-decreasing eigenvalue sequence satisfies the large $n$ asymptotic formula
\begin{equation}\label{AF2}
\lambda_n (\hat{J}^{\delta}_{\gamma}) =\beta n +(\gamma -1/2) (\beta -1) +\delta r _{\gamma,n}+ O(n^{-1}\ln n)
\end{equation}
with
\begin{equation}\label{AF2a}
r_{\gamma,n} = \sqrt{ \frac{\beta}{2\pi g n} } \, \cos \big(4\alpha n +
\hat{\theta}_\gamma \big),
\end{equation}
where $\beta$ is given by~\eqref{beta}, $\alpha$ by~\eqref{alpha} and
\begin{equation}\label{AF2b}
\hat{\theta}_\gamma =(\gamma -1/2)(4\alpha -\pi) + {\pi}/{4}.
\end{equation}
\end{theo}

In order to prove Theorem~\ref{thm'}, we move from $\ell^2(\mathbb N)$ to $\ell^2(\mathbb Z)$, the Hilbert space of square summable sequences $x: \mathbb Z \rightarrow\mathbb{C}$. For $s>0$ we denote
\[
\ell^{2,s}(\mathbb Z)=\left\{ x\in \ell^2(\mathbb Z):\, \sum_{k\in \mathbb Z} \, \, (1+k^2)^s \, |x(k)|^2 <\infty \right\}
\]
and define $\tilde{J}^{\delta}_{\gamma}$ as the linear map $\ell^{2,1}(\mathbb Z)\to \ell^{2}(\mathbb Z)$ given by
\begin{equation}\label{tildeJ}
\tilde J^{\delta}_\gamma e_k =\bigl(k+(-1)^k \delta \bigr) e_k +g(k+\gamma)e_{k+1}+g(k-1+\gamma) e_{k-1},\quad k\in \mathbb Z,
\end{equation}
where $\{ e_k\}_{k\in \mathbb Z}$ is the canonical basis of $\ell^2(\mathbb Z)$. We can identify $\ell^2(\mathbb N)$ with
\[
\{ x\in \ell^2(\mathbb Z): x(k)=0 \text{ for } k\in \mathbb Z \setminus \mathbb N \}
\]
and consider $\tilde J^{\delta}_\gamma$ as an extension of $\hat J^{\delta}_\gamma$. Using Theorem~1.1 in~\cite{Roz}, we find that the spectrum of $\tilde J^{\delta}_\gamma$ is composed of a non-decreasing sequence of eigenvalues $\{ \lambda_j (\tilde J^{\delta}_\gamma)\}_{j\in \mathbb Z} $, which can be labeled so that for any $N>0$ one has the estimate
\begin{equation}\label{lambdantildeJ}
\lambda_n (\tilde J^{\delta}_\gamma) =
\lambda_n (\hat J^{\delta}_\gamma) +O(n^{-N}) \text{ when } n\to \infty
\end{equation}


\section{Diagonalisation of \texorpdfstring{$\tilde J^{\delta}_\gamma$}{J delta gamma} when \texorpdfstring{$\delta =0$}{delta =0}}\label{sec:5}

The operator $\tilde J^{\delta}_\gamma$ with $\delta =0$ was investigated in~\cite{E}. Let $S$ be the shift $Se_j=e_{j+1}$ in $\ell^2(\mathbb Z)$ and denote $\Lambda :=\diag(j)_{j\in \mathbb Z}$. Then~\eqref{tildeJ} gives
\[
\tilde J^0_\gamma =\Lambda +g \left({S \big(\Lambda +\gamma \big) +
\big(\Lambda +\gamma \big) S^{-1} }\right) = \Lambda + g\left({ S(\Lambda +\gamma \big) +
\,\rm{h.c.} \, }\right)
\]
Let $\mathbb T =\mathbb R/2\pi \mathbb Z$ and let $\mathcal{F}_{\mathbb T}$ be the isometric isomorphism $ \Lrm^2(\mathbb T) \to \ell^2(\mathbb{Z})$ given by
\begin{equation}\label{F0}
(\mathcal{F}_{\mathbb T} f)(j) = \int_{-\pi}^{\pi} f(\theta)\, \erm^{-\irm j\theta} \, \frac{{\dd} \theta}{2\pi}.
\end{equation}
Then $L_\gamma :=\mathcal F_{\mathbb T}^{-1} \tilde J^0_\gamma \mathcal F_{\mathbb T}$ is the first order differential operator
\[
L_\gamma = \frac{1}{2} \left(\big(1+2g\cos \theta \big) \Biggl(- \irm
\frac{\dd}{{\dd} \theta} \Biggr) + \rm{h.c.}\right) +(1+2\gamma)\cos \theta
\]
and following~\cite{E}, we define the diffeomorphism of $]-\pi,\pi [$ given by
\begin{equation}\label{Phi}
\Phi (\theta) : =\int_0^\theta
\frac{\beta \, {\dd} \theta ' }{1+2g\cos \theta '} =2\arctan \left({\sqrt{\frac{1-2g}{1+2g}} \tan \Big(\frac{\theta }{2} \Big) }\right).
\end{equation}
The change of variable $\eta =\Phi (\theta)$ defines the unitary operator acting in $\Lrm^2(\mathbb T)$ according to the formula
\begin{equation}\label{}
(U_{\Phi} f)(\theta)= \Phi '(\theta)^{1/2} f({\Phi} (\theta))
\end{equation}
and the direct computation gives
\[
U^{-1}_{\Phi} \, L_\gamma U_{\Phi} =\beta \left({- \irm
\frac{\dd}{{\dd} \eta} + q_\gamma (\eta) }\right)
\]
with
\[
q_\gamma (\eta):=\beta^{-1}(1+2\gamma)\cos (\Phi^{-1}(\eta)).
\]
Let $\tilde q_\gamma$ be a primitive of $q_\gamma$. We compute
\[
\langle {q_\gamma} \rangle :=(\tilde q_\gamma (\pi)-\tilde q_\gamma (-\pi))/(2\pi) =
(\gamma -1/2)(1-1/\beta)
\]
and remark that $\eta \to \langle {q_\gamma} \rangle \eta -\tilde q_\gamma (\eta)$ is 2$\pi$-periodic, hence we can define $(f_{\gamma,j })_{j\in \mathbb Z}$, the orthonormal basis in ${\Lrm}^2(\mathbb T)$ given by
\begin{equation}
f_{\gamma,j }(\eta)= \erm^{\irm j\eta} \, \erm^{ \irm (\langle {q_\gamma} \rangle \eta -\tilde q_\gamma (\eta))}.
\end{equation}
Then (see~\cite{E}), for every $j\in \mathbb Z$, one has
\[
\beta \left({- \irm \frac{\dd}{{\dd} \eta} + q_\gamma }\right) f_{\gamma,j} =\beta (j+\langle {q_\gamma} \rangle)f_{\gamma,j}.
\]
Consequently, for every $j\in \mathbb Z$,
\begin{equation}\label{diag}
\tilde J^0_\gamma u_{\gamma,j} =d_{\gamma,j} u_{\gamma,j}
\end{equation}
holds with
\begin{gather}\label{ugj}
u_{\gamma,j}=\mathcal F_{\mathbb T} U_{\Phi}f_{\gamma,j},
\\
d_{\gamma,j} =\beta (j+\langle {q_\gamma}\rangle)=\beta j+(\gamma -1/2)(\beta -1).
\end{gather}


\section{Generalized rotating-wave approximation (GRWA)} \label{sec:6}

The idea of the GRWA consists in using the diagonal entries of a perturbation as the first correction for eigenvalues of a perturbed diagonal matrix and we refer to~\cite{Fer} for numerous examples of this approach. In order to apply this idea, we consider $\tilde J^\delta_\gamma$ as a perturbation of $\tilde J^0_\gamma$ and use the diagonalisation~\eqref{diag}. We remark that
\begin{equation}\label{61}
\tilde J^\delta_\gamma = \tilde J^0_\gamma + \delta V
\end{equation}
holds with $V:=\diag (\, (-1)^j)_{j\in \mathbb Z}$. Let $U_\gamma$ be the unitary operator in $\ell^2(\mathbb Z)$ defined by $U_\gamma e_j =u_{\gamma,j}$, where $\{ e_j\}_{j\in \mathbb Z}$ is the canonical basis of $\ell^2(\mathbb Z)$ and $\{ u_{\gamma,j}\}_{j\in \mathbb Z}$ is the basis given by~\eqref{ugj}. Then~\eqref{61} gives
\begin{equation}\label{62}
U^{-1}_\gamma \tilde J^\delta_\gamma U_\gamma =D_{\gamma} +\delta V_\gamma,
\end{equation}
where $D_{\gamma}=\diag (d_{\gamma,j})_{j\in \mathbb Z}$ and $V_\gamma =U^{-1}_\gamma V U_\gamma$. We claim that the asymptotic estimate~\eqref{AF2}--\eqref{AF2b} follows from
\begin{equation}\label{63}
\lambda_j (D_{\gamma} +\delta V_\gamma) =d_{\gamma,j} + \delta V_\gamma (j,j) + O(j^{-1}\ln j) \text{ when } j\to \infty
\end{equation}
where
\[
V_\gamma (j,j) =\langle {e_j, V_\gamma e_j } \rangle_{\ell^2(\mathbb Z)}=
\langle{u_{\gamma,j}, Vu_{\gamma,j}} \rangle_{\ell^2(\mathbb Z)}.
\]
Indeed, since $\mathcal F_{\mathbb T}^{-1} V \, \mathcal F_{\mathbb T} =T_\pi$ is the translation $\theta +2\pi \mathbb Z \to \theta +\pi +2\pi \mathbb Z$, we find that
\begin{equation}\label{64}
V_\gamma (j,j) =\langle {U_{\Phi} f_{\gamma,j}, T_\pi U_{\Phi} f_{\gamma,j}} \rangle_{{\Lrm}^2(\mathbb T)}=
\int_{-\pi}^\pi \erm^{\irm j(\Phi -T_\pi \Phi)(\theta)} p_\gamma (\theta)\, {\overline{(T_\pi p_\gamma)(\theta)}}\, \frac{{\dd} \theta }{2\pi}
\end{equation}
holds with\,
\[
p_\gamma (\theta):= \erm^{\irm \langle {q_\gamma}\rangle \Phi (\theta)-
\irm \widetilde q_\gamma (\Phi (\theta))}
\beta^{1/2} (1+2g \cos \theta)^{-1/2}
\]
and the stationary phase method gives
\begin{equation}\label{65}
V_\gamma (j,j) = {r}_{\gamma,j} +O(j^{-1}) \text{ when } j\to \infty
\end{equation}
where ${ r}_{\gamma,j}$ is given by~\eqref{AF2a}--\eqref{AF2b}. Thus~\eqref{AF2}--\eqref{AF2b} follow from~\eqref{63}, \eqref{65} and~\eqref{lambdantildeJ}.

The proof of~\eqref{63} is based on the approach of Yanovich~\cite{Tur3} and the explicit expressions of the entries
\begin{equation}
V_\gamma (j,k) ={\langle U_{\Phi} f_{\gamma,j}, T_\pi U_{\Phi} f_{\gamma,k}\rangle}_{{\Lrm}^2(\mathbb T)}.
\end{equation}

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