\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 15, pp. 1--30.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2004/15\hfil Heun equation II] {The Heun equation and the Calogero-Moser-Sutherland system II: Perturbation and algebraic solution} \author[Kouichi Takemura\hfil EJDE--2004/15\hfilneg] {Kouichi Takemura} \address{Kouichi Takemura \hfill\break Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan} \email{takemura@yokohama-cu.ac.jp} \date{} \thanks{Submitted May 15, 2003. Published February 5, 2004.} \subjclass[2000]{33E15, 81Q10} \keywords{Heun equation, Calogero-Moser-Sutherland system, \hfill\break\indent Inozemtsev model, perturbation, Kato-Rellich theory, trigonometric limit, Heun function, \hfill\break\indent algebraic solution} \begin{abstract} We apply a method of perturbation for the $BC_1$ Inozemtsev model from the trigonometric model and show the holomorphy of perturbation. Consequently, the convergence of eigenvalues and eigenfuncions which are expressed as formal power series is proved. We investigate also the relationship between $L^2$ space and some finite dimensional space of elliptic functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{prop}{Proposition}[section] \newtheorem{thm}[prop]{Theorem} \newtheorem{lemma}[prop]{Lemma} \newtheorem{cor}[prop]{Corollary} \section{Introduction} In this paper, we report some properties for eigenvalues and eigenfuncions of the $BC_1$ Inozemtsev model. Consequently, we obtain results on Heun function. The $BC_1$ Inozemtsev model is a one-particle model of quantum mechanics whose Hamiltonian is $$H= -\frac{d^2}{dx^2} + \sum_{i=0}^3 l_i(l_i+1)\wp (x+\omega_i), \label{Ino00}$$ where $\wp (x)$ is the Weierstrass $\wp$-function with periods $(1, \tau)$, $\omega _0=0$, $\omega _1=1/2$, $\omega_2=(1+\tau )/2$, $\omega_3=\tau /2$ are half-periods, and $l_i$, $(i=0,1,2,3)$ are coupling constants. This model is sometimes called the $BC_1$ elliptic Inozemtsev model, because the potential is described by use of elliptic functions. There are two evidences which ensure the importance of the $BC_1$ Inozemtsev model. The first one is equivalence to Heun equation, which will be explained in section \ref{HI}. The other one is that $BC_1$ quantum Inozemtsev model is a special $(N=1)$ case of $BC_N$ Inozemtsev model \cite{Ino}, which is a generic integrable quantum system with $B_N$ symmetry. In fact, classification of integrable quantum systems with $B_N$ symmetry was done by Ochiai, Oshima, and Sekiguchi \cite{OOS}, and it was shown that integrable quantum system with $B_N$ symmetry is $BC_N$ Inozemtsev model or its degenerate one. It is known that $BC_N$ Inozemtsev system contains the well-known Calogero-Moser-Sutherland system with $B_N$ symmetry as a special case. In this paper, we try to obtain physical eigenfunctions and eigenvalues of the $BC_1$ Inozemtsev model, and investigate their properties. Here a physical'' eigenfunction means that it is contained in an appropriate Hilbert space, which is often a space of square-integrable ($L^2$) functions. Note that roughly speaking the physical'' eigenfunction corresponds to the Heun function of Heun equation. Applying a method of perturbation is a possible approach to this problem, which was done in \cite{Tak,KT} for the Calogero-Moser-Sutherland system of type $A_N$. Now we explain this method shortly. Elliptic functions have a period $\tau$. By a trigonometric limit $p =\exp(\pi \sqrt{-1} \tau) \to 0$, the Hamiltonian of the $BC_1$ elliptic Inozemtsev model tends to the Hamiltonian of the $BC_1$ Calogero-Moser-Sutherland model, and it is known that eigenvalues and eigenstates of the $BC_1$ Calogero-Moser-Sutherland model are obtained explicitly by use of Jacobi polynomials. Based on eigenstates for the case $p=0$, we can obtain eigenvalues and eigenstates for the $BC_1$ elliptic Inozemtsev model $(p\neq 0)$ as formal power series in $p$. This procedure is sometimes called an algorithm of perturbation (see section \ref{sect:formalpert}). Generally speaking, convergence of the formal power series obtained by perturbation is not guaranteed a priori, but for the case of $BC_1$ elliptic Inozemtsev model, the convergence radius of the formal power series in $p$ is shown to be non-zero (see Corollary \ref{cor:conv}), and this perturbation is holomorphic. As a result, real-holomorphy of eigenvalues in $p$ and completeness of eigenfunctions is proved. Note that a partial result was obtained in part I \cite{Tak1} by applying Bethe Ansatz. There is another method to investigate eigenvalues and eigenstates of the $BC_1$ Inozemtsev model. If the coupling constants $l_0$, $l_1$, $l_2$, $l_3$ satisfy some equation, the Hamiltonian $H$ (see (\ref{Ino00})) preserves a finite dimensional space of doubly periodic functions which is related to the quasi-exact solvability \cite{Tur,GKO}. On a finite dimensional space, eigenvalues are calculated by solving the characteristic equation, which is an algebraic equation, and eigenfunctions are obtained by solving linear equations. In this sense, eigenvalues on a finite dimensional space are algebraic'', and eigenvalues and eigenfunctions on a finite dimensional space would be more explicit than ones on an infinite dimensional Hilbert space. In this paper, we also investigate relationship between Hilbert spaces ($L^2$ spaces) and invariant spaces of doubly periodic functions with respect to the action of the Hamiltonian $H$. In some cases, a finite dimensional invariant space becomes a subspace of the Hilbert space. Then it is shown under some assumption that the set of eigenvalues on the finite dimensional invariant space coincides with the set of small eigenvalues from the bottom on the Hilbert space. This paper is organized as follows. In section \ref{HI}, the relationship between the Heun equation and the $BC_1$ Inozemtsev system is clarified. Next we consider a trigonometric limit and review that eigenstates for the trigonometric model are given by hypergeometric (Jacobi) polynomials. We also explain how to apply an algorithm of perturbation in order to obtain formal eigenvalues and eigenfunctions for the $BC_1$ elliptic Inozemtsev model. In section \ref{sec:pertu}, we prove convergence of the algorithm of perturbation by applying Kato-Rellich theory. We also obtain several results related to Kato-Rellich theory. Although holomorphy of the eigenfunctions $\tilde{v}_m(x,p)$ in $p$ as elements of $L^2$ space is shown by Kato-Rellich theory, convergence of the eigenfunctions for each $x$ is not assured immediately. In section \ref{sec:holomo}, we show uniform convergence and holomorphy of the eigenfunctions $\tilde{v}_m(x,p)$ for $x$ on compact sets. In section \ref{sec:algfcn}, finite dimensional invariant subspaces of doubly periodic functions are investigated and relationship to the Hilbert space ($L^2$ space) is discussed. In section \ref{sec:nonneg}, we focus on the case $l_0, l_1, l_2, l_3 \in \mathbb{Z}_{\geq 0}$. In section \ref{sec:example}, examples are presented to illustrate results of this paper (especially section \ref{sec:algfcn}). In section \ref{sec:com}, we give some comments. In section \ref{sec:app}, some propositions are proved and definitions and properties of elliptic functions are provided. We note that some results of this paper are generalized to the case of the $BC_N$ Inozemtsev model (see \cite{Takq}). \section{Heun equation, trigonometric limit and algorithm of perturbation} \label{HI} \subsection{Heun equation and Inozemtsev system} It is known that the Heun equation admits an expression in terms of elliptic functions and this expression is closely related to the $BC_1$ Inozemtsev system \cite{Ron,OOS,OS,Tak1}. In this subsection, we will explain this. Let us recall the Hamiltonian of the $BC_1$ Inozemtsev model $$H= -\frac{d^2}{dx^2} + \sum_{i=0}^3 l_i(l_i+1)\wp (x+\omega_i), \label{Ino}$$ where $\wp (x)$ is the Weierstrass $\wp$-function with periods $(1, \tau)$, $\omega _0=0$, $\omega _1=1/2$, $\omega_2=(1+\tau )/2$, $\omega_3=\tau /2$ are half-periods, and $l_i$ $(i=0,1,2,3)$ are coupling constants. Assume that the imaginary part of $\tau$ is positive. Set \begin{gather*} e_i=\wp(\omega_i) \quad (i=1,2,3), \quad a=\frac{e_2-e_3}{e_1-e_3} , \\ \tilde{\Phi}(w)=w^{\frac{l_0+1}{2}}(w-1)^{\frac{l_1+1}{2}}(aw-1)^{\frac{l_2+1}{2}}. \end{gather*} Note that $a$ is nothing but the elliptic modular function $\lambda (\tau)$. We change a variable by $$w=\frac{e_1-e_3}{\wp (x)-e_3}. \label{wxtrans}$$ Then \label{Inotrans} \begin{aligned} \tilde{\Phi}(w)^{-1} \circ H \circ \tilde{\Phi}(w) &=-4(e_1-e_3)\Big\{ w(w-1)(aw-1) \big(\frac{d}{dw}\big) ^2 \\ &\quad +\frac{1}{2}\big( \frac{2l_0+3}{w}+ \frac{2l_1+3}{w-1} + \frac{a(2l_2+3)}{aw-1} \big) \frac{d}{dw} \Big\} +a\alpha \beta w +\tilde{q} \Big\}, \end{aligned} where $\tilde{q}= \big( \frac{a+1}{3}\sum_{i=0}^3 l_i(l_i +1)-a(l_0+l_2+2)^2 -(l_0+l_1+2)^2 \big)$, $\alpha =\frac{l_0+l_1+l_2+l_3+4}{2}$, $\beta =\frac{l_0+l_1+l_2-l_3+3}{2}$. Let $f(x)$ be an eigenfunction of $H$ with an eigenvalue $E$, i.e., $$(H-E) f(x)= \Big( -\frac{d^2}{dx^2} + \sum_{i=0}^3 l_i(l_i+1)\wp (x+\omega_i)-E\Big) f(x)=0. \label{InoEF}$$ From (\ref{Inotrans}) and (\ref{InoEF}), we obtain $$\Big(\big(\frac{d}{dw}\big) ^2 + \big( \frac{l_0+\frac{3}{2}}{w} +\frac{l_1+\frac{3}{2}}{w-1}+\frac{l_2+\frac{3}{2}}{w-\frac{1}{a}}\big) \frac{d}{dw} +\frac{\alpha \beta w -q}{w(w-1)(w-\frac{1}{a})} \Big)\tilde{f}(w)=0, \label{Heun}$$ where $\tilde{f}(\frac{e_1-e_3}{\wp (x)-e_3}) \tilde{\Phi}(\frac{e_1-e_3}{\wp (x)-e_3})=f(x)$ and $q=-\frac{1}{4a}\big( \frac{E}{e_1-e_3}+\tilde{q} \big)$. Note that the condition $$l_0+\frac{3}{2}+l_1+\frac{3}{2}+l_2+\frac{3}{2}=\alpha +\beta +1 \label{HeunFR}$$ is satisfied. Equation (\ref{Heun}) with condition (\ref{HeunFR}) is called the Heun equation \cite{Ron,SL}. It has four singular points $0$, $1$, $a^{-1}$, $\infty$, all the singular points are regular. The following Riemann's $P$-symbol show the exponents. \begin{equation*} P\begin{pmatrix} 0 & 1 & a^{-1} & \infty & & \\ 0 & 0 & 0 & \alpha & z & q \\ -l_0-\frac{1}{2} & -l_1-\frac{1}{2} & -l_2-\frac{1}{2} & \beta & & \end{pmatrix} \end{equation*} Up to here, we have explained how to transform the equation of the Inozemtsev model into the Heun equation. Conversely, if a differential equation of second order with four regular singular points on a Riemann sphere is given, we can transform it into equation (\ref{Heun}) with condition (\ref{HeunFR}) with suitable $l_i$ $(i=0,1,2,3)$ and $q$ by changing a variable $w \to \frac{a'w+b'}{c'w+d'}$ and a transformation $f \to w^{\alpha_1}(w-1)^{\alpha_2}(w-a^{-1})^{\alpha_3}f$. It is known that if $a \neq 0,1$ then there exists a solution $\tau$ to the equation $a=\frac{e_2-e_3}{e_1-e_3}$ ($e_i$ $(i=1,2,3)$ depend on $\tau$). Thus the parameter $\tau$ is determined. The values $e_1,e_2,e_3$ and $E$ are determined by turn. Hence we obtain a Hamiltonian of $BC_1$ Inozemtsev model with an eigenvalue $E$ starting from a differential equation of second order with four regular singular points on a Riemann sphere. \subsection{Trigonometric limit} \label{sect:trig} In this section, we will consider a trigonometric limit $(\tau \to \sqrt{-1} \infty)$. We introduce a parameter $p=\exp (\pi \sqrt{-1} \tau )$, then $p\to 0$ as $\tau \to \sqrt{-1} \infty$. (Note that the parameter $p$ is different from the one in \cite{Tak1}.) If $p \to 0$, then $e_1 \to \frac{2}{3}\pi^2$, $e_2 \to -\frac{1}{3}\pi^2$, $e_3 \to -\frac{1}{3}\pi^2$, and $a \to 0$, and the relation between $x$ and $w$ (see (\ref{wxtrans})) tends to $w=\sin ^2 \pi x$ as $p \to 0$. Set \begin{gather} H_T= -\frac{d^2}{dx^2} + l_0(l_0+1)\frac{\pi ^2}{\sin ^2\pi x} +l_1(l_1+1) \frac{\pi ^2}{\cos ^2\pi x} ,\label{triIno} \\ L_T = w(w-1)\Big\{ \frac{d^2}{dw^2} + \big( \frac{l_0+\frac{3}{2}}{w} + \frac{l_1+\frac{3}{2}}{w-1} \big)\frac{d}{dw} +\frac{(l_0+l_1+2)^2}{4w(w-1)} \Big\} \label{gauss}. \end{gather} Then $H \to H_T -\frac{\pi^2}{3}\sum_{i=0}^3 l_i (l_i +1)$ and equation (\ref{Heun}) tends to $(L_T -\frac{E}{\pi^2}-\frac{1}{3}\sum_{i=0}^3 l_i (l_i +1) )\tilde{f}(w)=0$ as $p \to 0$. The operator $H_T$ is nothing but the Hamiltonian of the $BC_1$ trigonometric Calogero-Moser-Sutherland model, and the equation $(L_T-C)\tilde{f}(w)=0$ ($C$ is a constant) is a Gauss hypergeometric equation. Now we solve a spectral problem for $H_T$ by using hypergeometric functions. We divide into four cases, the case $l_0>0$ and $l_1>0$, the case $l_0>0$ and $l_1=0$, the case $l_0=0$ and $l_1>0$, and the case $l_0=l_1=0$. For each case, we set up a Hilbert space $\mathbf{H}$, find a dense eigenbasis, and obtain essential selfadjointness of the gauge-transformed trigonometric Hamiltonian. The Hilbert space $\mathbf{H}$ plays an important role to show holomorphy of perturbation in $p$, which will be discussed in section \ref{sec:pertu}. We note that the case $l_0=0$ and $l_1>0$ comes down to the case $l_0>0$ and $l_1=0$ by setting $x \to x+\frac{1}{2}$. \subsubsection{The case $l_0>0$ and $l_1>0$} \label{sssec1} Set \begin{equation*} \Phi(x)=(\sin \pi x)^{l_0+1}(\cos \pi x)^{l_1+1}, \quad \mathcal{H}_T=\Phi(x)^{-1} H_T \Phi(x), \end{equation*} then the gauge transformed Hamiltonian $\mathcal{H}_T$ is expressed as $$\mathcal{H}_T= -\frac{d^2}{dx^2}-2\pi\Big( \frac{(l_0 +1)\cos \pi x}{\sin \pi x} - \frac{(l_1 +1)\sin \pi x}{\cos \pi x} \Big) \frac{d}{dx}+(l_0+l_1+2)^2\pi^2. \label{JacobiHam}$$ By a change of variable $w=\sin ^2 \pi x$, we have \label{Ghyper} \begin{aligned} \: & \mathcal{H}_T -\pi^2(2m+l_0+l_1+2)^2 \\ & = -4\pi ^2 \Big\{ w(1-w)\frac{d^2}{dw^2} \\ & \quad +\Big(\frac{2l_0+3}{2}-((l_0+l_1+2)+1)w\Big)\frac{d}{dw}+m(m+l_0+l_1+2) \Big\} \end{aligned} for each value $m$. Hence the equation $\mathcal{H}_T -\pi^2(2m+l_0+l_1+2)^2$ is transformed into a hypergeometric equation. Set $$\psi_m(x) = \psi^{(l_0,l_1)}_m(x) = \tilde{c}_m G_m \Big( l_0+l_1+2, \frac{2l_0+3}{2}; \sin ^2 \pi x\Big) \quad (m \in \mathbb{Z}_{\geq 0}), \label{Jacobipol}$$ where the function $G_m(\alpha, \beta; w)=_2 \! F_1(-m,\alpha+m;\beta;w)$ is the Jacobi polynomial of degree $m$ and $$\tilde{c}_m=\sqrt{\frac{\pi (2m+l_0+l_1+2)\Gamma (m+l_0+l_1+2) \Gamma (l_0+m+\frac{3}{2})}{m! \Gamma (m+l_1 +\frac{3}{2}) \Gamma (l_0+\frac{3}{2})^2}}$$ is a constant for normalization. Then $$\mathcal{H}_T \psi_m(x)=\pi^2(2m+l_0+l_1+2)^2\psi_m(x). \label{eqn:Htpsim}$$ We define the inner products $$\langle f,g\rangle =\int_{0}^{1} dx\overline{f(x)} g(x), \quad \langle f,g\rangle _{\Phi} =\int_{0}^{1} dx\overline{f(x)} g(x) |\Phi(x)|^2. \label{innerprod}$$ Then $\langle \psi_m(x), \psi_{m'}(x) \rangle _{\Phi} =\delta_{m,m'}$. Set \label{Hilb1} \begin{aligned} \mathbf{H} = \Big\{& f: \mathbb{R} \to \mathbb{C} : \mbox{measurable with } \int_{0}^{1} |f(x)| ^2 |\Phi (x)|^2 dx<+\infty, \\ & f(x)=f(x+1), \; f(x)=f(-x) \mbox{ a.e. }x \Big\} \end{aligned} and define an inner product on the Hilbert space $\mathbf{H}$ by $\langle \cdot , \cdot \rangle _{\Phi}$. Then the space spanned by functions $\{ \psi_m(x) | m\in \mathbb{Z}_{\geq 0} \}$ is dense in $\mathbf{H}$. For $f(x),g(x) \in \mathbf{H} \cap C^{\infty}(\mathbb{R} )$, we have \label{HTsymm} \begin{aligned} \langle \mathcal{H}_T f(x),g(x)\rangle _{\Phi} &= \langle \left(\mathcal{H}_T f (x)\right)\Phi(x) ,g(x)\Phi(x)\rangle = \langle H_T \left( f(x) \Phi(x)\right) ,g(x)\Phi(x)\rangle \\ &= \langle f(x) \Phi(x) ,H_T \left( g(x)\Phi(x)\right) \rangle =\langle f(x), \mathcal{H}_T g(x)\rangle _{\Phi}. \end{aligned} It follows that the operator $\mathcal{H}_T$ is essentially selfadjoint on the space $\mathbf{H}$. \subsubsection{The case $l_0>0$ and $l_1=0$} \label{sssec2} Set \begin{equation*} \Phi(x)=(\sin \pi x)^{l_0+1}, \quad \mathcal{H}_T=\Phi(x)^{-1} H_T \Phi(x), \end{equation*} then the gauge transformed Hamiltonian is expressed as \begin{equation*} \mathcal{H}_T= -\frac{d^2}{dx^2}-2\pi \Big(\frac{(l_0 +1)\cos \pi x}{\sin \pi x}\Big) \frac{d}{dx}+(l_0+1)^2\pi^2. \end{equation*} By a change of variable $w=\sin ^2 \pi x$, we have a hypergeometric differential equation, \label{GGhyper} \begin{aligned} \: &\mathcal{H}_T -\pi^2(2m+l_0+1)^2\\ &= -4\pi ^2 \Big\{ w(1-w)\frac{d^2}{dw^2}+\Big(\frac{2l_0+3}{2}-((l_0+1)+1)w\Big) \frac{d}{dw}+m(m+l_0+1) \Big\}. \end{aligned} for each $m$. Set $$\psi^G _m(x)= \tilde{c}^G_m C^{l_0+1}_m (\cos \pi x) \quad (m \in \mathbb{Z}_{\geq 0}), \label{Gegenpol}$$ where the function $C^{\nu}_m (z)=\frac{\Gamma (m+2\nu)}{m! \Gamma(2\nu)}\, _2 \! F_1(-m,m+2\nu ; \nu+\frac{1}{2} ;\frac{1-z}{2})$ is the Gegenbauer polynomial of degree $m$ and $\tilde{c}^G _m=\sqrt{\frac{2^{2l_0+1}(m+l_0+1)m! \Gamma (l_0+1)^2}{\Gamma (m+2l_0+2)}}$. Then $\mathcal{H}_T \psi^G_m(x)=\pi^2(m+l_0+1)^2\psi^G_m(x),$ and $\langle \psi^G_m(x), \psi^G_{m'}(x) \rangle _{\Phi} =\delta_{m,m'}$, where the inner product is defined as (\ref{innerprod}) for $\Phi (x)=(\sin \pi x )^{l_0+1}$. There are relations between Gegenbauer polynomials and Jacobi polynomials. More precisely, $\psi^G _{2m}(x) = \psi^{(l_0,-1)} _{m}(x)$ and $\psi^G _{2m+1}(x) = (\cos \pi x) \psi^{(l_0,0)} _{m}(x)$ $(m \in \mathbb{Z}_{\geq 0})$. Set \label{Hilbge} \begin{gathered} \begin{aligned} \mathbf{H} = \Big\{ &f: \mathbb{R} \to \mathbb{C} : \mbox{measurable with} \int_{0}^{1} |f(x)| ^2 |\Phi (x)|^2 dx<+\infty, \\ &\ f(x)=f(x+2), \; f(x)=f(-x) \mbox{ a.e. }x \Big\}, \end{aligned}\\ \mathbf{H}_+ = \{ f \in \mathbf{H} | f(x)=f(x+1) \mbox{ a.e. }x \} ,\\ \mathbf{H}_- = \{ f \in \mathbf{H} | f(x)=-f(x+1) \mbox{ a.e. }x \}, \end{gathered} and inner products on the Hilbert space $\mathbf{H}$ and its subspaces $\mathbf{H}_+$, $\mathbf{H}_-$ are given by $\langle \cdot , \cdot \rangle _{\Phi}$. Then $\mathbf{H}_+ \perp \mathbf{H}_-$ and $\mathbf{H}= \mathbf{H}_+ \oplus \mathbf{H}_-$. The space spanned by $\{ \psi^G _m(x) | m\in \mathbb{Z}_{\geq 0} \}$ is dense in $\mathbf{H}$. For $f(x),g(x) \in \mathbf{H} \cap C^{\infty}(\mathbb{R} )$, we have $\langle \mathcal{H}_T f(x),g(x)\rangle _{\Phi}=\langle f(x), \mathcal{H}_T g(x)\rangle _{\Phi}$ similarly to (\ref{HTsymm}), and it follows that the operator $\mathcal{H}_T$ is essentially selfadjoint on the space $\mathbf{H}$. Similar results hold for subspaces $\mathbf{H}_+$ and $\mathbf{H}_-$. In fact the space spanned by functions $\{ \psi_m(x) | m\in 2\mathbb{Z}_{\geq 0} \}$ (resp. $\{ \psi_m(x) | m\in 2\mathbb{Z}_{\geq 0} +1 \}$) is dense in $\mathbf{H}_+$ (resp. $\mathbf{H}_-$) and the operator $\mathcal{H}_T$ is essentially selfadjoint on the space $\mathbf{H}_+$ (resp. $\mathbf{H}_-$). \subsubsection{The case $l_0=0$ and $l_1>0$} \label{sssec21} Although the case $l_0=0$ and $l_1>0$ comes down to the case $l_0>0$ and $l_1=0$ by setting $x \to x+\frac{1}{2}$, we collect results for the $l_0=0$ and $l_1>0$ case for convenience. Set \begin{equation*} \Phi(x)=(\cos \pi x)^{l_1+1}, \quad \mathcal{H}_T=\Phi(x)^{-1} H_T \Phi(x), \end{equation*} then the gauge transformed Hamiltonian is expressed as \begin{equation*} \mathcal{H}_T= -\frac{d^2}{dx^2}-2\pi \Big(\frac{(l_1 +1)\sin \pi x}{\cos \pi x} \Big) \frac{d}{dx}+(l_1+1)^2\pi^2. \end{equation*} Now we set $$\psi^{G'} _m(x)= \tilde{c}^{G'}_m C^{l_1+1}_m (\sin \pi x) \quad (m \in \mathbb{Z}_{\geq 0}), \label{Gegenpol2}$$ where $C^{\nu}_m (z)$ is the Gegenbauer polynomial appeared in (\ref{Gegenpol}) and \\ $\tilde{c}^{G'} _m=\sqrt{\frac{2^{2l_1+1}(m+l_1+1)m! \Gamma (l_1+1)^2} {\Gamma (m+2l_1+2)}}$, then \begin{align*} \mathcal{H}_T \psi^{G'}_m(x)=\pi^2(m+l_1+1)^2\psi^{G'}_m(x), \end{align*} and $\langle \psi^{G'}_m(x), \psi^{G'}_{m'}(x) \rangle _{\Phi} =\delta_{m,m'}$, where the inner product is defined as (\ref{innerprod}). Set \begin{align*} \mathbf{H} &= \Big\{ f: \mathbb{R} \to \mathbb{C} : \mbox{measurable wiht } \int_{0}^{1} |f(x)| ^2 |\Phi (x)|^2 dx<+\infty, \\ &\quad f(x)=f(x+2), \; f(x)=f(-x+1) \mbox{ a.e. }x \Big\}, \\ \mathbf{H}_+ &= \{ f \in \mathbf{H} | f(x)=f(x+1) \mbox{ a.e. }x \} ,\\ \mathbf{H}_- &= \{ f \in \mathbf{H} | f(x)=-f(x+1) \mbox{ a.e. }x \}. \end{align*} Here the inner product on the Hilbert space $\mathbf{H}$ is given by $\langle \cdot , \cdot \rangle _{\Phi}$. Then $\mathbf{H}_+ \perp \mathbf{H}_-$ and $\mathbf{H}= \mathbf{H}_+ \oplus \mathbf{H}_-$. The space spanned by functions $\{ \psi_m^{G'}(x) | m\in \mathbb{Z}_{\geq 0} \}$ is dense in $\mathbf{H}$, and the operator $\mathcal{H}_T$ is essentially selfadjoint on the space $\mathbf{H}$. \subsubsection{The case $l_0=0$ and $l_1=0$} \label{sssec3} In this case, the trigonometric Hamiltonian is $H_T =-\frac{d^2}{dx^2}$. Set $\Phi(x)=1$, $\mathcal{H}_T= H_T =-\frac{d^2}{dx^2}$, $\psi_m(x) = \sqrt{2} \cos m\pi x$, $\varphi _m(x)=\sqrt{2} \sin m\pi x$ $(m\in \mathbb{Z}_{\geq 1})$, and $\psi_0(x) =1$. Then \begin{gather*} \mathcal{H}_T \psi_m(x)=\pi^2m^2\psi_m(x), \quad (m\in \mathbb{Z} _{\geq 0}), \\ \mathcal{H}_T \varphi _m(x)=\pi^2m^2\varphi_m(x), \quad (m\in \mathbb{Z} _{>0}), \\ \langle \psi_m(x), \psi_{m'}(x) \rangle _{\Phi} =\langle \varphi_m(x), \varphi_{m'}(x) \rangle _{\Phi} = \delta_{m,m'} , \\ \langle \psi_m(x), \varphi_{m'}(x) \rangle _{\Phi} = 0 , \end{gather*} where the inner product is defined by (\ref{innerprod}). Set \begin{align*} \mathbf{H} &=\Big\{ f: \mathbb{R} \to \mathbb{C} : \mbox{measurable with} \int_{0}^{1} |f(x)| ^2 dx<+\infty, \\ &\qquad f(x)=f(x+2) \mbox{ a.e. }x \Big\},\\ \mathbf{H}_1 &= \{ f(x) \in \mathbf{H} : f(x)=f(x+1), \; f(x)=f(-x) \mbox{ a.e. } x\}, \\ \mathbf{H}_2 &= \{ f(x) \in \mathbf{H} : f(x)=f(x+1), \; f(x)=-f(-x)\mbox{ a.e. } x\}, \\ \mathbf{H}_3 &= \{ f(x) \in \mathbf{H} : f(x)=-f(x+1), \; f(x)=f(-x)\mbox{ a.e. } x\}, \\ \mathbf{H}_4 &= \{ f(x) \in \mathbf{H} : f(x)=-f(x+1), \; f(x)=-f(-x)\mbox{ a.e. } x\}. \end{align*} Then the spaces $\mathbf{H}_i$ are pairwise orthogonal and $\mathbf{H} = \oplus _{i=1}^4 \mathbf{H}_i$. The space spanned by the functions $\{ \psi_m(x) | m\in \mathbb{Z}_{\geq 0} \}$ and $\{ \varphi_m(x) | m\in \mathbb{Z}_{\geq 1} \}$ is dense in $\mathbf{H}$ and the operator $\mathcal{H}_T$ is essentially selfadjoint on the space $\mathbf{H}$, and also on subspaces $\mathbf{H}_i$, $(i=1,2,3,4)$. \subsection{Perturbation on parameters $a$ and $p(=\exp (\pi \sqrt{-1} \tau ))$} \label{sect:formalpert}  As was explained in section \ref{sect:trig}, eigenvalues and eigenfunctions of the Hamiltonian $H$ are obtained explicitly for the case $p=0$. In this section, we apply a method of perturbation and have an algorithm for obtaining eigenvalues and eigenfunctions as formal power series in $p$. Since the functions $\wp(x+\omega_i)$ $(i=0,1,2,3)$ admit expansions (\ref{wpth}) and (\ref{wpth1}), the Hamiltonian $H$ (see (\ref{Ino})) admits the expansion $$H=H_T +C_T+\sum_{k=1}^{\infty} V_k(x) p^k, \label{Hamilp0}$$ where $H_T$ is the Hamiltonian of the trigonometric model defined in (\ref{triIno}), $V_k(x)$ $(k \in \mathbb{Z}_{\geq 1})$ are even periodic functions with period $1$, and $C_T= -\frac{\pi^2}{3}\sum_{i=0}^3 l_i(l_i+1)$ is a constant. Note that for each $k$ the function $V_k(x)$ is expressed as a finite sum of $\cos 2\pi n x$ $(n=0,\dots ,k)$. First we consider the case ($l_0 >0$, $l_1 > 0$), ($l_0 >0$, $l_1 =0$) or ($l_0 = 0$, $l_1 >0$). Set \begin{equation*} v_m = \begin{cases} \psi _m(x)\Phi(x) & ( l_0>0 , \; l_1>0) \\ \psi ^G _m(x)\Phi(x) & ( l_0>0 , \; l_1=0) \\ \psi ^{G'} _m(x)\Phi(x) & ( l_0=0 , \; l_1>0) \end{cases} \end{equation*} for $m\in \mathbb{Z}_{\geq 0}$. Then $v_m$ $(m\in \mathbb{Z}_{\geq 0})$ is a normalized eigenvector of $H_T$ on $\mathbf{H}$. Let $E_m$ be the eigenvalue of $H_T$ w.r.t. the eigenvector $v_m$, i.e., \begin{equation*} E_m=\begin{cases} (2m+l_0+l_1+2)^2 & ( l_0>0 , \; l_1>0) \\ (m+l_0+1)^2 & ( l_0>0 , \; l_1=0) \\ (m+l_1+1)^2 & ( l_0=0 , \; l_1>0). \end{cases} \end{equation*} Then we have $E_m\neq E_{m'}$, if $m \neq m'$. We will determine eigenvalues $E_m(p) = E_m+C_T+\sum_{k=1}^{\infty} E_{m}^{\{k\}}p^k$ and normalized eigenfunctions $v_m(p)= v_m+ \sum_{k=1}^{\infty} \sum_{m'} c_{m,m'}^{\{k\}}v_{m'}p^k$ of the operator $H_T +C_T+\sum_{k=1}^{\infty} V_k(x) p^k$ as formal power series in $p$. In other words, we will find $E_m(p)$ and $v_m(p)$ that satisfy equations \begin{gather} (H_T +C_T +\sum_{k=1}^{\infty} V_k(x) p^k )v_m(p) = E_m(p)v_m(p) ,\label{Hpertexp}\\ \langle v_m(p) , v_m(p) \rangle =1 , \label{Hpertexp1} \end{gather} as formal power series in $p$. First we calculate coefficients of $\sum_{m'} d_{m,m'}^{\{k\}}v_{m'}=V_k(x) v_{m}$. Since $V_k(x)$ is a finite sum of $\cos 2n\pi x$ $(n=0,\dots ,k)$ and the eigenvector $v_m$ is essentially a hypergeometric polynomial, coefficients $d_{m,m'}^{\{k\}}$ are obtained by applying the Pieri formula repeatedly. For each $m$ and $k$, $d_{m,m'}^{\{k\}}\neq 0$ for finitely many $m'$. Now we compute $E_{m}^{\{k\}}$ and $c_{m,m'}^{\{k\}}$ for $k \geq 1$. Set $c_{m,m'}^{\{0\}} =\delta _{m,m'}$. By comparing coefficients of $v_{m'}p^k$, it follows that conditions (\ref{Hpertexp}, \ref{Hpertexp1}) are equivalent to following relations \begin{align} & c_{m,m'}^{\{ k \}}= \frac{\sum_{k'=1}^{k} ( \sum _{m''}c_{m,m''}^{\{ k-k' \} }d_{m'',m'}^{\{ k' \}})-\sum_{k'=1}^{k-1} c_{m,m'}^{\{ k-k' \} }E_{m}^{\{ k' \}} } {E_m-E_{m'}} ,\quad (m' \neq m) \label{nondeg} \\ & c_{m,m}^{\{ k \}}=-\frac{1}{2} \Big(\sum_{k'=1}^{k-1} \sum_{m''} c_{m,m''}^{\{ k' \}}c_{m,m''}^{\{ k-k' \} } \Big) ,\\ & E_m^{\{ k \}} = \sum_{k'=1}^{k}\sum_{m''}c_{m,m''}^{\{ k-k' \} }d_{m'',m}^{\{ k' \}}- \sum_{k'=1}^{k-1} c_{m,m}^{\{ k-k' \} } E_m^{\{ k' \}}.\label{nondeg2} \end{align} Note that the denominator of (\ref{nondeg}) is non--zero because of non-degeneracy of eigenvalues. Then numbers $c_{m,m'}^{\{ k \}}$ and $E_m^{\{ k \}}$ are determined recursively from (\ref{nondeg} - \ref{nondeg2}) uniquely. It is shown recursively that for each $m$ and $k$, $\# \{ m'' | \: c_{m,m''}^{\{ k \}} \neq 0 \}$ is finite and the sums on (\ref{nondeg} - \ref{nondeg2}) in parameters $m''$ are indeed finite sums. Therefore we obtain eigenvalues'' $E_m(p)$ and eigenfunctions'' $v_m(p)$ of the operator $H$ as formal power series in $p$. At this stage, convergence is not discussed. Now consider the case $l_0=l_1=0$. Though there is degeneracy of eigenvalues on the full Hilbert space $\mathbf{H}$, the degeneracy disappears when the action of the Hamiltonian is restricted on $\mathbf{H}_i$ for each $i \in \{0,1,2,3 \}$ and the calculation of perturbation works compatibly on each space $\mathbf{H}_i$. Hence the calculation is valid for the case $l_0=l_1=0$. Let us discuss perturbation for equation (\ref{Heun}) on the parameter $a\left( =\frac{e_2-e_3}{e_1-e_3} \right)$. We consider the case $l_0>0$ and $l_1>0$. Write $$L=(aw-1)L_T+ aw(w-1)\big(l_2+\frac{3}{2}\big)\frac{d}{dw}+aq_1w+\mathcal{E}, \label{eqnL}$$ where \begin{gather} L_T= \left( w(w-1)\left(\frac{d^2}{dw^2}+\left( \frac{l_0+\frac{3}{2}}{w}+\frac{l_1+\frac{3}{2}}{w-1}\right) \frac{d}{dw}\right) +\frac{(l_0+l_1+2)^2}{4}\right), \label{eqnLT} \\ q_1= \frac{(l_0+l_1+l_2-l_3+3)(l_0+l_1+l_2+l_3+4)-(l_0+l_1+2)^2}{4}, \nonumber \\ \mathcal{E}=\frac{E}{4(e_1-e_3)}-\frac{(l_0+l_2+2)^2}{4}a-\sum_{i=0}^3\frac{l_i(l_i+1)e_3}{4(e_1-e_3)} .\nonumber \end{gather} Then the equation $L \tilde{f}(w)=0$ is equivalent to (\ref{Heun}). We are going to find eigenvalues and eigenfunctions of $L$ as perturbation on $a$. Set $\tilde{\psi}_m(w)={u}_m= _2\! F_1 (-m,m+l_0+l_1+2;\frac{2l_0 +3}{2};w)$, where $\sin^2\pi x= w$. We will use the following relations later which can be found in \cite{Ron}. \begin{gather*} L_T \tilde{\psi}_m(w) = \frac{(2m+l_0+l_1+2)^2}{4} \tilde{\psi}_m(w), \\ w \tilde{\psi}_m(w)= A_m \tilde{\psi}_{m+1}(w)+ B_m \tilde{\psi}_{m}(w)+C_m \tilde{\psi}_{m-1}(w), \\ w(w-1) \frac{d}{dw} \tilde{\psi}_m(w)= A'_m \tilde{\psi}_{m+1}(w)+ B'_m \tilde{\psi}_{m}(w)+C'_m \tilde{\psi}_{m-1}(w). \end{gather*} where \begin{gather*} A_m= -\frac{(m+l_0+l_1+2)(m+\frac{2l_0+3}{2})}{(2m+l_0+l_1+2)(2m+l_0+l_1+3)},\\ A'_m= -\frac{m(m+l_0+l_1+2)(m+\frac{2l_0+3}{2})}{(2m+l_0+l_1+2)(2m+l_0+l_1+3)},\\ B_m= \frac{2m(m+l_0+l_1+2)+\frac{2l_0+3}{2}(l_0+l_1+1)}{(2m+l_0+l_1+1) (2m+l_0+l_1+3)}, \\ B'_m= \frac{m(m+l_0+l_1+2)(l_0-l_1)}{(2m+l_0+l_1+1)(2m+l_0+l_1+3)}, \\ C_m= -\frac{m(m+\frac{2l_1+1}{2})}{(2m+l_0+l_1+2)(2m+l_0+l_1+1)}, \\ C'_m= \frac{m(m+l_0+l_1+2)(m+\frac{2l_1+1}{2})}{(2m+l_0+l_1+2)(2m+l_0+l_1+1)}. \end{gather*} Set $\mathcal{E}_m= \frac{(2m+l_0+l_1+2)^2}{4}$, $\mathcal{E}_m(a)=\mathcal{E}_m+\sum_{k=1}^{\infty} \mathcal{E}_{m}^{\{k\}} a^k$,\\ ${u}_m(a)= {u}_m+ \sum_{k=1}^{\infty} \sum_{m'} \tilde{c}_{m,m'}^{\{k\}}{u}_{m'}a^k$, and $\tilde{c}_{m,m}^{\{k\}}=0$ for $k\geq 1$ and all $m$. We will determine $\mathcal{E}_{m}^{\{k\}}$ and $\tilde{c}_{m,m'}^{\{k\}}$ $(k \geq 1)$ to satisfy $L \tilde{f}(w)=0$. Substituting in (\ref{eqnL}) with $\mathcal{E}= \mathcal{E}_m (a)$, the following relations are shown: \begin{gather} \tilde{c}_{m,m'}^{\{k\}} = \frac{1}{\mathcal{E}_m -\mathcal{E}_{m'}} \left( \tilde{c}_{m,m'+1}^{\{k-1\}} \tilde{C}_{m'+1}+ \tilde{c}_{m,m'}^{\{k-1\}} \tilde{B}_{m'}+ \tilde{c}_{m,m'-1}^{\{k-1\}} \tilde{A}_{m'-1} \right) \;\; (m' \neq m), \label{Tcjjp} \\ \mathcal{E}_{m}^{\{k\}}= \tilde{c}_{m,m+1}^{\{k-1\}} \tilde{C}_{m+1} + \delta_{k-1,0} \tilde{B}_{m}+ \tilde{c}_{m,m-1}^{\{k-1\}} \tilde{A}_{m-1} , \label{TEjk} \end{gather} where \begin{align*} \tilde{A}_{m}&= -(l_2+\frac{3}{2})A'_m-\mathcal{E}_mA_m-q_1A_m \\ & = \frac{(m+l_0+l_1+2)(m+\frac{2l_0+3}{2})(m+\frac{l_0+l_1+l_2-l_3+3}{2}) (m+\frac{l_0+l_1+l_2+l_3+4}{2})}{(2m+l_0+l_1+2)(2m+l_0+l_1+3)} , \end{align*} \begin{align*} \tilde{B}_{m}&= -(l_2+\frac{3}{2})B'_m-\mathcal{E}_mB_m-q_1B_m \\ &=-\Big(2m(m +l_0+l_1+2)+\frac{(2l_0+3)(l_0+l_1+1)}{2}) \big(m(m+l_0+l_1+2)\\ &\quad +\frac{(l_0+l_1+l_2-l_3+3)(l_0+l_1+l_2+l_3+4)}{4}\big)\Big)\\ &\quad\div \Big((2m+l_0+l_1+3)(2m+l_0+l_1+1)\Big)\\ &\quad -\frac{(l_2+\frac{3}{2})(l_0-l_1)(m+l_0+l_1+2)m}{(2m+l_0+l_1+3) (2m+l_0+l_1+1)} , \end{align*} \begin{align*} \tilde{C}_{m}&= -(l_2+\frac{3}{2})C'_m-\mathcal{E}_mC_m-q_1C_m \\ &= \frac{m(m+\frac{2l_1+1}{2})(m+\frac{l_0+l_1-l_2+l_3+1}{2}) (m+\frac{l_0+l_1-l_2-l_3}{2})}{(2m+l_0+l_1+2)(2m+l_0+l_1+1)}. \end{align*} Solving the recursive equations (\ref{Tcjjp},\ref{TEjk}), $\mathcal{E}_m(a)$ and ${u}_m(a)$ are obtained as formal power series in $a$. By expanding $e_1$, $e_2$, and $e_3$ as series in $a$, eigenvalues of the operator $L$ are obtained as formal series in $a$. Now we compare two expansions of eigenvalues ($E_m(p) \Leftrightarrow \mathcal{E}_m (a)$) and eigenfunctions ($v_m(p) \Leftrightarrow {u}_m(a)$). From the formula $a=16p\prod_{n=1}^{\infty}\big( \frac{1+p^{2n}}{1+p^{2n-1}} \big)^8$ (see \cite[\S 21.7]{WW}), it follows that $a$ is holomorphic in $p$ near $p=0$ and admits an expansion $a=16p +O(p^2)$. Hence the formal power series ${u}_m(a)$ and $\mathcal{E}_m (a)$ are expressed as the formal power series in $p$, and coefficients of $p^k$ on ${u}_m(a)$ (resp. $\mathcal{E}_m (a)$) are expressed as linear combinations of coefficients of $a^l$ $(l\leq k)$ on ${u}_m(a)$ (resp. $\mathcal{E}_m (a)$). Set $\widehat{v}_m(p)= (1-aw)^{-(l_2+1)/2}{u}_m(a)$ and $\widehat{E}_m (p)=\mathcal{E}_m (a)$, then they also satisfy equation (\ref{Hpertexp}) as formal power series in $p$. Since the coefficients are determined by the recursive relations uniquely, it follows that $\widehat{E}_m (p)= E_m(p)$ and $\widehat{v}_m(p) = C_m(p) v_m(p)$, where $C_m(p)$ is a formal power series in $p$. Note that $C_m(p)$ appears from a difference of the normalization. In summary, the perturbation on the variable $p$ is equivalent to the one on the variable $a$. \section{Perturbation on the $L^2$ space} \label{sec:pertu} Throughout this section, assume $l_0\geq 0$ and $l_1\geq 0$. \subsection{Holomorphic perturbation} In this subsection, we will use definitions and propositions written in Kato's book \cite{Kat} freely. The main theorem in this subsection is Theorem \ref{mainthmKato}. As an application, we show convergence of the formal power series of eigenvalues $E_m(p)$ (resp. $\mathcal{E}_{m}(a)$) in $p$ (resp. $a$) which are calculated by the algorithm of perturbation explained in section \ref{sect:formalpert}. We denote the gauge-transformed Hamiltonian of the $BC_1$ Inozemtsev model by $H(p)$, i.e., $$H(p)= \Phi(x)^{-1} \circ \Big( - \frac{d^2}{dx^2} + \sum_{i=0}^3 l_i(l_i+1)\wp (x+\omega_i) \Big) \circ \Phi(x), \label{eqn:gtrans}$$ where $$\Phi(x)= \begin{cases} (\sin \pi x)^{l_0+1}(\cos \pi x)^{l_1+1} & (l_0>0, l_1>0); \\ (\sin \pi x)^{l_0+1} & (l_0>0, l_1=0 );\\ (\cos \pi x)^{l_1+1} & (l_0=0, l_1>0 );\\ 1 & (l_0=0, l_1=0) , \end{cases}$$ and $p=\exp(\pi \sqrt{-1} \tau)$. Let $V_k(x)$ be the functions in (\ref{Hamilp0}), $C_T$ be the constant in (\ref{Hamilp0}), and $\mathcal{H}_T(=\Phi(x)^{-1} H_T \Phi(x))$ be the gauge-transformed Hamiltonian of the $BC_1$ Calogero-Moser-Sutherland model. Then the operator $H(p)$ is expanded as \begin{equation*} H(p)= \mathcal{H}_T+C_T+ \sum_{k=1}^{\infty} V_k(x)p^k. \end{equation*} Note that $H(0)=\mathcal{H}_T+C_T$. The functions $V_k(x)$ satisfy the following lemma. \begin{lemma} \label{lem:Vkx} Let $s$ be a real number satisfying $s>1$. Then there exists a constant $A$ such that $V_k(x)\leq As^k$ for all $k \in \mathbb{Z}_{\geq 1}$ and $x \in \mathbb{R}$. \end{lemma} \begin{proof} The functions $V_k(x)$ are determined by relations (\ref{wpth}). Write \label{majser} \begin{aligned} C(p)&=\sum_{k=1}^{\infty}V_kp^k\\ &=8\pi^2 \Big( \sum_{n=1}^{\infty} (l_0(l_0+1) +l_1(l_1+1)) \frac{2np ^{2n}}{1-p ^{2n}}\\ &\quad + \left( l_2(l_2+1)+l_3(l_3+1) \right) \big( \frac{n(p^{n}+p^{2n})}{1-p ^{2n}} \big) \Big) . \end{aligned} Then the function $\sum _{k=1}^{\infty } V_k(x)$ is uniformly evaluated by coefficients of majorant series $C(p)=\sum_{k=1}^{\infty}V_kp^k$ for $x\in \mathbb{R}$. Since the convergence radius of (\ref{majser}) in $p$ is $1$, we obtain the lemma. \end{proof} \begin{prop} The operator $H(p)$ $(-11$. From Lemma \ref{lem:Vkx}, we have $\| V_k(x) f \|\leq As^k \| f\|$, because the measure of an interval $(0,1)$ is $1$. On the other hand, the operators $V_k(x)$ are defined on the whole space $\mathbf{H}$ and symmetric. Hence the proposition follows from Kato-Rellich theorem \cite[VII-\S 2.2, Theorem 2.6]{Kat}. \end{proof} For the case $p=0$, the operator $\tilde{H}(0)$ coincides with the gauge-transformed Hamiltonian of the trigonometric $BC_1$ Calogero-Moser-Sutherland model up to constant. All eigenfunctions of the operator $\tilde{H}(0)$ were obtained explicitly in section \ref{sect:trig}. The spectrum $\sigma (\tilde{H}(0))$ contains only isolated spectra and the multiplicity of each spectrum is $1$ or $2$. Then the resolvent $R(\zeta, \tilde{H}(0))=(\zeta -\tilde{H}(0))^{-1}$ is compact for $\zeta \not \in \sigma (\tilde{H}(0))$. From Theorem 2.4 in \cite[VII-\S 2.1]{Kat} and Proposition \ref{prop:holA} in this paper, we obtain the following statement. \begin{prop} \label{prop:compresol} The operator $\tilde{H}(p)$ has compact resolvent for all $p$ such that $-10}$ such that $\tilde{E}_m (p)$ and $\tilde{v}_m (p,x)$ are expanded as the series in $p$ and they converge on $|p|<\epsilon _m$. The convergence of $\tilde{v}_m (p,x)$ is as an element in $\mathbf{H}$. Eigenvalues $\tilde{E}_m (p)$ and eigenfunctions $\tilde{v}_m (p,x)$ satisfy the following relations for $|p|<\epsilon _m$, \begin{gather*} \Big( H_T +C_T +\sum_{k=1}^{\infty} V_k(x) p^{k} \Big) \Phi(x) \tilde{v}_m (p,x)= \tilde{E}_m (p)\Phi(x)\tilde{v}_m (p,x) ,\\ \langle \Phi(x) \tilde{v}_m (p,x), \Phi(x)\tilde{v}_m (p,x)\rangle =1 . \end{gather*} These equations are the same as (\ref{Hpertexp}, \ref{Hpertexp1}). From uniqueness of the coefficients obtained by perturbation, it is seen that $\tilde{E}_m (p)= E_m(p)$ and $\Phi(x)\tilde{v}_m (p,x)= v_m(p)$. Hence the convergence of $E_m(p)$ and $v_m(p)$ are shown. \end{proof} We also obtain the holomorphy of perturbation on the variable $a$ from Theorem \ref{mainthmKato}, because $p$ is holomorphic in $a$ near $a=0$ (see section \ref{sect:formalpert}). \begin{cor} \label{cor:conv} Let $\mathcal{E}_m(a)$ $(m\in \mathbb{Z}_{\geq 0})$ (see section \ref{sect:formalpert}) be the eigenvalue and ${u}_m(a)$ be the eigenvector of the Heun operator $L$ (see (\ref{eqnL})) obtained by the algorithm for perturbation from $L_T$ (see (\ref{eqnLT})). If $|a|$ is sufficiently small then the power series $\mathcal{E}_m(a)$ converges, and the power series ${u}_m(a)$ converges as an element in $L^2$ space. \end{cor} \subsection{Properties of the eigenvalue} In this subsection, we will show some properties of eigenvalues. First we discuss the multiplicity of eigenvalues. Under some assumptions, it is seen that eigenvalues never stick together. We also introduce some inequalities for eigenvalues. \begin{thm} \label{thm:multone} Assume $l_0\geq 1/2$ or $l_1\geq 1/2$. Let $\tilde{E}_m(p)$ $(m\in \mathbb{Z}_{\geq 0})$ be the eigenvalues of $\tilde{H}(p)$ defined in Theorem \ref{mainthmKato}. Then $\tilde{E}_m(p)\neq \tilde{E}_{m'}(p)$ $(m\neq m')$ for $-10}$ such that $\tilde{H}(p_0)$ has exactly one eigenvalue in the interval $(\tilde{E}_m(p_0)-\epsilon _1,\tilde{E}_m(p_0)+\epsilon _1)$. Since the operators $\tilde{H}(p)$ $(-10}$ such that $\tilde{H}(p)$ has exactly one eigenvalue in the interval $(\tilde{E}_m(p_0)-\epsilon _1/2,\tilde{E}_m(p_0)+\epsilon _1/2)$ for $p$ such that $p_0-\epsilon_2 0}$ such that $\tilde{E}_m(p)\neq \tilde{E}_{m'}(p)$ for $0<|p-p_0|<\epsilon$. Hence if $|p-p_0|(\neq 0)$ is sufficiently small then values $\tilde{E}_m(p),\tilde{E}_{m'}(p)$ belong to the interval $(\tilde{E}_m(p_0)-\epsilon _1/2,\tilde{E}_m(p_0)+\epsilon _1/2)$ and $\tilde{E}_m(p) \neq \tilde{E}_{m'}(p)$. This shows that $\tilde{H}(p)$ has no less than two eigenvalues in the interval $(\tilde{E}_m(p_0)-\epsilon _1,\tilde{E}_m(p_0)+\epsilon _1)$, and it contradicts. Therefore, we obtain the theorem. \end{proof} \begin{cor} \label{cor:ineq} Assume $l_0\geq 1/2$ or $l_1\geq 1/2$, then $\tilde{E}_m(p)< \tilde{E}_{m'}(p)$ for $m0}$. Hence $\frac{d}{dp}\left( \sum_{k=1}^{\infty} V_{2k}(x)p^{2k}\right)\geq 0$ for $0\leq p<1$ and $\frac{d}{dp}\left( \sum_{k=1}^{\infty} V_{2k}(x)p^{2k}\right)\leq 0$ for $-10$ such that $\|W(p)\|0}$ such that $\Gamma _m$ contains only one element $E_m$ of the set $\sigma (\tilde{H}(0))$ inside $\Gamma _m$ and the projection $P_m(p)=P_{\Gamma_m}(p)$ satisfies $\| P_m(p)-P_m(0) \|<1$ for $|p|0 ,\: l_1>0$), ($l_0>0 ,\: l_1=0$) or ($l_0=0 ,\: l_1>0$), multiplicity of every eigenvalue of the operator $\tilde{H}(0)$ on the Hilbert space $\mathbf{H}$ is one. Then a function $P_m(p)v_m$ for each $m$ is an eigenvector of the operator $\tilde{H}(p)$ and admits an expression $$P_m(p)v_m= c_{(p,m)} \tilde{v}_m(p,x), \label{pmvmcmvm}$$ ($c_{(p,m)}$ is a constant) for sufficiently small $|p|$, because the operator $\tilde{H}(p)$ preserves the space $P_m(p) \mathbf{H}$ and $\tilde{v}_m(p,x)$ is an eigenvector of $\tilde{H}(p)$ with the eigenvalue $\tilde{E}_m(p)$ (see Theorem \ref{mainthmKato}). For the case $l_0=l_1=0$, the Hilbert space $\mathbf{H}$ is decomposed into $\mathbf{H}_1\oplus \mathbf{H}_2\oplus \mathbf{H}_3\oplus \mathbf{H}_4$ (see section \ref{sssec3}). For each space, there is no degeneracy of eigenvalues for the operator $\tilde{H}(0)$. Hence the expression (\ref{pmvmcmvm}) is also valid. As for the expansion of $P_m(p)v_m$ in $v_{m'}$ $({m'} \in \mathbb{Z}_{\geq 0})$, the following proposition which is analogous to \cite[Proposition 5.11]{KT} is shown. \begin{prop} \label{prop:pmvm} Let $|p|<1$. Write $P_m(p)v_m = \sum_{m'} s_{m,{m'}} v_{m'}$. For each $m$ and $C\in \mathbb{R} _{>1}$, there exists $C'\in \mathbb{R}_{>0}$ and $p_{\star }$ such that coefficients $s_{m,{m'}}$ satisfy $|s_{m,{m'}}| \leq C'(C|p|)^{|m-{m'}|/2}$ for all ${m'} \in \mathbb{Z}_{\geq 0}$ and $p$ $(|p|0}$ and $R \in (0,1)$. If $r'$ satisfies $00}$ there exists $p_0 \in \mathbb{R}_{>0}$ such that the series $v_m(p)$ converges absolutely uniformly for $(p,x) \in [-p_0 , p_0]\times B_r$, where $B_r= \{ x\in \mathbb{C} | | \Im x | \leq r\}$. \end{thm} \begin{proof} We prove for the case $l_0>0$ and $l_1>0$. Write $P_m(p)v_m = \sum_{m'} s_{m,{m'}} v_{m'}$ and apply Proposition \ref{prop:pmvm} for the case $C=2$. Then there exists $C'\in \mathbb{R}_{>0}$ and $p_{\star }$ such that coefficients $s_{m,{m'}}$ satisfy $|s_{m,{m'}}| \leq C'(2|p|)^{|m-{m'}|/2}$ for all $p$, ${m'}$ such that $|p|0}$ such that $|s_{m,{m'}}| \leq A (\sqrt{2|p|})^{m'}$ for $|p|\leq p_{0}(0$ and $l_1=0$) or ($l_0=0$ and $l_1>0$) or $(l_0=l_1=0)$, we can prepare alternative propositions to Propositions \ref{prop:pmvm} and \ref{prop:psival}, and can prove the theorem similarly. \end{proof} \section{Algebraic eigenfunctions} \label{sec:algfcn} \subsection{Invariant subspaces of doubly periodic functions} \label{sec:invsp} If the coupling constants $l_0$, $l_1$, $l_2$, $l_3$ satisfy some equation, the Hamiltonian $H$ (see (\ref{Ino})) preserves a finite dimensional space of doubly periodic functions. In this section, we look into a condition for existence of the finite dimensional invariant space of doubly periodic functions with respect to the action of the Hamiltonian $H$ (see Proposition \ref{findim}). After that, we investigate relationship between invariant spaces of doubly periodic functions and $L^2$ spaces. \begin{prop} \label{findim} Let $\tilde{\alpha}_i$ be a number such that $\tilde{\alpha}_i= -l_i$ or $\tilde{\alpha}_i= l_i+1$ for each $i\in \{ 0,1,2,3\}$. Assume $-\sum_{i=0}^3 \tilde{\alpha}_i /2\in \mathbb{Z}_{\geq 0}$ and set $d=-\sum_{i=0}^3 \tilde{\alpha}_i /2$. Let $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ be the $d+1$ dimensional space spanned by $\Big\{ (\frac{\sigma_1(x)}{\sigma(x)})^{\tilde{\alpha }_1} (\frac{\sigma_2(x)}{\sigma(x)})^{\tilde{\alpha }_2}(\frac{\sigma_3(x)}{\sigma(x)} )^{\tilde{\alpha }_3}\wp(x)^n \: | \:n=0, \dots ,d \Big\}$, where $\sigma (x)$ is the Weierstrass sigma-function defined in (\ref{wpsigzeta}) and $\sigma_i(x)$ ($i=1,2,3$) are the co-sigma functions defined in (\ref{cosigma}). Then the Hamiltonian $H$ (see (\ref{Ino})) preserves the space $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$. \end{prop} \begin{proof} Set $z=\wp(x)$, $\widehat{\Phi}(z)=(z-e_1)^{\tilde{\alpha}_1/2} (z-e_2)^{\tilde{\alpha}_2/2}(z-e_3)^{\tilde{\alpha}_3/2}$, and $\widehat{H}= \widehat{\Phi}(z)^{-1} \circ H \circ\widehat{\Phi}(z)$. Then \begin{align*} \widehat{H}= & -4(z-e_1)(z-e_2)(z-e_3) \Big\{ \frac{d^2}{dz^2}+ \sum_{i=1}^3\frac{\tilde{\alpha}_i+\frac{1}{2}}{z-e_i} \frac{d}{dz}\Big\} \\ & +\big\{ -\left(\tilde{\alpha}_1+\tilde{\alpha}_2+\tilde{\alpha}_3-l_0\right) \left(\tilde{\alpha}_1+\tilde{\alpha}_2+\tilde{\alpha}_3+l_0+1\right) z\nonumber \\ & + e_1(\tilde{\alpha}_2+\tilde{\alpha}_3)^2+e_2(\tilde{\alpha}_1 +\tilde{\alpha}_3)^2+e_3(\tilde{\alpha}_1+\tilde{\alpha}_2)^2 \big\} \nonumber . \end{align*} Let $\widehat{V}_{d+1}$ be the space of polynomials in $z$ with degree at most $d$. From formulas (\ref{sigmai}), it is enough to show that the operator $\widehat{H}$ preserves the space $\widehat{V}_{d+1}$. The action of the Hamiltonian is written as \label{ze2} \begin{aligned} \widehat{H}(z-e_2)^r &= -4\Big( (r+\gamma_1)(r+\gamma_2) (z-e_2)^{r+1} \\ & \quad +((e_2-e_3)(r+\tilde{\alpha}_2+\tilde{\alpha}_1)r+(e_2-e_1) (r+\tilde{\alpha}_2+\tilde{\alpha}_3)r+q')(z-e_2)^r \\ & \quad+ r\big(r+\tilde{\alpha}_2-\frac{1}{2}\big)(e_2-e_3)(e_2-e_1)(z-e_2)^{r-1}\Big), \end{aligned} where $q'=-\frac{1}{4}\left(e_1(\tilde{\alpha}_2+\tilde{\alpha}_3)^2 +e_2(\tilde{\alpha}_1+\tilde{\alpha}_3)^2+e_3(\tilde{\alpha}_1 +\tilde{\alpha}_2)^2\right)+e_2\gamma_1\gamma_2$, \\ $\gamma_1=(\tilde{\alpha}_1+\tilde{\alpha}_2+\tilde{\alpha}_3-l_0)/2$ and $\gamma_2=(\tilde{\alpha}_1+\tilde{\alpha}_2+\tilde{\alpha}_3+l_0+1)/2$. Hence the operator $\widehat{H}$ preserves the space of polynomials in $z$. Since $\tilde{\alpha}_0= -l_0$ or $l_0+1$, it follows that $(r+\gamma_1)(r+\gamma_2)=0$ for $r=d$, $(d=-\sum_{i=0}^3 \tilde{\alpha}_i/2)$. Hence the coefficient of $(z-e_2)^{r+1}$ on the right hand side of (\ref{ze2}) is zero for the case $r=d$. Therefore the operator $\widehat{H}$ preserves the space $\widehat{V}_{d+1}$. \end{proof} \begin{prop} \label{prop:fintrig} With the notation in Proposition \ref{findim}, assume $d=-\sum_{i=0}^3 \tilde{\alpha}_i/2 \in \mathbb{Z}_{\geq 0}$. For the trigonometric case $(p=0)$, the eigenvalues of the Hamiltonian $H$ on the finite dimensional space $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ are written as $\{ \pi^2 (2r+ \tilde{\alpha}_0+ \tilde{\alpha}_1)^2-\frac{\pi^2}{3} \sum_{i=0}^3 l_i(l_i+1)\} _{r=0, \dots ,d}$. \end{prop} \begin{proof} As $p\to 0$, we have $e_1\to 2\pi^2/3$, $e_2\to -\pi^2/3$, and $e_3\to -\pi^2/3$. In this case the coefficient of $(z-e_2)^{r-1}$ on the right hand side of (\ref{ze2}) is zero for all $r$. Hence the operator $H$ acts triangularly. Then the eigenvalues appear on diagonal elements. By a straightforward calculation, the coefficient of $(z-e_2)^{r}$ on the right hand side of (\ref{ze2}) is written as \begin{equation*} \pi^2 (2r+ \tilde{\alpha}_2+ \tilde{\alpha}_3)^2-\frac{\pi^2}{3} \sum_{i=0}^3 l_i(l_i+1). \end{equation*} Here we used relations $\tilde{\alpha}_i^2- \tilde{\alpha}_i= l_i(l_i+1)$ $(i=0,1,2,3)$. Therefore the eigenvalues of the Hamiltonian $H$ on the finite dimensional space $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ are $\{ \pi^2 (2r+ \tilde{\alpha}_2+ \tilde{\alpha}_3)^2-\frac{\pi^2}{3} \sum_{i=0}^3 l_i(l_i+1)\} _{r=0, \dots ,d}$. By replacing $r \to d-r$, we obtain the proposition. \end{proof} \subsection{Algebraic eigenfunctions on the Hilbert space} \label{sec:algL2} In this subsection we investigate a relationship between the Hilbert space $\mathbf{H}$ and the finite dimensional space $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ which was defined in the previous subsection. Throughout this subsection, assume $l_0\geq 0$ and $l_1 \geq 0$. Let us consider the case $\tilde{\alpha}_i \in \{ -l_i , l_i+1 \}$ $(i=0,1,2,3)$ and $d=-\sum_{i=0}^3 \tilde{\alpha}_i/2 \in \mathbb{Z}_{\geq 0}$. It is easily confirmed that if $\tilde{\alpha}_0 \geq 0$ and $\tilde{\alpha}_1 \geq 0$ then any function $f(x)$ in $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ is square-integrable on the interval $(0,1)$. Set $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}= \{ \frac{f(x)}{\Phi(x)} | f(x) \in V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3 }\}$, where $\Phi(x)$ is the ground state of the trigonometric model which was defined in section \ref{sect:trig}. \begin{prop} \label{prop:finL2} Let $\tilde{\alpha}_i \in \{ -l_i, l_i+1\}$ for $i=0,1,2,3$. If $\tilde{\alpha}_0 \geq 0$, $\tilde{\alpha}_1 \geq 0$ and $d=-\sum_{i=0}^3 \tilde{\alpha}_i /2\in \mathbb{Z}_{\geq 0}$, then $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3} \subset \mathbf{H}$. For the case ($l_0>0$ and $l_1=0$) or ($l_0=0$ and $l_1>0$) we have $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3} \subset \mathbf{H}_i$ for $i=+$ or $-$, and for the case $l_0=l_1=0$ we have $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3} \subset \mathbf{H}_i$ for $i=1,2,3$ or $4$. \end{prop} \begin{proof} We consider four cases, i.e., the case $l_0>0$, $l_1>0$, the case $l_0>0$, $l_1=0$ case, the case $l_0=0,\; l_1>0$, and the case $l_0=l_1=0$. First, we prove the proposition for the case $l_0>0$, $l_1>0$. The numbers $\tilde{\alpha}_0$ and $\tilde{\alpha}_1$ must be chosen as $\tilde{\alpha}_0= l_0+1$ and $\tilde{\alpha}_1= l_1+1$ from the condition $\tilde{\alpha}_0 ,\tilde{\alpha}_1 ,l_0 ,l_1 >0$. Now we check that, if $f(x) \in \tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$, then $f(x)$ satisfies the definition of the Hilbert space $\mathbf{H}$ (see (\ref{Hilb1})). Square-integrability of the function $f(x)$ follows from the condition $\tilde{\alpha}_0\geq 0$ and $\tilde{\alpha}_1\geq 0$. Periodicity and symmetry of $f(x)$ follow from the condition $-\sum_{i=0}^3 \tilde{\alpha}_i /2 \in \mathbb{Z}$. Hence $f(x)\in \mathbf{H}$. For the case $l_0>0$ and $l_1=0$, $\tilde{\alpha}_0$ and $\tilde{\alpha}_1$ are chosen as $\tilde{\alpha}_0= l_0+1$ and ($\tilde{\alpha}_1= 0$ or $1$). Then $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3} \subset \mathbf{H}_i$ for $i=+$ or $-$ is shown similarly. Note that the sign of $i$ is determined by whether functions in $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ are periodic or antiperiodic (for details see section \ref{sec:L2l0l12}). For the other cases, the proofs are similar. \end{proof} \subsection*{Remark} A function in $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ is multi-valued in general, and so we should specify branches of the function. In our case, the analytic continuation of the function near the real line should be performed along paths passing through the upper half plane. \smallskip In the finite dimensional space $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$, eigenvalues are calculated by solving a characteristic equation, which is an algebraic equation, and eigenfunctions are obtained by solving linear equations. In this sense, eigenvalues in the finite dimensional space are algebraic''. Now we figure out properties of the spaces $\tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ contained in the Hilbert space $\mathbf{H}$. We divide into four cases. \subsubsection{The case $l_0>0$ and $l_1>0$} Let $\tilde{\alpha}_2 \in \{ -l_2, l_2+1 \}$ and $\tilde{\alpha}_3 \in \{ -l_3, l_3+1 \}$. From Proposition \ref{prop:finL2}, if $d=-(l_0+l_1)/2-1-(\tilde{\alpha}_2 +\tilde{\alpha}_3 )/2\in \mathbb{Z}_{\geq 0}$ then the $d+1$ dimensional vector space $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ is a subspace of the Hilbert space $\mathbf{H}$. We will show that the set of eigenvalues of the gauge-transformed Hamiltonian $H(p)$ (see (\ref{eqn:gtrans})) on the space $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ is the set of small eigenvalues of $H(p)$ on the Hilbert space $\mathbf{H}$ from the bottom. \begin{lemma} For the trigonometric case $p=\exp(\pi \sqrt{-1} \tau) = 0$, if $d=-(l_0+l_1)/2-1-\tilde{\alpha}_2 -\tilde{\alpha}_3 \in \mathbb{Z}_{\geq 0}$ then the set of eigenvalues of the gauge-transformed Hamiltonian $H(0)$ on the finite dimensional space $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}|_{p=0}$ coincides with the set of small eigenvalues of $H(0)$ on the Hilbert space $\mathbf{H}$ from the bottom. In other words, the $m$-th smallest eigenvalue of $H(0)$ on $\mathbf{H}$ is also an eigenvalue on $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}|_{p=0}$, if and only if $m\leq d+1$. \end{lemma} \begin{proof} From Proposition \ref{prop:fintrig}, eigenvalues of the trigonometric Hamiltonian $H$ on the finite dimensional space $V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ are written as $\{ \pi^2 (2r+l_0+l_1+2)^2-\frac{\pi^2}{3} \sum_{i=0}^3 l_i(l_i+1) \} _{r=0, \dots ,d}$. On the other hand, from equality (\ref{eqn:Htpsim}) and the limit $H \to H_T-\frac{\pi^2}{3} \sum_{i=0}^3 l_i(l_i+1)$ as $p\to 0$, eigenvalues of the gauge-transformed trigonometric Hamiltonian $H(0)=\Phi(x)H \Phi(x)^{-1}|_{p=0}$ on the Hilbert space $\mathbf{H}$ are written as $\{ \pi^2 (2r+l_0+l_1+2)^2-\frac{\pi^2}{3} \sum_{i=0}^3 l_i(l_i+1)\} _{r\in \mathbb{Z}_{\geq 0}}$. Therefore the lemma follows. \end{proof} From the previous lemma, eigenvalues of the trigonometric gauge-transformed Hamiltonian $H(0)$ on the finite dimensional space $V_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ are $\{ \tilde{E}_r(0) \: | \: r=0,\dots ,d \}$, where the values $\tilde{E}_r(p)$ are defined in Theorem \ref{mainthmKato}. It is obvious that the eigenvalues of the operator $H(p)$ on the finite dimensional space $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ are continuous in $p$. Hence if $-1 0$) or ($l_0>0$ and $l_1\geq 1/2$), then the set of eigenvalues of the gauge-transformed Hamiltonian $H(p)$ on the finite dimensional space $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ coincides with the set of small eigenvalues of $H(p)$ on the Hilbert space $\mathbf{H}$ from the bottom. In other words, the $m$-th smallest eigenvalue of $H(p)$ on $\mathbf{H}$ is also an eigenvalue on $\tilde{V}_{l_0+1, l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3}$, if and only if $m\leq d+1$. \end{thm} \subsubsection{The case $l_0>0$ and $l_1=0$} \label{sec:L2l0l12} Let $\tilde{\alpha}_2 \in \{ -l_2, l_2+1 \}$ and $\tilde{\alpha}_3 \in \{ -l_3, l_3+1 \}$. If $d=-(l_0+1)/2-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, then the $d+1$ dimensional space $\tilde{V}_{l_0+1, 0, \tilde{\alpha}_2, \tilde{\alpha}_3}$ is a subspace of the space $\mathbf{H}_+$, and if $d=-(l_0+2)/2-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, then the $d+1$ dimensional space $\tilde{V}_{l_0+1, 1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ is a subspace of the space $\mathbf{H}_-$. Similarly to Theorem \ref{thm:lowest}, the following statement are shown.\\ $\bullet$ If $d=-(l_0+1)/2-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, $l_0\geq 1/2$, and $-10$} \label{sec:L2l0l13} Let $\tilde{\alpha}_2 \in \{ -l_2, l_2+1 \}$ and $\tilde{\alpha}_3 \in \{ -l_3, l_3+1 \}$. If $d=-(l_1+1)/2-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, then $\tilde{V}_{0,l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{0,l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3} =d+1$. If $d=-(l_1+2)/2-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, then $\tilde{V}_{1,l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3} \subset \mathbf{H}_-$ and $\dim \tilde{V}_{1,l_1+1, \tilde{\alpha}_2, \tilde{\alpha}_3} =d+1$. We can confirm similar statements to the ones in section \ref{sec:L2l0l12}. \subsubsection{The case $l_0=0$ and $l_1=0$} Let $\tilde{\alpha}_2 \in \{ -l_2, l_2+1 \}$ and $\tilde{\alpha}_3 \in \{ -l_3, l_3+1 \}$. If $d=-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, then $\tilde{V}_{0, 0, \tilde{\alpha}_2, \tilde{\alpha}_3}\subset \mathbf{H}_1$, $\tilde{V}_{1, 1, \tilde{\alpha}_2, \tilde{\alpha}_3}\subset \mathbf{H}_2$, $\dim \tilde{V}_{0, 0, \tilde{\alpha}_2, \tilde{\alpha}_3} =d+1$, and $\dim \tilde{V}_{1, 1, \tilde{\alpha}_2, \tilde{\alpha}_3}=d$. If $d=-1/2-(\tilde{\alpha}_2 +\tilde{\alpha}_3)/2 \in \mathbb{Z}_{\geq 0}$, then $\tilde{V}_{1, 0, \tilde{\alpha}_2, \tilde{\alpha}_3}\subset \mathbf{H}_4$, $\tilde{V}_{0, 1, \tilde{\alpha}_2, \tilde{\alpha}_3}\subset \mathbf{H}_3$, and $\dim \tilde{V}_{1, 0, \tilde{\alpha}_2, \tilde{\alpha}_3} =\dim \tilde{V}_{0, 1, \tilde{\alpha}_2, \tilde{\alpha}_3}=d+1$. \subsection{The case of nonnegative integral coupling constants} \label{sec:nonneg} If the coupling constants $l_0, l_1, l_2, l_3$ are nonnegative integers, the model satisfies some special properties. Specifically, it admits the Bethe Ansatz method \cite{Tak1} and the potential has the finite-gap property \cite{TV,Smi,Tak3}. In this subsection, we reproduce several results more explicitly for the case $l_0, l_1, l_2, l_3$ are nonnegative integers. Note that some results were obtained in \cite{Tak1}. Throughout this subsection, assume $l_i \in \mathbb{Z}_{\geq 0}$ for $i=0,1,2,3$. Assume $\tilde{\beta}_i\in \mathbb{Z}$ $(i=0,1,2,3)$ and $-\sum_{i=0}^3 \tilde{\beta}_i /2 \in \mathbb{Z}_{\geq 0}$. Let $V_{\tilde{\beta}_0, \tilde{\beta}_1, \tilde{\beta}_2, \tilde{\beta}_3}$ be a vector space spanned by $\big\{ (\frac{\sigma_1(x)}{\sigma(x)})^{\tilde{\beta }_1} (\frac{\sigma_2(x)}{\sigma(x)})^{\tilde{\beta }_2} (\frac{\sigma_3(x)}{\sigma(x)})^{\tilde{\beta }_3}\wp(x)^n\big\} _{n=0, \dots ,-\sum_{i=0}^3 \tilde{\beta}_i /2 }$ and we set $\tilde{V}_{\tilde{\beta}_0, \tilde{\beta}_1, \tilde{\beta}_2, \tilde{\beta}_3}= \{ \frac{f(x)}{\Phi(x)} | f(x) \in V_{\tilde{\beta}_0, \tilde{\beta}_1, \tilde{\beta}_2, \tilde{\beta}_3 }\}$, where $\Phi(x)$ was defined in section \ref{sect:trig}. Let $\tilde{\alpha}_i \in \{ -l_i , l_i+1\}$ $(i=0,1,2,3)$ and \begin{gather*} U_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}= \begin{cases} V_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}, & \sum_{i=0}^3 \tilde{\alpha}_i/2 \in \mathbb{Z}_{\leq 0} ;\\ V_{1-\tilde{\alpha}_0, 1-\tilde{\alpha}_1, 1-\tilde{\alpha}_2, 1-\tilde{\alpha}_3} , & \sum_{i=0}^3 \tilde{\alpha}_i /2\in \mathbb{Z}_{\geq 2} ;\\ \{ 0 \} ,& \mbox{otherwise}, \end{cases} \\ \tilde{U}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}= \begin{cases} \tilde{V}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3} , & \sum_{i=0}^3 \tilde{\alpha}_i /2\in \mathbb{Z}_{\leq 0}; \\ \tilde{V}_{1-\tilde{\alpha}_0, 1-\tilde{\alpha}_1, 1-\tilde{\alpha}_2, 1-\tilde{\alpha}_3} ,& \sum_{i=0}^3 \tilde{\alpha}_i /2\in \mathbb{Z}_{\geq 2};\\ \{ 0 \} ,& \mbox{otherwise}. \end{cases} \end{gather*} If $l_0 +l_1 +l_2 +l_3$ is even, then the Hamiltonian $H$ (see (\ref{Ino})) preserves the spaces \\ $U_{-l_0,-l_1,-l_2,-l_3}$, $U_{-l_0 ,-l_1,l_2+1 ,l_3+1}$, $U_{-l_0,l_1+1,-l_2,l_3+1}$, $U_{-l_0 ,l_1+1,l_2+1,-l_3}$. The \ gauge-transformed Hamiltonian $H(p)$ (see (\ref{eqn:gtrans})) preserves the spaces $\tilde{U}_{-l_0,-l_1,-l_2,-l_3}$, $\tilde{U}_{-l_0 ,-l_1,l_2+1 ,l_3+1}$, $\tilde{U}_{-l_0,l_1+1,-l_2,l_3+1}$, $\tilde{U}_{-l_0 ,l_1+1,l_2+1,-l_3}$. If $l_0 +l_1 +l_2 +l_3$ is odd, then the Hamiltonian $H$ preserves the spaces $U_{-l_0,-l_1,-l_2,l_3+1}$, $U_{-l_0 ,-l_1,l_2+1,-l_3}$, $U_{-l_0 ,l_1+1,-l_2,-l_3}$, $U_{l_0+1,-l_1,-l_2,-l_3}$, and the gauge-transformed Hamiltonian $H(p)$ preserves the spaces $\tilde{U}_{-l_0,-l_1,-l_2,l_3+1}$, $\tilde{U}_{-l_0 ,-l_1,l_2+1,-l_3}$, $\tilde{U}_{-l_0 ,l_1+1,-l_2,-l_3}$, and \break $\tilde{U}_{l_0+1,-l_1,-l_2,-l_3}$. Now we present results on the inclusion of $\tilde{U}_{\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\alpha}_2, \tilde{\alpha}_3}$ in a Hilbert space. We consider eight cases. \subsubsection{The case $l_0>0 ,\: l_1>0$ and $l_0+l_1+l_2+l_3$ is even} If $l_2+l_3-l_0-l_1\geq 2$, then $\tilde{V}_{l_0+1, l_1+1,-l_2, -l_3} \subset \mathbf{H}$ and $\dim \tilde{V}_{l_0+1, l_1+1,-l_2, -l_3} = (l_2+l_3-l_0-l_1)/2$. \subsubsection{The case $l_0>0 ,\: l_1>0$ and $l_0+l_1+l_2+l_3$ is odd} If $l_2-l_0-l_1-l_3\geq 3$, then $\tilde{V}_{l_0+1, l_1+1,-l_2, l_3+1} \subset \mathbf{H}$ and $\dim \tilde{V}_{l_0+1, l_1+1,-l_2, l_3+1} = (l_2-l_0-l_1-l_3-1)/2$. If $l_3-l_0-l_1-l_2\geq 3$, then $\tilde{V}_{l_0+1, l_1+1,l_2+1, -l_3} \subset \mathbf{H}$ and $\dim \tilde{V}_{l_0+1, l_1+1,l_2+1, -l_3} = (l_3-l_0-l_1-l_2-1)/2$. \subsubsection{The case $l_0>0 , \: l_1=0$ and $l_0+l_1+l_2+l_3$ is even} If $l_2+l_3-l_0\geq 2$, then the space $\tilde{V}_{l_0+1, 1,-l_2, -l_3}$ is a subspace of the space $\mathbf{H}_- (\subset \mathbf{H})$ and $\dim \tilde{V}_{l_0+1, 1,-l_2, -l_3} = (l_2+l_3-l_0)/2$. If $l_2-l_0-l_3\geq 2$, then $\tilde{V}_{l_0+1, 0,-l_2, l_3+1} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{l_0+1, 0,-l_2, l_3+1} = (l_2-l_0-l_3)/2$. If $l_3-l_0-l_2\geq 2$, then $\tilde{V}_{l_0+1, 0,l_2+1, -l_3} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{l_0+1, 0,l_2+1, -l_3} = (l_3-l_0-l_2)/2$. \subsubsection{The case $l_0>0 ,\: l_1=0$ and $l_0+l_1+l_2+l_3$ is odd} If $l_2+l_3-l_0\geq 1$, then $\tilde{V}_{l_0+1, 0,-l_2, -l_3} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{l_0+1, 0,-l_2, -l_3} = (l_2+l_3-l_0+1)/2$. If $l_2-l_0-l_3\geq 3$, then $\tilde{V}_{l_0+1, 1,-l_2, l_3+1} \subset \mathbf{H}_-$ and $\dim \tilde{V}_{l_0+1, 1,-l_2, l_3+1} =(l_2-l_0-l_3-1)/2$. If $l_3-l_0-l_2\geq 3$, then $\tilde{V}_{l_0+1, 1,l_2+1, -l_3} \subset \mathbf{H}_-$ and $\dim \tilde{V}_{l_0+1, 1,l_2+1, -l_3} = (l_3-l_0-l_2-1)/2$. \subsubsection{The case $l_0=0 ,\: l_1>0$ and $l_0+l_1+l_2+l_3$ is even} If $l_2+l_3-l_1\geq 2$, then $\tilde{V}_{ 1,l_1+1, -l_2, -l_3} \subset \mathbf{H}_-$ and $\dim \tilde{V}_{1, l_1+1, -l_2, -l_3} = (l_2+l_3-l_1)/2$. If $l_2-l_1-l_3\geq 2$, then $\tilde{V}_{0, l_1+1, -l_2, l_3+1} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{0, l_1+1, -l_2, l_3+1} = (l_2-l_1-l_3)/2$. If $l_3-l_1-l_2\geq 2$, then $\tilde{V}_{0, l_1+1, l_2+1, -l_3} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{0, l_1+1, l_2+1, -l_3} = (l_3-l_1-l_2)/2$. \subsubsection{The case $l_0=0 ,\: l_1>0$ and $l_0+l_1+l_2+l_3$ is odd} If $l_2+l_3-l_1\geq 1$, then $\tilde{V}_{0, l_1+1, -l_2, -l_3} \subset \mathbf{H}_+$ and $\dim \tilde{V}_{0, l_1+1,-l_2, -l_3} = (l_2+l_3-l_1+1)/2$. If $l_2-l_1-l_3\geq 3$, then $\tilde{V}_{1, l_1+1,-l_2, l_3+1} \subset \mathbf{H}_-$ and $\dim \tilde{V}_{1, l_1+1,-l_2, l_3+1} = (l_2-l_1-l_3-1)/2$. If $l_3-l_1-l_2\geq 3$, then $\tilde{V}_{1, l_1+1, l_2+1, -l_3} \subset \mathbf{H}_-$ and $\dim \tilde{V}_{1, l_1+1,l_2+1, -l_3} = (l_3-l_1-l_2-1)/2$. \subsubsection{The case $l_0=l_1=0$ and $l_0+l_1+l_2+l_3$ is even} In this case, the spaces $\tilde{V}_{0, 0, -l_2, -l_3}, \quad \tilde{V}_{1, 1, -l_2, -l_3},$ \begin{equation*} \left\{ \begin{array}{ll} \tilde{V}_{1, 0, -l_2, l_3+1}, \quad \tilde{V}_{0, 1, -l_2, l_3+1} & (l_2>l_3) \\ \tilde{V}_{1, 0, l_2+1, -l_3}, \quad \tilde{V}_{0,1, l_2+1, -l_3} & (l_2l_3) \\ \tilde{V}_{1, 1, l_2+1, -l_3}, \quad \tilde{V}_{0,0, l_2+1, -l_3} & (l_21$and$C|p|<1$, there exists$C''\in \mathbb{R}_{>0}$such that$|\tilde{t}_{m,{m'}}| \leq C''(C|p|)^{\frac{|{m'} -m|+1}{2}}$. \end{prop} \begin{proof} Since the normalized Jacobi polynomials form a complete orthonormal system, it follows that$\tilde{t}_{m,{m'}}= \langle \psi_ {m'}(x) ,(\sum_{k=1}^{\infty}V_k(x)p^{k})\psi _m(x)\rangle _{\Phi }$. If$k<|m'-m|$, then$ \langle V_k(x)p^{k}\psi _m(x), \psi_ {m'}(x) \rangle _{\Phi }=0by Lemma \ref{anallem2} and orthogonality. Therefore, \begin{align*} |\tilde{t}_{m,{m'}}| & = \Big| \Big\langle \psi_ {m'}(x), \Big(\sum_{k=1}^{\infty}V_k(x)p^{k}\Big) \psi _m(x) \Big\rangle _{\Phi } \Big|\\ & = \Big| \Big\langle \psi_{m'}(x),\Big(\sum_{k=|m'-m|}^{\infty}V_k(x)p^{k}\Big) \psi _m(x) \Big\rangle _{\Phi } \Big| \\ & = \Big|\int _0^1 \sum_{k=|m'-m|}^{\infty}V_k(x)p^{k}\psi _m(x) \overline{\psi_ {m'}(x)} \Phi(x) ^2 dx \Big| \\ & \leq \sup_{x\in [0,1]} \Big| \sum_{k=|m'-m|}^{\infty}V_k(x)p^{k} \Big| \int_0^1 |\psi _m(x) \psi_ {m'}(x) \Phi(x)^2 | dx \\ & \leq \sum_{k \geq |m'-m|}V_k|p|^{k}. \end{align*} whereV_k$is defined by (\ref{majser}). For the case$m'=m$, we obtain$|\tilde{t}_{m,m}| \leq \sum_{k \geq 1}V_k|p|^{k}$. Since the convergence radius of the series$\sum_n V_n p^n$is equal to$1$, for each$C$such that$C>1$and$C|p|<1$there exists$C''\in \mathbb{R} _{>0}$such that$|\tilde{t}_{m,{m'}}| \leq C''(C|p|)^{|{m'} -m|}(m\neq {m'})$and$|\tilde{t}_{m,m}| \leq C''(C|p|) \leq C''(C|p|)^{1/2}$. From an inequality$(C|p|)^{|{m'} -m|} \leq (C|p|)^{\frac{|{m'} -m|+1}{2}}$for$|{m'}-m| \in \mathbb{Z}_{\geq 1}$, we obtain the proposition. \end{proof} \begin{prop} \label{analprop11} Let$D$be a positive number. Suppose$\mathop{\rm dist}(\zeta, \sigma (\tilde{H}(0)))\geq D$. Write$(\tilde{H}(p)-\zeta )^{-1} \psi _m(x)=\sum_{{m'}}t_{m,{m'}}\psi_ {m'}(x)$, where$(\tilde{H}(p)-\zeta )^{-1}$is defined in (\ref{Neus}). For each$m \in \mathbb{Z}_{\geq 0}$and$C\in \mathbb{R} _{>1}$, there exists$C'\in \mathbb{R} _{>0}$and$p_0\in \mathbb{R} _{>0}$which do not depend on$\zeta $(but depend on$D$) such that$t_{m,{m'}}$satisfy $$|t_{m,{m'}}| \leq C'(C|p|)^{|{m'} -m|/2}, \label{proptlm}$$ for all$p$,$m'$such that$|p|0})$and set$X:=(\zeta -\tilde{H}(0))^{-1}(\sum_{k=1}^{\infty}V_k(x)p^{k})$. From expansion (\ref{Neus}), there exists$p_1\in \mathbb{R}_{>0}$such that an inequality$\| X \| <1/2$holds for all$p$and$\zeta$such that$|p|D$. Then we have$(\sum_{i=0}^{\infty}X^i)(\tilde{H}(0)-\zeta )^{-1}= (\tilde{H}(p)-\zeta )^{-1}$. If we write$\sum_{{m'}} c'_{{m'}}\psi_ {m'}(x)=(\tilde{H}(0)-\zeta )^{-1} \sum_{{m'}}c_{{m'}}\psi_ {m'}(x)$, then$|c'_{{m'}}| \leq D^{-1}|c_{{m'}}|$for each${m'}$. Write$X\psi _m(x)=\sum_{{m'}}\check{t}_{m,{m'}}\psi_ {m'}(x)$. By combining with Proposition \ref{analprop1}, we obtain that for each$C$such that$C>1$and$\frac{C+1}{2} p_1<1$, there exists$C''\in \mathbb{R} _{>0}$which does not depend on$\zeta $(but depend on$D$) such that$|\check{t}_{m,{m'}}| \leq C''(\frac{C+1}{2}|p|)^{(|{m'} -m|+1)/2}$for$|p|0}$and$p_0$which do not depend on$\zeta $(but depend on$D$) such that$t^{\star}_{m,{m'}}$are well-defined by$\sum_{k=0}^{\infty} X^k \mathbf{e}_{m}= \sum_{{m'}\in \mathbb{Z}} t^{\star}_{m,{m'}}\mathbf{e}_{{m'}}$and satisfy $$|t^{\star}_{m,{m'}}| \leq C^{\star}\Big(\frac{C+1}{2}|p|\Big)^{|{m'} -m|/2}, \label{proptlms}$$ for all$p$and${m'}$such that$|p|1}$, there exist$C'\in \mathbb{R} _{>0}$and$p_{\ast }\in \mathbb{R} _{>0}$such that$s_{m,{m'}}$satisfy $$|s_{m,{m'}}| \leq C'(C|p|)^{|{m'} -m|/2}, \label{propslm}$$ for all$p$and$m'$such that$|p|1}$there exists$C_{\ast }\in \mathbb{R} _{>0}$and$p_{\ast }\in \mathbb{R} _{>0}$which do not depend on$\zeta (\in \Gamma_m) $such that$t_{m,{m'}}$satisfy$|t_{m,{m'}}(\zeta )| \leq C_{\ast } (C|p|)^{\frac{|{m'} -m|}{2}}$for all$p$,$m'$such that$|p|0}$and$R \in (0,1)$. If$r'$satisfies$00$and$l_1>0$. For the other cases, they are proved similarly. We introduce a Rodrigues-type formula for the Jacobi polynomials \begin{equation*} p_m(w)= \frac{(-1)^m}{m!}w^{-l_0-1/2}(1-w)^{-l_1-1/2}\big( \frac{d}{dw} \big) ^m \left( w^{l_0+m+1/2}(1-w)^{l_1+m+1/2} \right). \end{equation*} Then$p_m(\sin ^2 \pi x)=d_m \psi _m(x)$, where$d_m=\sqrt{\frac{\Gamma (m+l_0+3/2)\Gamma (m+l_1+3/2)}{\pi m! (2m+l_0+l_1+2)\Gamma (m+l_0+l_1+2)}}$. From the Stirling's formula, we have$(d_m)^{1/m}\to 1$as$m \to \infty $. Hence it is sufficient to show that the power series$\sum_{m=0}^{\infty}c_m p_m(\sin ^2 \pi x)$converges uniformly absolutely inside the zone$-r' \leq \Im x\leq r'$. The generating function of the Jacobi polynomials$p_m (w)$is written as $$\sum_{m=0}^{\infty} p_m(w)\xi^m= \frac{1}{S\big( \frac{1+\xi+S}{2} \big) ^{l_0+1/2} \big( \frac{1-\xi+S}{2} \big) ^{l_1+1/2}}, \label{genfunct}$$ where$S=\sqrt{(1+\xi)^2-4\xi w}$. Now we set$\sum_{m=0}^{\infty} \tilde{q}_m(y)\xi^m = \frac{1}{\sqrt{(1-\xi)^2-4y\xi}}$, \\$\sum_{m=0}^{\infty} \tilde{q}^{(a)}_m(y)\xi^m = \frac{1}{(1-\xi +\sqrt{(1-\xi)^2-4y\xi})^a}$, and $$\sum_{m=0}^{\infty} \tilde{p}_m(y)\xi^m = \frac{1}{(\sqrt{(1-\xi)^2-4y\xi})} \frac{1}{(1-\xi +\sqrt{(1-\xi)^2-4y\xi})^{l_0+l_1+1}}. \label{genpny}$$ Then it is shown that, if$a>0$, then$\tilde{q}_m(y)$and$\tilde{q}^{(a)}_m(y)$are polynomials in$y$of degree$m$with nonnegative coefficients. Hence$\tilde{p}_m(y)$is also a polynomial in$y$of degree$m$with nonnegative coefficients. Set$p_m(y)= \sum_{k=0}^m p_m^{(k)} y^k$and$\tilde{p}_m(y)= \sum_{k=0}^m \tilde{p}_m^{(k)} y^k$. From formulas (\ref{genfunct}, \ref{genpny}) and the nonnegativity, we obtain$|p_m^{(k)}|\leq \tilde{p}_m^{(k)}$for all$m$and$k$. From the inequality$|\sin^2\pi x | \leq \big| \big(\frac{e^{\pi r'}+e^{-\pi r'}}{2}\big)^2 \big|$for$-r'\leq \Im x \leq r', it is seen that \begin{aligned} |p_m(\sin^2\pi x)| &\leq \sum_{k=0}^m \big| p_m^{(k)} ( \sin^2\pi x )^k \big| \leq \sum_{k=0}^m \tilde{p}_m^{(k)} | \sin^2\pi x |^k \\ &\leq \Big|\tilde{p}_m\Big( \big(\frac{e^{\pi r'}+e^{-\pi r'}}{2}\big)^2\Big)\Big| \end{aligned}\label{ineq:pn} for allm\in \mathbb{Z} _{\geq 0}$and$x$such that$-r'\leq \Im x \leq r'$. On the other hand, the series$\sum_{m=0}^{\infty} \tilde{p}_m\big ((\frac{e^{\pi r'}+e^{-\pi r'}}{2})^2 \big) \xi ^m$, with respect to the variable$\xi$, has radius of convergence$e^{-2\pi r'}$; because the singular point of the right hand side of (\ref{genpny}) which is closest to the origin is located on the circle$|\xi |=e^{-2\pi r'}$. Let$r''$be a positive number such that$r'