| Izvestiya: Mathematics |
|
|
|
|
|
Difference analogue of the Treibich–Verdier operator G. S. Mauleshovaab, A. E. Mironovab a Novosibirsk State University b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Bibliographic databases:
 § 1. IntroductionIn this paper, we continue to study the relationship between one-dimensional finite-gap Schrödinger operators and their difference analogues. In [1], it was shown that the one-dimensional finite-gap Schrödinger operator commuting with some ordinary differential operator
$$
\begin{equation}
L_{2g+1} = \partial^{2g+1}_x + v_{2g}(x)\, \partial^{2g}_x + \dots + v_0(x),
\end{equation}
\tag{2}
$$
extends to a second-order difference operator of the form
$$
\begin{equation}
\widetilde{L}_2 = \frac{T_\varepsilon^2}{\varepsilon^2} +A(x,\varepsilon) \, \frac{T_\varepsilon}{\varepsilon}+V(x,\varepsilon),
\end{equation}
\tag{3}
$$
commuting with some difference operator of order $2g+1$
$$
\begin{equation}
\widetilde{L}_{2g+1} = \frac{T^{2g+1}_\varepsilon}{\varepsilon^{2g+1}} + u_{2g}(x,\varepsilon) \, \frac{T^{2g}_\varepsilon}{\varepsilon^{2g}} + \dots + u_0(x,\varepsilon),
\end{equation}
\tag{4}
$$
where $T_{\varepsilon}$ is the shift operator by $\varepsilon$, $T_{\varepsilon}f(x)=f(x+\varepsilon)$, and
$$
\begin{equation*}
\widetilde{L}_2 = L_2 + O(\varepsilon), \qquad \widetilde{L}_{2g+1} = L_{2g+1}+ O(\varepsilon).
\end{equation*}
\notag
$$
In general, the potential of a finite-gap Schrödinger operator $u(x)$ is expressed in terms of the theta function of the Jacobian variety of a hyperelliptic spectral curve [2], but in the special cases when the hyperelliptic curve covers an elliptic curve, the theta functional formula for $u(x)$ can be reduced to a formula expressing the potential $u(x)$ in terms of elliptic functions. A remarkable example of such an operator is the Treibich–Verdier operator.
$$
\begin{equation*}
L_2 = \partial_x^2 - \sum_{j=0}^3 s_j(s_j+1)\wp(x-\omega_j),\qquad s_j\in{\mathbb N}\cup\{0\}, \quad s=(s_0,s_1,s_2,s_3)\neq0.
\end{equation*}
\notag
$$
Here, $\Lambda=\mathbb{C}/\{2\omega_1{\mathbb Z} +2\omega_2{\mathbb Z}\}$ is the lattice of periods of the Weierstrass elliptic function $\wp(x)$ and $\omega_0=0$, $\omega_3 = \omega_1 + \omega_2$. The Treibich–Verdier operator has been studied in many papers, for example, in [3]–[9]. The main result of the present paper is as follows. Consider the function $A(x,\varepsilon)$ defined by
$$
\begin{equation*}
A(x,\varepsilon) = a(\varepsilon)\prod_{j=0}^3 A_j(x,\varepsilon),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
A_j(x,\varepsilon) = \prod_{k=0}^{g_j}\biggl(1+\frac{\zeta(x-\omega_j-(2 k+1) \varepsilon)-\zeta(x-\omega_j+(2 k+1) \varepsilon)}{\zeta(\varepsilon)+\zeta((4 k+1) \varepsilon)} \biggr)
\end{equation}
\tag{5}
$$
for odd $s_j = 2 g_j + 1$ and
$$
\begin{equation}
A_j(x,\varepsilon) =\prod_{k=1}^{g_j}\biggl(1+\frac{\zeta(x-\omega_j-2 k \varepsilon)-\zeta(x-\omega_j+2 k \varepsilon)}{\zeta(\varepsilon)+\zeta((4 k-1) \varepsilon)} \biggr)
\end{equation}
\tag{6}
$$
for even $s_j = 2 g_j$. Here, $a(\varepsilon)$ is an arbitrary meromorphic function, $\zeta(x)$ is the Weierstrass elliptic function, $\zeta'(x)=-\wp(x)$. The following result holds. Theorem 1. The difference operator commutes with some difference operator of odd order. If $a(\varepsilon) = -2/\varepsilon + O(\varepsilon^3)$, then An analogue of Theorem 1 for the Lamé operator was proved in [10].In § 2, we recall some results on one-dimensional finite-gap Schrödinger operators. In § 3, we recall some results on commuting difference operators of rank one. In § 4, we give a proof of Theorem 1. § 2. Finite-gap Schrödinger operatorsIn [11], [12], it was proved that a one-dimensional real smooth periodic Schrödinger operator $L_2$ is finite-gap if and only if $L_2$ commutes with some operator of the form (2). Now, a finite-gap operator is often understood to be an operator $L_2$ that satisfies the condition $[L_2,L_{2g+1}]=0$. The theory of one-dimensional finite-gap Schrödinger operators is closely related to that of finite-gap (algebraic-geometric) solutions of the Korteweg–de Vries equation (see [2], [11]–[13]). The common eigenfunction of the operators $L_2$ and $L_{2g+1}$ is the Baker–Akhiezer function $\psi$. We have The point $P=(z,w)$ lies on the spectral curve $\Gamma$ given by where $\psi=\psi(x,P)$. The function $\psi$ has an essential singularity at the infinite point of the spectral curve $\Gamma$. The potential of the Schrödinger operator has the form (see [2])
$$
\begin{equation*}
u(x)=2\, \partial_x^2\log\theta(xU_1+U_2)+\mathrm{const},
\end{equation*}
\notag
$$
where $\theta$ is the theta function of the Jacobian variety of the spectral curve $\Gamma$, $U_1,U_2\in \mathbb {C}^g$ are some vectors. The following factorization holds:
$$
\begin{equation*}
L_2 - z = (\partial_x+\chi_0(x,P))(\partial_x-\chi_0(x,P))
\end{equation*}
\notag
$$
(see [13]), where Unlike $\psi,$ the function $\chi_0$ is a rational function on $\Gamma$, which has $g$ simple poles (depending on $x$) and a simple pole at infinity
$$
\begin{equation*}
\chi_0 = \frac{R_x}{2 R}+\frac{w}{R}, \qquad R(x,z)=z^g+\alpha_{g-1}(x) z^{g-1}+\dots+\alpha_0(x).
\end{equation*}
\notag
$$
The polynomial $R$ satisfies the equation
$$
\begin{equation}
w^2 = F_g(z) = (z - u) R^2 + \frac{1}{4}R_x^2 - \frac{1}{2}R R_{xx}.
\end{equation}
\tag{10}
$$
Differentiating (10) with respect to $x$ and dividing by $R$, we obtain Among the finite-gap Schrödinger operators one can single out operators with elliptic (doubly periodic) potentials. Examples of such operators are, as already mentioned, the Lamé operator and the Treibich–Verdier operator. There is a criterion for a potential to be finite-gap (Theorem 1.1 in [14]). This criterion was reformulated in [15] as follows. Theorem 2. The Schrödinger operator $\partial^2_x+u(x)$ with an elliptic potential $u(x)$ is finite-gap if and only if the Laurent decomposition of $u(x)$ near each pole $x_0 \in \mathbb{C}$ has the form The finite-gap property of the Treibich–Verdier operator follows from Theorem 2. § 3. Commuting difference operators of orders $2$ and $2g+1$A new class of commutative rings of discrete operators of the form
$$
\begin{equation}
\widehat{L}_s=T^{s} + \widehat{u}_{s-1}(n)T^{s-1} + \dots + \widehat{u}_0(n), \qquad n \in \mathbb{Z},
\end{equation}
\tag{13}
$$
was introduced in [16], where $T$ is the shift operator, $Tf(n)=f(n+1)$, $f\colon \mathbb{Z} \to \mathbb{C}$. Note that $T$ appears in $\widehat{L}_s$ only with positive powers, whereas $T$ can appear in the commuting operators from [17]–[19] with both positive and negative powers. These commutative rings are reconstructed from the following spectral data
$$
\begin{equation*}
S=\{\Gamma, \gamma_1,\dots,\gamma_g, q, k^{-1}, P_n\},
\end{equation*}
\notag
$$
where $\Gamma$ is a Riemann surface of genus $g$, $\gamma=\gamma_1+\cdots+\gamma_g$ is a non-special divisor on $\Gamma$, $q\in \Gamma$ is a marked point, $k^{-1}$ is a local parameter near $q$, $P_n\in\Gamma$, $ n \in \mathbb Z$ is a set of points in general position. The common eigenfunction of the operators is the Baker–Akhiezer function $\psi_n(P)$, $P\in\Gamma$, which has the following properties. 1. The function $\psi_n(P)$ is meromorphic on $\Gamma$ with a divisor of zeros and poles of the form
$$
\begin{equation*}
\begin{alignedat}{2} &\gamma_1(n)+\dots+\gamma_g(n)+P_1+\dots+P_n-\gamma_1-\dots-\gamma_g-nq &\quad &\text{for } \ n > 0, \\ &\gamma_1(n)+\dots+\gamma_g(n)-P_1-\dots-P_n-\gamma_1-\dots-\gamma_g-nq &\quad &\text{for } \ n < 0. \end{alignedat}
\end{equation*}
\notag
$$
2. $\psi_0(P) = 1$. 3. $\psi_n(P)=k^n+O(k^{n-1})$ near $q$. For any meromorphic function $g(P)$ on $\Gamma$ with a unique pole of order $s$ at $q$ with decomposition $g(P)=k^s+O(k^{s-1})$, there is a unique operator of the form (13) such that Consider the operator commuting with some operator $\widehat{L}_{2g+1}$, and suppose that the spectral curve of the pair $\widehat{L}_2,\widehat{L}_{2g+1}$ is a hyperelliptic curve of the form (9). Then
$$
\begin{equation*}
\widehat{L}_2-z =(T+U_n)^2+W_n - z = (T+U_n+U_{n+1}+\chi_n(P))(T-\chi_n(P))
\end{equation*}
\notag
$$
(see [16]), where
$$
\begin{equation*}
\begin{gathered} \, \chi=\frac{\psi_{n+1}(P)}{\psi_n(P)}=\frac{S_n}{Q_n}+\frac{w}{Q_n}, \\ S_n(z)=-U_nz^g+\delta_{g-1}(n)z^{g-1}+\dots+\delta_0(n), \qquad Q_n=-\frac{S_{n-1}+S_n}{U_{n-1}+U_n}, \end{gathered}
\end{equation*}
\notag
$$
$\delta_j(n)$ are some functions. The functions $U_n$, $W_n$, $S_n$ satisfy
$$
\begin{equation}
(U_n+U_{n+1})(S_n-S_{n+1}) - (z-U^2_n-W_n)Q_n + (z-U^2_{n+1}-W_{n+1})Q_{n+2}=0.
\end{equation}
\tag{15}
$$
If $S_n$ obeys equation (15), then $S_n$ automatically satisfies equation (14) for some polynomial $F_g(z)$. Equation (14) is equivalent to the commutation condition for $\widehat{L}_2$ and $\widehat{L}_{2g+1}$. Particular solutions of equations (14), (15) lead to the following explicit examples of commuting operators of order 2 and $2g+1$. The extension of discrete operators $\widehat{L}_2$, $\widehat{L}_{2g+1}$ to commuting difference operators of the form (3), (4) with a spectral curve of the form (9) was studied in [1]. In the case of operator (3), the decomposition takes place
$$
\begin{equation*}
\begin{gathered} \, \widetilde{L}_2-z=\biggl(\frac{T_\varepsilon}{\varepsilon}+A(x,\varepsilon) +\widetilde{\chi}(x+\varepsilon,\varepsilon,z) \biggr) \biggl(\frac{T_\varepsilon}{\varepsilon}-\widetilde{\chi}(x,\varepsilon,z) \biggr), \\ \widetilde{\chi}=\frac{S(x,\varepsilon,z)}{Q(x,\varepsilon,z)}+\frac{w}{Q(x,\varepsilon,z)}, \end{gathered}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{gathered} \, S(x,\varepsilon,z)=-\delta_g(x,\varepsilon)z^g+\delta_{g-1}(x,\varepsilon)z^{g-1}+\dots+\delta_0(x,\varepsilon), \\ \nonumber Q(x,\varepsilon,z)=-\frac{S(x-\varepsilon,\varepsilon,z)+S(x,\varepsilon,z)}{A(x-\varepsilon,\varepsilon)}, \qquad A(x,\varepsilon) = \delta_g(x,\varepsilon)+\delta_g(x+\varepsilon,\varepsilon), \end{gathered}
\end{equation}
\tag{16}
$$
$\delta_j(x,\varepsilon)$ are some functions. The polynomial $S$ satisfies
$$
\begin{equation}
\begin{gathered} \, F_g(z)=S^2(x,\varepsilon,z)+Q(x,\varepsilon,z)Q(x+\varepsilon,\varepsilon,z)(z-V(x,\varepsilon)), \end{gathered}
\end{equation}
\tag{17}
$$
$$
\begin{equation}
\begin{aligned} \, &(S(x,\varepsilon,z)-S(x+\varepsilon,\varepsilon,z))A(x,\varepsilon)-Q(x,\varepsilon,z)(z-V(x,\varepsilon)) \nonumber \\ &\qquad+Q(x+2 \varepsilon,\varepsilon,z)(z-V(x+\varepsilon,\varepsilon))=0. \end{aligned}
\end{equation}
\tag{18}
$$
In the case of the difference operator (7) ($V(x,\varepsilon)=\wp(\varepsilon)$), equations (17), (18) take the form
$$
\begin{equation}
\begin{gathered} \, \!F_g(z)\,{=}\, S^2(x,\varepsilon,z)\,{+}\,\frac{(S(x\,{-}\,\varepsilon,\varepsilon,z)+ S(x,\varepsilon,z))(S(x,\varepsilon,z) + S(x\,{+}\,\varepsilon,\varepsilon,z))}{A(x-\varepsilon,\varepsilon) A(x,\varepsilon)}(z\,{-}\,\wp(\varepsilon)), \end{gathered}
\end{equation}
\tag{19}
$$
$$
\begin{equation}
\begin{aligned} \, &S(x+\varepsilon,\varepsilon,z) - S(x,\varepsilon,z) \nonumber \\ &\ +(z-\wp(\varepsilon))\biggl(\frac{S(x+\varepsilon,\varepsilon,z) +S(x+2\varepsilon,\varepsilon,z)}{A(x,\varepsilon)A(x+\varepsilon,\varepsilon)}-\frac{S(x-\varepsilon,\varepsilon,z)+S(x,\varepsilon,z)}{A(x-\varepsilon,\varepsilon)A(x,\varepsilon)}\biggr) = 0. \end{aligned}
\end{equation}
\tag{20}
$$
Equations (17) and (18) are difference analogues of equations (10) and (11). Similarly, as for $\widetilde{L}_{2}-z$, we have the expansion
$$
\begin{equation*}
\widetilde{L}_{2g+1}-w = \widetilde{L}_{2g} \biggl(\frac{T_\varepsilon}{\varepsilon} -\widetilde{\chi}(x,\varepsilon,z)\biggr),
\end{equation*}
\notag
$$
where $\widetilde{L}_{2g}$ is some difference operator whose coefficients depend on $x$, $\varepsilon$, and $P$. To prove Theorem 1, it suffices to prove that equation (20) has a solution $S$ of the form (16). Example 3. In the case of an elliptic spectral curve $\Gamma$ given by § 4. Proof of Theorem 1Let $\mathcal{R}(x,\varepsilon,z)$ denote the left-hand side of equation (20),
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{R}(x,\varepsilon,z) = S(x+\varepsilon,\varepsilon,z) - S(x,\varepsilon,z) \\ &\quad + (z-\wp(\varepsilon))\biggl(\frac{S(x+\varepsilon,\varepsilon,z) +S(x+2\varepsilon,\varepsilon,z)}{A(x,\varepsilon)A(x+\varepsilon,\varepsilon)} -\frac{S(x-\varepsilon,\varepsilon,z)+S(x,\varepsilon,z)}{A(x-\varepsilon,\varepsilon)A(x,\varepsilon)}\biggr). \end{aligned}
\end{equation*}
\notag
$$
The following theorem holds. Theorem 3. Equation (20) has a solution of the form where $P_i^j(\varepsilon,z)$ is some polynomial with respect to $z$. Proof. We first claim that the function $A(x,\varepsilon)$ is even with respect to $x$, Let us recall that
$$
\begin{equation*}
\zeta(-u)=-\zeta(u),\qquad \zeta(u+2\omega_1+2\omega_2)=\zeta(u)+2\zeta(w_1)+2\zeta(w_2).
\end{equation*}
\notag
$$
For odd $s_j = 2 g_j + 1$, we have by (5)
$$
\begin{equation*}
\begin{aligned} \, &A_j(-x,\varepsilon) = \prod_{k=0}^{g_j}\biggl(1+\frac{\zeta(-x-\omega_j-(2 k+1) \varepsilon)-\zeta(-x-\omega_j+(2 k+1) \varepsilon)}{\zeta(\varepsilon)+\zeta((4 k+1) \varepsilon)} \biggr) \\ &\qquad=\prod_{k=0}^{g_j}\biggl(1+\frac{\zeta(x+\omega_j-(2 k+1) \varepsilon)-\zeta(x+\omega_j+(2 k+1) \varepsilon)}{\zeta(\varepsilon)+\zeta((4 k+1) \varepsilon)} \biggr) \\ &\qquad=\prod_{k=0}^{g_j}\biggl(1+\frac{\zeta(x-\omega_j-(2 k+1) \varepsilon + 2 \omega_j)-\zeta(x-\omega_j+(2 k+1) \varepsilon+2 \omega_j)}{\zeta(\varepsilon)+\zeta((4 k+1) \varepsilon)} \biggr) \\ &\qquad=\prod_{k=0}^{g_j}\biggl(1+\frac{\zeta(x-\omega_j-(2 k+1) \varepsilon)-\zeta(x-\omega_j+(2 k+1) \varepsilon)}{\zeta(\varepsilon)+\zeta((4 k+1) \varepsilon)} \biggr) = A_j(x,\varepsilon). \end{aligned}
\end{equation*}
\notag
$$
That $A_j(x,\varepsilon)$ is even for $s_j = 2 g_j$ is proved similarly.
We need the following lemma. Proof. Note that
$$
\begin{equation*}
S(-x,\varepsilon,z) = S(x+\varepsilon,\varepsilon,z).
\end{equation*}
\notag
$$
Indeed,
$$
\begin{equation*}
\begin{aligned} \, &S_j(-x,\varepsilon,z)=\sum_{k=1}^{s_j}P_k^j\bigl(\zeta(-x-\omega_j-k\varepsilon)-\zeta(-x-\omega_j+(k-1)\varepsilon)\bigr) +P^j_0(\varepsilon,z) \\ &\qquad=\sum_{k=1}^{s_j}P_k^j\bigl(\zeta(x+\omega_j-(k-1)\varepsilon)-\zeta(x+\omega_j+k\varepsilon)\bigr) +P^j_0(\varepsilon,z) \\ &\qquad=\sum_{k=1}^{s_j}P_k^j\bigl(\zeta(x-\omega_j-(k-1)\varepsilon + 2\omega_j)-\zeta(x-\omega_j+k\varepsilon +2\omega_j)\bigr) +P^j_0(\varepsilon,z) \\ &\qquad=\sum_{k=1}^{s_j}P_k^j\bigl(\zeta(x-\omega_j-(k-1)\varepsilon)-\zeta(x-\omega_j+k\varepsilon)\bigr) +P^j_0(\varepsilon,z) = S_j(x+\varepsilon,\varepsilon,z). \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\begin{alignedat}{2} S(-x,\varepsilon,z) &= S(x+\varepsilon,\varepsilon,z), &\qquad S(-x+\varepsilon,\varepsilon,z) &= S(x,\varepsilon,z), \\ S(-x-\varepsilon,\varepsilon,z) &= S(x+2\varepsilon,\varepsilon,z), &\qquad S(-x+2\varepsilon,\varepsilon,z) &= S(x-\varepsilon,\varepsilon,z). \end{alignedat}
\end{equation*}
\notag
$$
Since $A(x,\varepsilon)$ is odd, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{R}(-x,\varepsilon,z) = S(-x+\varepsilon,\varepsilon,z) - S(-x,\varepsilon,z) \\ &\ \ + (z\,{-}\,\wp(\varepsilon))\biggl(\frac{S(-x+\varepsilon,\varepsilon,z) +S(-x+2\varepsilon,\varepsilon,z)}{A(-x,\varepsilon)A(-x+\varepsilon,\varepsilon)} \,{-}\,\frac{S(-x-\varepsilon,\varepsilon,z)\,{+}\,S(-x,\varepsilon,z)}{A(-x-\varepsilon,\varepsilon) A(-x,\varepsilon)} \biggr) \\ &\ = S(x,\varepsilon,z) - S(x+\varepsilon,\varepsilon,z) \\ &\ \ + (z\,{-}\,\wp(\varepsilon))\biggl(\frac{S(x,\varepsilon,z) +S(x-\varepsilon,\varepsilon,z)}{A(x,\varepsilon)A(x-\varepsilon,\varepsilon)} \,{-}\,\frac{S(x+2\varepsilon,\varepsilon,z)\,{+}\,S(x+\varepsilon,\varepsilon,z)}{A(x+\varepsilon,\varepsilon) A(x,\varepsilon)} \biggr) \\ &\ = - \mathcal{R}(x,\varepsilon,z). \end{aligned}
\end{equation*}
\notag
$$
This proves Lemma 1. We will need an explicit form of the poles of the function $\mathcal{R}$ with respect to $x$. Lemma 2. The function $\mathcal{R}(x,\varepsilon,z)$ is a $\Lambda$-periodic meromorphic function with respect to $x$ with simple poles (modulo $\Lambda$), which can only be at the points Proof. The double periodicity of $\mathcal{R}(x,\varepsilon,z)$ follows from the double periodicity of $A(x,\varepsilon)$ and $S(x,\varepsilon,z),$ and the lattice of periods coincides with the lattice $\Lambda = \{2\omega_1 \mathbb{Z}+2\omega_2 \mathbb{Z}\}$. From Lemma 1 it follows that if $\mathcal{R}(x,\varepsilon,z)$ has a pole at the point $x=x_{0}$, then there is also a pole at the point $x=-x_{0}$, and
$$
\begin{equation*}
\operatorname*{Res}_{x=x_0}\mathcal{R}(x,\varepsilon,z) = \operatorname*{Res}_{x=-x_0}\mathcal{R}(x,\varepsilon,z).
\end{equation*}
\notag
$$
Consider the function
$$
\begin{equation*}
B_j(x,\varepsilon) = \frac{1}{A_j(x,\varepsilon) A_j(x+\varepsilon,\varepsilon)}.
\end{equation*}
\notag
$$
According to Lemma 2 in [10],
$$
\begin{equation*}
B_0(x,\varepsilon) =b_0(\varepsilon)\bigl(\zeta(s_0 \varepsilon)-\zeta((s_0+1)\varepsilon) -\zeta(x)+\zeta(x+\varepsilon)\bigr),
\end{equation*}
\notag
$$
where $b_0(\varepsilon)$ is some function of $\varepsilon$, the explicit form of which is not important for the proof of Theorem 1.
A similar analysis shows that
$$
\begin{equation*}
B_j(x,\varepsilon) = b_j(\varepsilon)\bigl(\zeta(s_j \varepsilon)-\zeta((s_j+1)\varepsilon) -\zeta(x-\omega_j)+\zeta(x-\omega_j+\varepsilon)\bigr),
\end{equation*}
\notag
$$
where $b_j(\varepsilon)$ is some function of $\varepsilon$.
Hence
$$
\begin{equation*}
B(x,\varepsilon) = \frac{1}{A(x,\varepsilon) A(x+\varepsilon,\varepsilon)} = \prod_{j=0}^3B_j(x,\varepsilon).
\end{equation*}
\notag
$$
Now the formula for $\mathcal{R}(x,\varepsilon,z)$ assumes the form
$$
\begin{equation*}
\begin{aligned} \, \mathcal{R}(x,\varepsilon,z) &= S(x+\varepsilon,\varepsilon,z) - S(x,\varepsilon,z) \\ &\qquad + (z-\wp(\varepsilon))\bigl(B(x,\varepsilon)(S(x+\varepsilon,\varepsilon,z) +S(x+2\varepsilon,\varepsilon,z)) \\ &\qquad\qquad-B(x-\varepsilon,\varepsilon)(S(x-\varepsilon,\varepsilon,z)+S(x,\varepsilon,z))\bigr). \end{aligned}
\end{equation*}
\notag
$$
The function $\mathcal{R}(x,\varepsilon,z)$ can be written as
$$
\begin{equation*}
\mathcal{R}(x,\varepsilon,z) = \sum_{j=0}^3 \mathcal{R}_j(x,\varepsilon,z),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{aligned} \, \mathcal{R}_j(x,\varepsilon,z) &= S_j(x+\varepsilon,\varepsilon,z) - S_j(x,\varepsilon,z) \nonumber \\ &\qquad+ (z-\wp(\varepsilon))\bigl(B(x,\varepsilon)(S_j(x+\varepsilon,\varepsilon,z)+S_j(x+2\varepsilon,\varepsilon,z)) \nonumber \\ &\qquad\qquad-B(x-\varepsilon,\varepsilon)(S_j(x-\varepsilon,\varepsilon,z)+S_j(x,\varepsilon,z))\bigr). \end{aligned}
\end{equation}
\tag{23}
$$
Like $\mathcal{R}(x,\varepsilon,z)$, the function $\mathcal{R}_j(x,\varepsilon,z)$ is odd
$$
\begin{equation*}
\mathcal{R}_j(-x,\varepsilon,z) = - \mathcal{R}_j(x,\varepsilon,z).
\end{equation*}
\notag
$$
Substituting $S_j(x,\varepsilon,z)$ and $B(x,\varepsilon)$ into (23), we find that
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{R}_j(x,\varepsilon,z) = \sum_{k=1}^{s_j}P_k^j\big(\zeta(x-\omega_j-(k-1)\varepsilon) \\ &\quad-\zeta(x-\omega_j+k\varepsilon)-\zeta(x-\omega_j-k\varepsilon)+\zeta(x-\omega_j+(k-1)\varepsilon)\bigr) \\ &\quad+(z-\wp(\varepsilon)) \biggl(\prod_{j=0}^3b_j(\varepsilon)\bigl(\zeta(s_j \varepsilon)-\zeta((s_j+1)\varepsilon)-\zeta(x-\omega_j)+\zeta(x-\omega_j+\varepsilon)\bigr) \\ &\quad\qquad\times\biggl( \sum_{k=1}^{s_j}P_k^j\big(\zeta(x-\omega_j-(k-1)\varepsilon) -\zeta(x-\omega_j+k\varepsilon) \\ &\quad\qquad\qquad+\zeta(x-\omega_j-(k-2)\varepsilon)-\zeta(x-\omega_j+(k+1)\varepsilon)\big) 2P^j_0\biggr) \\ &\quad-\prod_{j=0}^3b_j(\varepsilon)\bigl(\zeta(s_j \varepsilon) -\zeta((s_j+1)\varepsilon) -\zeta(x-\omega_j-\varepsilon)+\zeta(x-\omega_j)\bigr) \\ &\quad\qquad\times\biggl( \sum_{k=1}^{s_j}P_k^j \bigl(\zeta(x-\omega_j-(k+1)\varepsilon) -\zeta(x-\omega_j+(k-2)\varepsilon) \\ &\qquad\qquad\qquad+\zeta(x-\omega_j-k\varepsilon)-\zeta(x-\omega_j+(k-1)\varepsilon)\bigr) + 2P^j_0\biggr)\biggr). \end{aligned}
\end{equation*}
\notag
$$
We claim that $\mathcal{R}_j(x,\varepsilon,z)$ has no second-order poles. A second-order pole of the function $\mathcal{R}_j(x,\varepsilon,z)$ can only arise in terms of the form
$$
\begin{equation}
\begin{aligned} \, &(z-\wp(\varepsilon)) \biggl(\prod_{j=0}^3b_j(\varepsilon)\bigl(\zeta(s_j \varepsilon)-\zeta((s_j+1)\varepsilon)-\zeta(x-\omega_j)+\zeta(x-\omega_j+\varepsilon)\bigr) \nonumber \\ &\qquad\times \biggl( \sum_{k=1}^{s_j}P_k^j \bigl(\zeta(x-\omega_j-(k-1)\varepsilon)-\zeta(x-\omega_j+k\varepsilon) \nonumber \\ &\qquad\qquad+\zeta(x-\omega_j-(k-2)\varepsilon)-\zeta(x-\omega_j+(k+1)\varepsilon)\bigr) + 2P^j_0\biggr)\biggr) \end{aligned}
\end{equation}
\tag{24}
$$
and
$$
\begin{equation}
\begin{aligned} \, &-(z-\wp(\varepsilon)) \biggl(\prod_{j=0}^3b_j(\varepsilon)\bigl(\zeta(s_j \varepsilon)-\zeta((s_j+1)\varepsilon)-\zeta(x-\omega_j-\varepsilon)+\zeta(x-\omega_j)\bigr) \nonumber \\ &\qquad\times\biggl( \sum_{k=1}^{s_j}P_k^j \bigl(\zeta(x-\omega_j-(k+1)\varepsilon)-\zeta(x-\omega_j+(k-2)\varepsilon) \nonumber \\ &\qquad\qquad+\zeta(x-\omega_j-k\varepsilon)-\zeta(x-\omega_j+(k-1)\varepsilon)\bigr) + 2P^j_0\biggr)\biggr) \end{aligned}
\end{equation}
\tag{25}
$$
at the points
$$
\begin{equation*}
x=\omega_j, \ \omega_j-\varepsilon,\ \omega_j+\varepsilon.
\end{equation*}
\notag
$$
Notice that the first factor in (24) has a first-order pole at $x=\omega_j-\varepsilon,$ but the second factor does not have a pole at this point, since the terms in the second factor responsible for this pole are canceled,
$$
\begin{equation*}
\begin{aligned} \, &P_1^j\bigl(\zeta(x-\omega_j-(1-1)\varepsilon)-\zeta(x-\omega_j+\varepsilon) \\ &\qquad\qquad+\zeta(x-\omega_j-(1-2)\varepsilon)-\zeta(x-\omega_j-(1+1)\varepsilon)\bigr)+2P_0^j \\ &\qquad= P_1^j\bigl(\zeta(x-\omega_j)-\zeta(x-\omega_j+2\varepsilon)\bigr)+2P_0^j. \end{aligned}
\end{equation*}
\notag
$$
A similar analysis shows that (25) has no pole at $x=\omega_j+\varepsilon$. Thus, a second-order pole of $\mathcal{R}_j(x,\varepsilon,z)$ can only arise at the point $x=\omega_j$. The second-order pole in (24) at $x=\omega_j$ is controlled by the term
$$
\begin{equation*}
-(z-\wp(\varepsilon)) \prod_{j=0}^3b_j(\varepsilon) \zeta(x-\omega_j)(P_1^j\zeta(x-\omega_j) +P_2^j\zeta(x-\omega_j)),
\end{equation*}
\notag
$$
and in (25) the same term, but with a different sign, is responsible for this pole
$$
\begin{equation*}
-(z-\wp(\varepsilon)) \prod_{j=0}^3b_j(\varepsilon) \zeta(x-\omega_j)(-P_1^j\zeta(x-\omega_j)-P_2^j\zeta(x-\omega_j)).
\end{equation*}
\notag
$$
Therefore, $\mathcal{R}_j(x,\varepsilon,z)$ has no second-order poles.
Poles of the form (22) are present at the first term $\mathcal{R}_j(x,\varepsilon,z),$ as well as at (24) and (25). Further, the second factor in (24) has a first-order pole at $x=\omega_j-(s_j+ 1)\varepsilon$, which is not in the list (22). However, note that at this point the first factor in (24) vanishes. Hence $\mathcal{R}_j(x,\varepsilon,z)$ has no pole at $x=\omega_j-(s_j+1)\varepsilon$. Similarly, $\mathcal{R}_j(x,\varepsilon,z)$ has no pole at the point $x=\omega_j+(s_j+1)\varepsilon$. Hence the function $\mathcal{R}(x,\varepsilon,z)$ can have only simple poles at points (22). This proves Lemma 2. By Lemma 1 and $\Lambda$-periodicity of $\mathcal{R}(x,\varepsilon,z)$,
$$
\begin{equation}
\operatorname*{Res}_{\omega_j+k\varepsilon}\mathcal{R}(x,\varepsilon,z) = \operatorname*{Res}_{-\omega_j-k\varepsilon}\mathcal{ R}(x,\varepsilon,z) = \operatorname*{Res}_{\omega_j-k\varepsilon}\mathcal{R}(x,\varepsilon,z),
\end{equation}
\tag{26}
$$
where $k =1,\dots,s_j$, $0\leqslant j\leqslant 3$. The residues of the function $\mathcal{R}_j(x,\varepsilon,z)$ at the points $ \omega_j+\varepsilon, \omega_j+2\varepsilon, \dots, \omega_j+s_j\varepsilon$ have the form
$$
\begin{equation*}
\begin{aligned} \, &\operatorname*{Res}_{\omega_j+\varepsilon}\mathcal{R}_j(x,\varepsilon,z) = \biggl(P^j_2 - P^j_1 + (z -\wp(\varepsilon)) \biggl((P^j_3 + P^j_2)B(\omega_j+\varepsilon,\varepsilon) \\ &\qquad+\prod_{d=0,\, d\neq j}^3B_d(\omega_d,\varepsilon) \biggl(\sum_{k=1}^{s_j}P^j_k\bigl(\zeta(\omega_d-\omega_j-k\varepsilon) -\zeta(\omega_d-\omega_j+(k-1)\varepsilon) \\ &\qquad\qquad+\zeta(\omega_d-\omega_j-(k+1)\varepsilon)-\zeta(\omega_d-\omega_j+k\varepsilon)\bigr) +2P^j_0\biggr)\biggr)\biggr), \\ &\operatorname*{Res}_{\omega_j+m\varepsilon}\mathcal{R}_j(x,\varepsilon,z) = P_{m+1}^j-P_m^j + (z-\wp(\varepsilon)) \bigl((P_{m+2}^j + P_{m+1}^j)B(\omega_j+m\varepsilon,\varepsilon) \\ &\qquad-(P_m^j + P_{m-1}^j)B(\omega_j+(m-1)\varepsilon,\varepsilon)\bigr), \qquad m = 2,\dots,s_j. \end{aligned}
\end{equation*}
\notag
$$
In the last formula, we assume that $P_m^j = 0$ for $m>s_j$. Since $\operatorname*{Res}_{\omega_j+m\varepsilon}\mathcal{R}_j(x,\varepsilon,z) = 0$, we have
$$
\begin{equation}
\begin{aligned} \, P^j_{m-1} &= \frac1{B(\omega_j+(m-1)\varepsilon,\varepsilon)(z-\wp(\varepsilon))}\bigl( P^j_{m+1}-P^j_m \nonumber \\ &\qquad+(z-\wp(\varepsilon)) \bigl((P^j_{m+2} + P^j_{m+1})B(\omega_j+m\varepsilon,\varepsilon) - P^j_mB(\omega_j+(m-1)\varepsilon,\varepsilon)\bigr)\bigr). \end{aligned}
\end{equation}
\tag{27}
$$
In particular,
$$
\begin{equation*}
P^j_{s_j-1} = -\frac{P^j_{s_j}\bigl(1 + (z-\wp(\varepsilon)) B(\omega_j+(s_j-1)\varepsilon,\varepsilon)\bigr)}{B(\omega_j+(s_j-1)\varepsilon,\varepsilon)(z-\wp(\varepsilon))}.
\end{equation*}
\notag
$$
Since $\operatorname*{Res}_{\omega_j+\varepsilon}\mathcal{R}_j(x,\varepsilon,z)=0$, we have
$$
\begin{equation}
\begin{aligned} \, P^j_0 &= -\frac{1}{2(z - \wp(\varepsilon))\prod_{d=0, \, d\neq j}^3B_d(\omega_d,\varepsilon)} \nonumber \\ &\qquad\times \biggl(P^j_2 - P^j_1+ (z - \wp(\varepsilon))\biggl((P^j_2 + P^j_3) B(\omega_j+\varepsilon,\varepsilon) \nonumber \\ &\qquad\qquad+\prod_{d=0, \, d\neq j}^3B_d(\omega_d,\varepsilon) \sum_{k=1}^{s_j}P^j_k\bigl(\zeta(\omega_d-\omega_j-k\varepsilon) -\zeta(\omega_d-\omega_j+(k-1)\varepsilon) \nonumber \\ &\qquad\qquad +\zeta(\omega_d-\omega_j-(k+1)\varepsilon) -\zeta(\omega_d-\omega_j+k\varepsilon)\bigr)\biggr)\biggr). \end{aligned}
\end{equation}
\tag{28}
$$
So, from the recurrence formulas (27) and (28) we can express $P^j_0,\dots, P^j_{s_j-1}$ in terms of $P^j_{s_j}$. From the obtained formulas it follows that if we put then $P^j_i$ for $0\leqslant i < s_j$ are polynomials of degree $s_j$ with respect to $z$.From (26) we have
$$
\begin{equation*}
\operatorname*{Res}_{\omega_j-\varepsilon}\mathcal{R}_j(x,\varepsilon,z) = \dots = \operatorname*{Res}_{\omega_j-s_j\varepsilon}\mathcal{R}_j(x,\varepsilon,z) = 0.
\end{equation*}
\notag
$$
Hence $\mathcal{R}_j(x,\varepsilon,z)$ has no poles, and so, since $\mathcal{R}_j(x,\varepsilon,z)$ is odd, we find that Consequently, $\mathcal{R}(x,\varepsilon,z) = 0$. This proves Theorem 3. In the case of the expansion $a(\varepsilon) = -2/\varepsilon + O(\varepsilon^3)$, a direct calculation shows that
$$
\begin{equation*}
A = -\frac{2}{\varepsilon} - \biggl(\sum_{j=0}^3 s_j(s_j+1)\wp(x-\omega_j)\biggr) \varepsilon + O(\varepsilon^3)
\end{equation*}
\notag
$$
(a similar analysis was given in [10] for the Lamé operator). This implies (8). Theorem 1 is proved.
Citation in format AMSBIB
\Bibitem{MauMir26} |
|
||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|