Discrete Dynamics of the Tyurin Parameters and Commuting Difference Operators

We study commuting difference operators of rank two. In the case of hyperelliptic spectral curves an equation which is equivalent to the Krichever – Novikov equations on Tyurin parameters is obtained. With the help of this equation examples of operators corresponding to hyperelliptic spectral curves of arbitrary genus are constructed. Among these examples there are operators with polynomial and trigonometric coefficients.
If two difference operators
$$ L_4=\sum^{2}_{i=-2} u_i(n)T^i, \qquad L_{4g+2}=\sum^{2g+1}_{i=-(2g+1)}v_i(n)T^i,\qquad u_2=v_{2g+1}=1 $$
commute, where $T$ — shift operator, then there is a nonzero polynomial $F(z,w)$ such that $F(L_4,L_{4g+2})=0.$ The polynomial $F$ defines the spectral curve of $L_4,L_{4g+2}$
$$ \Gamma=\{(z,w)\in {\mathbb C}^2| F(z,w)=0\}. $$
The common eigenvalues are parametrized by the spectral curve
$$ L_4\psi=z\psi, \quad L_{4g+2}\psi=w\psi, (z,w)\in \Gamma. $$
The rank of the pair $L_4,L_{4g+2}$ is called the dimension of the space of common eigenfunctions for fixed eigenvalues
$$ l={\rm dim}\{\psi:L_4\psi=z\psi, \ \ L_{4g+2}\psi=w\psi,\ \ (z,w)\in \Gamma.\} $$
The curve $\Gamma$ admits a holomorphic involution
$$\sigma:\Gamma\rightarrow\Gamma,{\ }{\ }{\ }\sigma(z,w)=\sigma(z,-w).$$
The common eigenfunctions $L_4$ and $L_{4g+2}$ satisfy the equation
$$ \psi_{n+1}(P)=\chi_1(n,P)\psi_{n-1}(P)+\chi_2(n,P)\psi_n(P), $$
The functions $\chi_1(n,P)$ and $\chi_2(n,P)$ are rational on $\Gamma$ and have $2g$ simple poles depending on $n$. In addition the function $\chi_2(n,P)$ has a simple pole in $q$. For finding $L_4$ and $L_{4g+2}$ it is sufficient to find $\chi_1$ and $\chi_2.$
The following theorems are proved.
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Theorem 1. If
$$ \chi_1(n,P)=\chi_1(n,\sigma(P)),\qquad \chi_2(n,P)=-\chi_2(n,\sigma(P)), $$
then $L_4$ has the form
$$ L_4=(T+V_nT^{-1})^2+W_n, $$
herewith
$$ \chi_1=-V_n\frac{Q_{n+1}}{Q_{n}},\qquad \chi_2=\frac{w}{Q_n}, $$
where
$$ Q_n(z)=z^g+\alpha_{g-1}(n)z^{g-1}+\ldots+\alpha_0(n). $$
Functions $V_n, W_n, Q_n$ satisfy the following equation
$$ F_g(z)=Q_{n-1}Q_{n+1}V_n+Q_{n}(Q_{n+2}V_{n+1}+Q_{n+1}(z-V_n-V_{n+1}-W_n)). $$

Theorem 2. The operator
$$ L_4=(T+(r_3n^3+r_2n^2+r_1n+r_0)T^{-1})^2+g(g+1)r_3n $$
commutes with a difference operator $L_{4g+2}$ of order $4g+2$, where $r_0, r_1, r_2, r_3$ — parameters, $r_3 \neq 0$.
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Theorem 3. The operator
$$ L_4=(T+(r_1a^n+r_0)T^{-1})^2+(a^{2g+1}-a^{g+1}-a^g+1)r_1a^{n-g} $$
commutes with a difference operator $L_{4g+2},$ where $r_0, r_1, a$ are parameters such that $r_1 \neq 0,$ $a \neq 0,$ $a^{2g+1}-a^{g+1}-a^g+1 \neq 0$.
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Theorem 4. The operator
$$ L_4=(T+(r_1\cos(n)+r_0)T^{-1})^2-4r_1\sin(\frac{g}{2})\sin(\frac{g+1}{2})\cos(n+\frac{1}{2}) $$
commutes with a difference operator $L_{4g+2},$ where $r_0, r_1$ — parameters, $r_1 \neq 0.$