In this paper, the third in a series, we continue to study the generalized Lam\'{ e} equation H$(n_0,n_1,n_2,n_3;B)$ with the Darboux-Treibich-Verdier potential \begin{ equation*} y^{ \prime \prime }(z)=\bigg[ \sum_{ k=0}^{ 3}n_{ k}(n_{ k}+1)\wp(z+\tfrac{ \omega_{ k}}{ 2}|\tau)+B\bigg] y(z),\quad n_{ k}\in \mathbb{ Z}_{ \geq0} \end{ equation*} and a related linear ODE with additional singularities $\pm p$ from the monodromy aspect.We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann-Hilbert correspondence from the set \[\{ \text{ H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{ C}\}\cup \{ \text{ H}(n_0+2,n_1,n_2,n_3;B) | B\in\mathbb{ C}\}\] to the set of group representations $\rho:\pi_1(E_{ \tau})\to SL(2,\mathbb{ C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $\tau=\frac12+i\frac{ \sqrt{ 3}}{ 2}$, there are $B_1,B_2$ such that the monodromy representations of H$(1,0,0,0;B_1)$ and H$(4,0,0,0;B_2)$ are { \bf the same}, namely the Riemann-Hilbert correspondence from the set \[\{ \text{ H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{ C}\}\cup \{ \text{ H}(n_0+3,n_1,n_2,n_3;B) | B\in\mathbb{ C}\}\] to the set of group representations is { \bf not} necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on $\mathbb{ CP}^1$ with finite singularities.
Categories: math.CA