I'm doing quantum mechanics and I have an eigenfunction which is a theta function. I then discretised it, since I want see if I can find the eigenvalues for the discrete case by finding the eigenfunctions for the continuum and discretising them then applying translation operators for derivatives.
I have a theta function of the form $$\theta_3(z|\tau)$$ where $$z=\frac{ivL}{N}-\frac{ik_1}{N}+\frac{k_2}{N};$$ $$\tau=\frac{i}{N},$$ and $k_1$, $k_2=0,1,..., N-1$ with $v$ and $L$ being constants. Using the periodicity of the theta function
$$\theta_3(z+\tau|\tau)=\exp(-\pi i\tau-2\pi iz)\theta_3(z|\tau);$$ and $$\theta_3(z-\tau|\tau)=\exp(-\pi i\tau+2\pi iz)\theta_3(z|\tau)$$ where the above condition come from $k_1\mapsto k_1\pm 1$. My question is thus: for the other translation I acquire the resulting theta functions $$\theta_3(z\pm i\tau|\tau),$$ is there a rule for this such that $$\theta_3(z\pm i\tau|\tau)=A(z,\tau_\pm)\theta_3(z|\tau)?$$
I've tried looking at the series representation and computing it explicitly, but no luck. Am I just being hopeful, when in fact, there is no way to find the neat form of the $z\pm i\tau$ periodicity condition?
This leads to another question: how does one use an eigenfunction which is theta function in the Schrödinger equation to find eigenvalues? The derivatives aren't defined since it creates a sum like $\sum_{m\in\mathbb{Z}}\alpha m\hspace{1mm}\mathrm{e}^{\hspace{1mm}\beta m^2+m\gamma}$ where $\alpha$, $\beta$ and $\gamma$ are constants, which as far as I know can't be computed...